/r tA VECTOR CALCULUS WITH APPLICATIONS TO PHYSICS BY JAMES BYRNIE SHAW PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF ILLINOIS ILLUSTRATED NEW YORK D. VAN NOSTRAND COMPANY Eight Warren Street 1922 s Copyright, 1922 By D. Van Nostrand Company All rights reserved, including that of translation into foreign languages, including the Scandinavian Printed in the United States of America PREFACE. This volume embodies the lectures given on the subject to graduate students over a period of four repetitions. The point of view is the result of many years of consideration of the whole field. The author has examined the various methods that go under the name of Vector, and finds that for all purposes of the physicist and for most of those of the geometer, the use of quaternions is by far the simplest in theory and in practice. The various points of view are mentioned in the introduction, and it is hoped that the es- sential differences are brought out. The tables of com- parative notation scattered through the text will assist in following the other methods. The place of vector work according to the author is in the general field of associative algebra, and every method so far proposed can be easily shown to be an imperfect form of associative algebra. From this standpoint the various discussions as to the fundamental principles may be under- stood. As far as the mere notations go, there is not much difference save in the actual characters employed. These have assumed a somewhat national character. It is un- fortunate that so many exist. The attempt in this book has been to give a text to the mathematical student on the one hand, in which every physical term beyond mere elementary teims is carefully defined. On the other hand for the physical student there will be found a large collection of examples and exercises which will show him the utility of the mathematical meth- ods. So very little exists in the numerous treatments of the day that does this, and so much that is labeled vector iii 505384 IV PREFACE analysis is merely a kind of short-hand, that it has seemed very desirable to show clearly the actual use of vectors as vectors. It will be rarely the case in the text that any use of the components of vectors will be found. The triplexes in other texts are very seldom much different fiom the ordi- nary Cartesian forms, and not worth learning as methods. The difficulty the author has found with other texts is that after a few very elementary notions, the mathematical student (and we may add the physical student) is suddenly plunged into the profundities of mathematical physics, as if he were familiar with them. This is rarely the case, and the object of this text is to make him familiar with them by easy gradations. It is not to be expected that the book will be free from errors, and the author will esteem it a favor to have all errors and oversights brought to his attention. He desires to thank specially Dr. C. F. Green, of the University of Illinois, for his careful assistance in reading the proof, and for other useful suggestions. Finally he has gathered his material widely, and is in debt to many authors for it, to all of whom he presents his thanks. James Byrnie Shaw. Urbana, III., July, 1922. TABLE OF CONTENTS. Chapter I. Introduction 1 Chapter II. Scalar Fields 18 Chapter III. Vector Fields 23 Chapter IV. Addition of Vectors 52 Chapter V. Vectors in a Plane 62 Chapter VI. Vectors in Space 94 Chapter VII. Applications 127 1. The Scalar of two Vectois 127 2. The Vector of two Vectors 136 3. The Scalar of three Vectors 142 4. The Vector of three Vectors 143 Chapter VIII. Differentials and Integrals 145 1. Differentiation as to one Scalar Parameter .... 145 Two Parameters 151 2. Differentiation as to a Vector 155 3. Integration 196 Chapter IX. The Linear Vector Function 218 Chapter X. Deformable Bodies 253 Strain 253 Kinematics of Displacement 265 Stress 269 Chapter XL Hydrodynamics 287 VECTOR CALCULUS CHAPTER I INTRODUCTION 1. Vector Calculus. By this term is meant a system of mathematical thinking which makes use of a special class of symbols and their combinations according to certain given laws, to study the mathematical conclusions resulting from data which depend upon geometric entities called vectors, or physical entities representable by vectors, or more generally entities of any kind which could be repre- sented for the purposes under discussion by vectors. These vectors may be in space of two or three or even four or more dimensions. A geometric vector is a directed segment of a straight line. It has length (including zero) and direc- tion. This is equivalent to saying that it cannot be de- fined merely by one single numerical value. Any problem of mathematics dependent upon several variables becomes properly a problem in vector calculus. For instance, analytical geometry is a crude kind of vector calculus. Several systems of vector calculus have been devised, differing in their fundamental notions, their notation, and their laws of combining the symbols. The lack of a uniform. notation is deplorable, but there seems little hope of the adoption of any uniform system soon. Existing systems have been rather ardently promoted by mathematicians of the same nationality as their authors, and disagreement exists as to their relative simplicity, their relative directness, and their relative logical exactness. These disagreements arise sometimes merely with regard to the proper manner of representing certain combinations of the symbols, or other matters which are purely matters of convention; 1 2 YKCTOR CALCULUS sometimes they are due to different views as to what are the import an1 things to find expressions for; and sometimes they are due to more fundamental divergences of opinion as to the real character of the mathematical ideas underlying any system of this sort. We will in- dicate these differences and dispose of them in this work. 2. Bases. We may classify broadly the various systems of vector calculus as geometric and algebraic. The former is to be found wherever the desire is to lay emphasis on the spatial character of the entities we are discussing, such as the line, the point, portions of a plane, etc. The latter lays emphasis on the purely algebraic character of the entities with which the calculations are made, these entities being similar to the positive and negative, and the imag- inary of ordinary algebra. For the geometric vector systems, the symbolism of the calculus is really nothing more than a short-hand to enable one to follow certain operations upon real geometric elements, with the possi- bility kept always in mind that these entities and the operations may at any moment be called to the front to take the place of their short-hand representatives. For the algebraic systems, the symbolism has to do with hypernumbers, that is, extensions of the algebraic negative and imaginary numbers, and does not pretend to be the translation of actual operations which can be made visible, any more than an ordinary calculation of algebra could be paralleled by actual geometric or physical operations. If these distinctions are kept in mind the different points of view become intelligible. The best examples of geo- metric systems are the Science of Extension of Grassmann, with its various later forms, the Geometry of Bynames of Study, the Geometry of Lines of Saussure, and the Geometry of Feuillets of Cailler. The best examples of algebraic systems are the Quaternions of Hamilton, Dyadics of Gibbs, INTRODUCTION ,3 Multenions of McAulay, Biquaternions of Clifford, Tri- quaternions of Combebiac, Linear Associative Algebra of Peirce. Various modifications of these exist, and some mixed systems may be found, which will be noted in the proper places. The idea of using a calculus of symbols for writing out geometric theorems perhaps originated with Leibniz, 1 though what he had in mind had nothing to do with vector calculus in its modern sense. The first effective algebraic vector calculus was the Quaternions of Hamilton 2 (1843), the first effective geometric vector calculus was the Ausdehn- ungslehre of Grassmann 3 (1844). They had predecessors worthy of mention and some of these will be noticed. 3. Hypernumbers. The real beginning of Vector Cal- culus was the early attempt to extend the idea of number. The original theory of irrational number was metric, 4 and defined irrationals by means of the segments of straight lines. When to this was added the idea of direction, so that the segments became directed segments, what we now call vectors, the numbers defined were not only capable of being irrational, but they also possessed quality, and could be negative or positive. Ordinary algebra is thus the first vector calculus. If we consider segments with direction in a plane or in space of three dimensions, then we may call the numbers they define hypernumbers. The source of the idea was the attempt to interpret the imaginary which had been created to furnish solutions for any quadratic or cubic. The imaginary appears early in Cardan's work. 5 For instance he gives as solution of the problem of separating 10 into two parts whose product is 40, the values 5 + V — 15, and 5 — V — 15. He considered these numbers as impossible and of no use. Later it was dis- covered that in the solution of the cubic by Cardan's formula there appeared the sum of two of these impossible 4 VWCTOfl CALCULUS values when the answer actually was real. Bombelli #;ive as the solution of the cubic r 3 = 15x + 4 the form ^(2 + V - 121) + ^(2 - V - 121) = 4. These impossible numbers incited much thought and there came about several attempts to account for them and to interpret them. The underlying question was essen- tially that of existence, which at that time was usually sought for in concrete cases. The real objection to the negative number was its inapplicability to objects. Its use in a debit and credit account would in this sense give it existence. Likewise the imaginary and the complex num- ber, and later others, needed interpretation, that is, applica- tion to physical entities. 4. Wessel, a Danish surveyor, in 1797, produced a satisfactory method 7 of defining complex numbers by means of vectors in a plane. This same method was later given by Argand 8 and afterwards by Gauss 9 in connection with various applications. Wessel undertook to go farther and in an analogous manner define hypernumbers by means of directed segments, or vectors, in space of three dimen- sions. He narrowly missed the invention of quaternions. In 1813 Servois 10 raised the question whether such vectors might not define hypernumbers of the form . p cos a + q cos (3 + r cos y and inquired what kind of non-reals p, q, r would be. He did not answer the question, however, and Wessel's paper remained unnoticed for a century. 5. Hamilton gave the answer to the question of Servois as the result of a long investigation of the whole problem. 11 He first considered algebraic couples, that is to say in our terminology, hypernumbers needing two ordinary numerical INTRODUCTION 5 values to define them, and all possible modes of combining them under certain conditions, so as to arrive at a similar couple or hypernumber for the product. He then con- sidered triples and sets of numbers in general. Since — 1 and i = V — 1 are roots of unity, he paid most attention to definitions that would lead to new roots of unity. His fundamental idea is that the couple of numbers (a, b) where a and b are any positive or negative numbers, rational or irrational, is an entity in itself and is therefore subject to laws of combination just as are single numbers. For instance, we may combine it with the other couple (x, y) in two different ways : (a, b) + (x, y) = (a + x, b + y) (a, 6) X (x, y) = {ax — by, ay + bx). In the first case we say we have, added the couples, in the second case that we have multiplied them. It is possible to define division also. In both cases if we set the couple on the right hand side equal to {u, v) we find that dujdx — dv/dy, dujdy = — dv/dx. Pairs of functions u, v which satisfy these partial differential equations Hamilton called conjugate functions. The partial differential equations were first given by Cauchy in this connection. The particular couples €l = (1, 0), € 2 = (0, 1) play a special role in the development, for, in the first place, any couple may be written in the form (a, b) = aei + be 2 and the notation of couples becomes superfluous; in the second place, by defining the products of ei and e 2 in various ways we arrive at various algebras of couples. The general C> VECTOR CALCULUS definition would be, using the • for X, €l'€i = Cin€i + Cii 2 € 2 , €i'€ 2 = Ci2i€i + ^12262, €2'€i = C2ll€i + C212€2> «2 * €2 = C221«l + C222€2- By varying the choice of the arbitrary constants c, and Hamilton considered several different cases, different algebras of couples could be produced. In the case above the c's are all zero except Cm = 1, C122 = 1, C212 — 1, C221 = — 1. From the character of 4 it may be regarded as entirely identical with ordinary 1, and it follows therefore that e 2 may be regarded as identical with the V — 1. On the other hand we may consider €1 to be a unit vector pointing to the right in the plane of vectors, and c 2 to be a unit vector perpendicular to ei. We have then a vector calculus practically identical with Wessel's. The great merit of Hamilton's investigation lies of course in its generality. He continued the study of couples by a similar study of triples and then quadruples, arriving thus at Quaternions. His chief difference in point of view from those who followed him and who used the concept of couple, triple, etc. {Mul- tiple we will say for the general case), is that he invariably defined one product, whereas others define usually several. 6. Multiples. There is a considerable tendency in the current literature of vector calculus to use the notion of multiple. A vector is usually designated by a triple as (x, y, z), and usually such triple is called a vector. It is generally tacitly understood that the dimensions of the numbers of the triple are the same, and in fact most of the products defined would have no meaning unless this homogeneity of dimension were assumed to hold. We find products defined arbitrarily in several ways. For instance, the scalar product of the triples (a, b, c) and (x, y, z) INTRODUCTION 7 is =fc (ax + by + cz), the sign depending upon the person giving the definition; the vector product of the same two triples is usually given as the triple (bz — cy, ex — az, ay — bx). It is obvious at once that a great defect of such definitions is that the triples involved have no sense until the significance of the first number, the second number, and the third number in each triple is understood. If these depend upon axes for their meaning, then the whole calculus is tied down to such axes, unless, as is usually done, the expressions used in the definitions are so chosen as to be in some respects independent of the particular set of axes chosen. When these expressions are thus chosen as invariants under given transformations of the axes we arrive at certain of the well-known systems of vector analysis. The transformations usually selected to furnish the profitable expressions are the group of orthog- onal transformations. For instance, it was shown by Burkhardt 12 that all the invariant expressions or invariant triples are combinations of the three following : ax + by + cz, (bz — cy, ex — az, ay — bx), (al + bm + cn)x + (am — bl)y + (an — cl)z, (bl — am)x + (al -f- bm + cn)y + (bn — cm)z, (cl — ari)x + (cm — bn)u + (al + bm + cn)z. A study of vector systems from this point of view has been made by Schouten. 13 7. Quaternions. In his first investigations, Hamilton was chiefly concerned with the creation of systems of hypernumbers such that each of the defining units, similar to the ei and € 2 above, was a root of unity. 14 That is, the process of multiplication by iteration would bring back the multiplicand. He was actually interested in certain special 8 VECTOR CALCULUS cases of abstract groups, 15 and if he had noticed the group property his researches would perhaps have extended into the whole field of abstract groups. In quaternions he found a set of square roots of — 1, which he designated by i, j, k, connected with his triples though belonging to a set of quad- ruples. In his Lectures on Quaternions, the first treatise he published on the subject, he chose a geometrical method of exposition, consequently many have been led to think of quaternions as having a geometric origin. However, the original memoirs show that they were reached in a purely algebraic way, and indeed according to Hamilton's philoso- phy were based on steps of time as opposed to geometric steps or vectors. The geometric definition is quite simple, however, and not so abstract as the purely algebraic definition. Ac- cording to this idea, numbers have a metric definition, a number, or hypernumber, being the ratio of two vectors. If the vectors have the same direction we arrive at the ordinary numerical scale. If they are opposite we arrive at the negative numbers. If neither in the same direction nor opposite we have a more general kind of number, a hypernumber in fact, which is a quaternion, and of which the ordinary numbers and the negative numbers are merely special cases. If we agree to consider all vectors which are parallel and in the same direction as equivalent, that is, call them free vectors, then for every pair of vectors from the origin or any fixed point, there is a quaternion. Among these quaternions relations will exist, which will be one of the objects of study of later chapters. 8. Mobius was one of the early inventors of a vector calculus on the geometric basis. In his Barycentrisch.es Kalkul 16 he introduced a method of deriving points from other points by a process called addition, and several INTRODUCTION 9 applications were made to geometry. The barycentric calculus is somewhat between a system of homogeneous coordinates and a real vector calculus. His addition was used by Grassmann. 9. Grassmann in 1844 published his treatise called Die lineale Ausdehnungslehre 17 in which several different proc- esses called multiplication are used for the derivation of geometric entities from other geometric entities. These processes make use of a notation which is practically a sort of short-hand for the geometric processes involved. Grassmann considered these various kinds of multiplication abstractly, leaving out of account the meaning of the elements multiplied. His methods apply to space of N dimensions. In the symmetric multiplication it is possible to interchange any two of the factors without affecting the result. In the circular multiplication the order may be changed cyclically. In the lineal multiplication all the laws hold as well for any factors which are linear combina- tions of the hypernumbers which define the base, as for those called the base. He studies two species of circular multiplication. If the defining units of the base are ex, e 2 , e 3 • • •€„, then we have in the first variety of circular multipli- cation the laws €l 2 + € 2 2 + 6 3 2 + • • • + € n 2 = 0, €i€j = €j€i. In the second variety we have the laws ei 2 = 0, e/ = 0, • - • e n 2 = 0, Mi = 0, *+j. In the lineal genus of multiplication he studies two species, in the first, called the algebraic multiplication, we have the law My = *fii for all i, j. while in the second, called the exterior multiplication, the interchange of any two factors changes the sign of the 10 VECTOR CALCULUS result. Of the latter there are two varieties, the progressive multiplication in which the number of dimensions of the geometric figure which is the product is the sum of the dimensions of the factors, while in the other, called re- gressive multiplication, the dimension of the product is the difference between the sum of the dimensions of the factors and N the dimension of the space in which the operation takes place. From the two varieties he deduces another kind called interior multiplication. If we confine our thoughts to space of three dimensions, defined by points, and if €1, e 2 , e 3 , e 4 are such points, the progressive exterior product of two, as €1, e 2 , is ei€ 2 and represents the segment joining them if they do not coincide. The product is zero if they coincide. The product of this into a third point € 3 is ei€ 2 e 3 and represents the parallelogram with edges €162, ei€ 3 and the other two parallel to these respectively. If all three points are in a straight line the product is zero. The exterior progressive product c 1 e 2 e 3 € 4 represents the parallelepiped with edges €ie 2 , €ie 3 , €i€ 4 and the opposite parallel edges. The regressive exterior product of €i€ 2 and €ie 3 € 4 is their common point €1. The regressive product of €ie 2 e 3 and €ie 2 € 4 is their common line €ie 2 . The complement of €1 is defined to be € 2 e 3 e 4 , and of €i€ 2 is e 3 fct, and of €i€ 2 e 3 is € 4 . The interior product of any expression and another is the progressive or regressive product of the first into the complement of the other. For instance, the interior product of €1 and e 2 is the progressive product of €1 and €i€ 3 e 4 which vanishes. The interior product of e 2 and e 2 is the product of e 2 and eie 3 e 4 which is € 2 eie 3 e 4 . The interior product of €j€ 2 e 3 and ei€ 4 is the product of €ie 2 e 3 and € 2 e 3 which would be regressive and be the line e 2 e 3 . We have the same kinds of multiplication if the expres- sions e are vectors and not points, and they may even be INTRODUCTION 1 1 planes. The interpretation is different, however. It is easy to see that Grassmann's ideas do not lend themselves readily to numerical application, as they are more closely related to the projective transformations of space. In fact, when translated, most of the expressions would be phrased in terms of intersections, points, lines and planes, rather than in terms of distances, angles, areas, etc. 10. Dyadics were invented by Gibbs, 18 and are of both the algebraic and the geometric character. Gibbs has, like Hamilton, but one kind of multiplication. If we have given two vectors a, (3 from the same point, their dyad is a(3. This is to be looked upon as a new entity of two dimensions belonging to the point from which the vectors are drawn. It is not a plane though it has two dimensions, but is really a particular and special kind of dyadic, an entity of two- dimensional character, such that in every case it can be considered to be the sum of not more than three dyads. Gibbs never laid any stress on the geometric existence of the dyadic, though he stated definitely that it was to be considered as a quantity. His greatest stress, however, was upon the operative character of the dyadic, its various combinations with vectors being easily interpretable. The simplest interpretation is from its use in physics to represent strain. Gibbs also pushed his vector calculus into space of many dimensions, and into triadic and higher forms, most of which can be used in the theory of the elasticity of crystals. The scalar and vector multiplication he considered as functions of the dyadic, rather than as multiplications, and there are corresponding functions of triadics and higher forms. In this respect his point of view is close to that of Hamilton, the difference being in the use of the dyadic or the quaternion. 11. Other forms of vector calculus can be reduced to 3 12 VECTOR CALCULUS these or to combinations of parts of these. The differences are usually in the notations, or in the basis of exposition. Notations for One Vector Greek letters, Hamilton, Tait, Joly, Gibbs. Italics, Grassmann,_Peano, Fehr, Ferraris, Macfarlane. Heun writes a, b, c. Old English or German letters, Maxwell, Jaumann, Jung, Foppl, Lorentz, Gans, Abraham, Bucherer, Fischer, Sommerfeld. Clarendon type, Heaviside, Gibbs, Wilson, Jahnke, Timer- ding, Burali-Forti, Marcolongo. Length of a vector T ( ), Hamilton, Tait, Joly. | | , Gans, Bucherer, Timerding. Italic corresponding to the ve ctor letter, Wilson, Jaumann, &ing, Fischer, Jahnke. Corresponding small italic, Macfarlane. Mod. ( ), Peano, Burali-Forti, Marcolongo, Fehr. Unit of a vector U ( ), Hamilton, Tait, Joly, Peano. Clarendon small, Wilson. ( )i, Bucherer, Fischer. Corresponding Greek letter, Macfarlane. Some write the vector over the length. Square of a vector ( ) 2 . The square is usually positive except in Quaternions, where it is negative. Reciprocal ( ) -1 , Hamilton, Tait, Joly, Jaumann. tt , Hamilton, Tait, Joly, Fischer, Bucherer. CHAPTER II SCALAR FIELDS 1. Fields. If we consider a given set of elements in space, we may have for each element one or more quantities determined, which can be properly called functions of the element. For instance, at each point in space we may have a temperature, or a pressure, or a density, as of the air. Or for every loop that we may draw in a given space we may have a length, or at some fixed point a potential due to the loop. Again, we may have at each point in space a velocity which has both direction and length, or an electric intensity, or a magnetic intensity. Not to multiply examples unnecessarily, we can see that for a given range of points, or lines, or other geometric elements, we may have a set of quantities, corresponding to the various elements of the range, and therefore constituting a function of the range, and these quantities may consist of numerical values, or of vectors, or of other hypernumbers. When they are of a simple numerical character they are called scalars, and the function resulting is a scalar function. Examples are the density of a fluid at each point, the density of a distribution of energy, and similar quantities consisting of an amount of some entity per cubic centimeter, or per square centimeter, or per centimeter. EXAMPLES (1) Electricity. The unit of electricity is the coulomb, connected with the absolute units by the equations 1 coulomb = 3 • 10° electrostatic units == 10 -1 electromagnetic units. 13 14 VECTOR CALCULUS The density of electricity is its amount in a given volume, area, op length divided by the volume, area, or length respectively. The dimensions of electricity will be repre- sented by [9], and for its amount the symbol 9 will be used. For the volume density we will use e, for areal density e' , for linear density e". If the distribution may be considered to be continuous, we may take the limits and find the density at a point. (2) Magnetism. Considering magnetism to be a quan- tity, we will use for the unit of measurement the maxwell, connected with the absolute units by the equation 1 maxwell = 3-10 10 electrostatic units = 1 electromagnetic unit. Sometimes 10 8 maxwells is called a weber. The symbol for magnetism will be $, the dimensions [$], the densities m, m', m". (3) Action. This quantity is much used in physics, the principle of least action being one of the most important fundamental bases of modern physics. The dimensions of action are [93>], the symbol we shall use is A, and the unit might be a quantum, but for practical purposes a joule-second is used. In the case of a moving particle the action at any point depends upon the path by which the particle has reached the point, so that as a function of the points of space it has at each point an infinity of values. A function which has but a single value at a point will be called monodromic, but if it has more than one value it will be called polydromic. The action is therefore a polydromic function. We not only have action in the motion of par- ticles but we find it as a necessary function of a momentum field, or of an electromagnetic field. (4) Energy. The unit of energy is the erg or the joule SCALAR FIELDS 15 = 10 7 ergs. Its dimensions are [G^T 7-1 ], its symbol will beW. (5) Activity. This should not be confused with action. It is measured in watts, symbol J, dimensions [Q$T~ 2 ]. (6) Energy-density. The symbol will be U, dimensions (7) Activity-density. The symbol will be Q, dimensions pi- 3 r 2 ]. (8) Mass. The symbol is M, dimensions [0$77r 2 ]. The unit of mass is the gram. A distribution of mass is usually called a distribution of matter. (9) Density of mass. The symbol will be c, dimensions (10) Potential of electricity. Symbol V, dimensions (11) Potential of magnetism. Symbol N, dimensions [02 7 - 1 ]. (12) Potential of gravity. Symbol P, dimensions [G^T 7-1 ]. 2. Levels. Points at which the function has the same value, are said to define a level surface of the function. It may have one or more sheets. Such surfaces are usually named by the use of the prefixes iso and equi. For instance, the surfaces in a cloud, which have all points at the same temperature, are called isothermal surfaces; surfaces which have points at the same pressure are called isobaric surfaces; surfaces of equal density are isopycnic surfaces; those of equal specific volume (reciprocal of the density) are the iso- steric surfaces; those of equal humidity are isohydric surfaces. Likewise for gravity, electricity, and magnetism we have equipotential surfaces. 3. Lamellae. Surfaces are frequently considered for which we have unit difference between the values of the function for the successive surfaces. These surfaces and 16 VECTOR CALCULUS the space between them constitute a succession of unit lamellae. If we follow a line from a point A to a point B, the number of unit lamellae traversed will give the difference between the two values of the function at the points A and B. If this is divided by the length of the path we shall have the mean rate of change of the function along the path. If the path is straight and the unit determining the lamellae is made to decrease indefinitely, the limit of this quotient at any point is called the derivative of the function at that point in the given direction. The derivative is ap- proximately the number of unit lamellae traversed in a unit distance, if they are close together. 4. Geometric Properties. Monodromic levels cannot in- tersect each other, though any one may intersect itself. Any one or all of the levels may have nodal lines, conical points, pinch-points, and the other peculiarities of geo- metric surfaces. These singularities usually depend upon the singularities of the congruence of normals to the surface. In the case of functions of two variables, the scalar levels will be curves on the surface over which the two variables are defined. Their singularities may be any that can occur in curves on surfaces. 5. Gradient. The equation of a level surface is found by setting the function equal to a constant. If, for in- stance, the point is located by the coordinates x, y, z and the function is f(x, y, z), then the equation of any level is u = /(*> V> z ) = C. If we pass to a neighboring point on the same surface we have du = f{x + dx, y -f- dy, z + dz) — f{x, y, z) = 0. We may usually find functions df/dx, bf\a\ df/dz, SCALAR FIELDS 17 functions independent of dx, dy, dz, such that du — dfjdx • dx + df/dy • dy + df/dz • dz. Now the vector from the first point to the second has as the lengths of its projections on the axes: dx, dy, dz; and if we define a vector whose projections are dfjdx, df/dy, df/dz, which we will call the Gradient of f, then the con- dition du = is the condition that the gradient of / shall be perpendicular to the differential on the surface. Hence, if we represent the gradient of / by v/, and the differential change from one point to the other by dp, we see that dp is any infinitesimal tangent on the surface and v/ is along the normal to the surface. It is easy to see that if we differen- tiate u in a direction not tangent to a level surface of u we shall have du = df/dx-dx + df/dy •<&,+ df/dz -dz = dC. If the length of the differential path is ds then we shall have* du/ds = projection of^fon the unit vector in the direction of dp. The length of the vector v/ is sometimes called the gradient rather than the vector itself. Sometimes the negative of the expression used here is called the gradient. When the three partial derivatives of / vanish for the same point, the intensity of the gradient, measured by its length, is zero, and the direction becomes indeterminate from the first differentials. At such points there are singu- larities of the function. At points where the function becomes infinite, the gradient becomes indeterminate and such points are also singular points. 6. Potentials. The three components of a vector at a point may be the three partial derivatives of the same function as to the coordinates, in which case the vector may be looked upon as the gradient of the integral func- * Since dxjds, dyjds, dzjds are the direction-cosines of dp. 18 VECTOR CALCULUS tion, which is called a potential junction, or sometimes a force function. For instance, if the components of the velocity satisfy the proper conditions, the velocity is the gradient of a velocity 'potential. These conditions will be discussed later, and the vector will be freed from dependence upon any axes. 7. Relative Derivatives. In case there are two scalar functions at a point, we may have use for the concept of the derivative of one with respect to the other. This is defined to be the quotient of the intensity of the gradient of the first by that of the second, multiplied by the cosine of their included angle. If the unit lamellae are constructed, it is easy to see from the definition that the relative deriva- tive of the first as to the second will be the limit of the average or mean of the number of unit sheets of the first traversed from one point to another, along the normal of the second divided by the number of unit sheets of the second traversed at the same time. For instance, if we draw the isobars for a given region of the United States and the simultaneous isotherms, then in passing from a point A to a point B if we traverse 24 isobaric unit sheets and 10 isothermal unit sheets, the average is 2.4 isobars per isotherm. ^ 8. Unit-Tubes. If there are two scalar functions in the field, and the unit lamellae are drawn, the unit sheets will usually intersect so as to divide the space under considera- tion into tubes whose cross-section will be a curvilinear parallelogram. Since the area of such parallelogram is approximately dsids2 esc 0, where dsi is the distance from a unit sheet of the function u to the next unit sheet, and ds 2 the corresponding distance for the function v, while 6 is the angle between the surfaces; and since we have, Tyu being the intensity of the gradient SCALAR FIELDS 19 of u, and T^/v the intensity of the gradient of v, dsi - 1/TVu, ds 2 = 1/Tw the area of the parallelogram will be l/(TyuTvv sin 6). Consequently if we count the parallelograms in any plane Fig. 1. cross-section of the two sets of level surfaces, this number is an approximate value of the expression T^uT^Jv sin 6 X area parallelogram when summed over the plane cross-section. That is to say, the number of these tubes which stand perpendicular to the plane cross-section is the approximate integral of the expression T^uT^v sin 6 over the area of the cross-section. These tubes are called unit tubes for the same reason that the lamellae are called unit lamellae. In counting the tubes it must be noticed whether the successive surfaces crossed correspond to an increasing or to a decreasing value of u or of v. It is also clear that when sin 6 is everywhere the integral must be zero. In such case the three Jacobians d(u, v)/d(y, z), d(u, v)/d(z, x), d{u, v)/d(x, y) 20 VECTOR CALCULUS are each equal zero, and this is the^condition that u is a function of v. In case the plane of cross-section is the x, y plane, the first two expressions vanish anyhow, since u, v are functions of x, y only. It is clear if we take the levels of one of the functions, say u, as the upper and lower parts of the boundary of the cross-section, that in passing from one of the other sides of the boundary along each level of u the number of unit tubes we encounter from that side of the boundary to the opposite side is the excess of the value of v on the second side over that on the first side. If then we count the dif- ferent tubes in the successive lamellae of u between the two sides of the cross-section we shall have the total excess of those on the second side over those on the first side. That is to say, the number of unit tubes or the integral over the area bounded by level 1 and level 2 of u, and any other two lines which cross these two levels so as to produce a simple area between, is the excess of the sum between the two levels of the values of v on one side over the same sum between the two levels of u on the other side. These graphical solutions are used in Meteorology. This gives the excess of the integral J vdu along the second line between the two levels of u, over the same in- tegral along the first line. It represents the increase of this integral in a change of path from one line to the other. For instance if the integral is energy, the number of tubes is the amount of energy stored or released in the passage from one line to the other, as in a cyclone. The number of tubes for any closed path is the approximate integral I rdu around the path. , SCALAR FIELDS 21 EXERCISES. 1. If the density varies as the distance from a given axis, what are the isopycnic surfaces? 2. A rotating fluid mass is in equilibrium under the force of gravity, the hydrostatic pressure, and the centrifugal force. What are the levels? Show that the field of force is conservative. 3. The isobaric surfaces are parallel planes, and the isopycnic surfaces are parallel planes at an angle of 10° with the isobaric planes. What is the rate of change of pressure per unit rate of change of density along a line at 45° with the isobaric planes? 4. If the pressure can be stated as a function of the density, what conditions are necessary? Are they sufficient? What is the interpreta- tion with regard to the levels? 5. Three scalar functions have a functional relation if their Jacobian vanishes. What does this mean with regard to their respective levels? 6. If the isothermal surfaces are spheres with center at the earth's center, the temperature sheets for decrease of one degree being 166.66 feet apart, and if the isobaric levels are similar spheres, the pressure being given by log B = log B, - 0.0000177 (a - z ), where B is the pressure at z feet above the surface of the earth, what is the relative derivative of the temperature as to the pressure, and the pressure as to the temperature? 7. To find the maximum of u(x, y, z) we set du = 0. If there is also a condition to be fulfilled, v(x, y, z) = 0, then dv = also. These two equations in dx, dy, dz must be satisfied for all compatible values of dx, dy, dz, and we must therefore have du du du _ _ dy # dv dv_ dx' dy' dz' ~ dx' dy' dz } which is equivalent to the single vector equation Vw = wyv. What does this mean in terms of the levels : ; The unit tubes? If there is also another equation of condition l(x, y, z) =0 then also dt = and the Jacobian of the three functions u, v, t must equal zero. Interpret. 8. On the line of intersection of two levels of two different functions the values of both functions remain constant. If we differentiate a third function along the locus in question, the differential vanishing everywhere, what is the significance? 22 VECTOR CALCULUS 9. If a field of force has a potential, then a fluid, subject to the force and such that its pressure is a function of the density and the tempera- ture, will have the equipotential levels for isobaric levels also. The density will be the derivative of the pressure relative to the potential. Show therefore that equilibrium is not possible unless the isothermals are also the levels of force and of pressure. [p = p(c, T), and vp = cvv = PcVc + prvT. If then vc = 0, cvv = prVT.] 10. If the full lines below represent the profiles of isobaric sheets, and the dotted lines the profiles of isosteric sheets, count the unit tubes between the two verticals, and explain what the number means. If they were equipotentials of gravity and isopycnic surfaces, what would the number of unit tubes mean? Fig. 2. 11. If u = y — 12x 3 and v = y + x 2 + \x, find Vw and w and TvuTw -sin 6, and integrate the latter over the area between x = f x = 1, y = 0, y = 12. Draw the lines. 12. If u = ax + by + cz and v = x 2 -f- if + z 2 , find vw and vv and TyuTvvsm 6 and integrate the latter expression over the surface of a cylinder whose axis is in the direction of the z axis. Find the deriva- tive of each relative to the other. CHAPTER III VECTOR FIELDS 1. Hypercomplex Quantity. In the measurement of quantity the first and most natural invention of the mind was the ordinary system of integers. Following this came the invention of fractions, then of irrational numbers. With these the necessary list of numbers for mere measure- ment of similar quantities is closed, up to the present time. Whether it will be necessary to invent a further extension of number along this line remains for the future to show. In the attempt to solve equations involving ordinary numbers, it became necessary to invent negative numbers and imaginary numbers. These were known and used as fictitious numbers before it was noticed that quantities also are of a negative or an "imaginary" character. We find instances everywhere. In debit and credit, for ex- ample, we have quantity which may be looked upon as of two different kinds, like iron and time, but the most logical conception is to classify debits and credits together in the single class balance. One's balance is what he is worth when the debits and credits have been compared. If the preponderance is on the side of debit we consider the balance negative, if on the side of credit we consider the balance positive. Likewise, we may consider motion in each direc- tion of the compass as in a class by itself, never using any conception of measurement save the purely numerical one of comparing things which are exactly of the same kind together. But it is more logical, and certainly more general, to consider motions in all directions of the compass and of any distances as all belonging to a single class of quantity. 23 24 VECTOR CALCULUS In that case the comparison of the different motions leads us to the notion of complex numbers. When Wessel made his study of the vectors in a plane he was studying the hypernumbers we usually call "the complex field." The hypernumbers had been studied in themselves before, but were looked upon (rightly) as being creations of the mind and (in that sense correctly) as having no existence in what might be called the real world. However, their deduction from the vectors in a plane showed that they were present as relations of quantities which could be considered as alike. Again when Steinmetz made use of them in the study of the relations of alternating currents and electromotive forces, it became evident that the so-called power current and wattless current could be regarded as parts of a single complex current, and similarly for the electromotive forces. The laws of Ohm and Kirchoff could then be generalized so as to be true for the new complex quantities. In this brief history we find an example of the interaction of the develop- ments of mathematics. The inventions of mathematics find instances in natural phenomena, and in some cases furnish new conceptions by which natural phenomena can be regarded as containing elements that would ordinarily be completely overlooked. In space of three (or more) dimensions, the vectors issuing from a point in all directions and of all lengths furnish quantities which may be considered to be all of the same kind, on one basis of classification. Therefore, they will define certain ratios or relations which may be called hypernumbers. This is the class of hypernumbers we are particularly concerned with, though we shall occa- sionally notice others. Further, any kind of quantity which can be represented completely for certain purposes by vectors issuing from a point we will call vector quantity. VECTOR FIELDS 25 Such quantities, for instance, are motions, velocities, accelerations, at least in the Newtonian mechanics, forces, momenta, and many others. The object of VECTOR CAL- CULUS is to study these hypernumbers in relation to their corresponding quantities, and to derive an algebra capable of handling them. We do not consider a vector as a mere triplex of ordinary numbers. Indeed, we shall consider two vectors to be identical when they represent or can represent the same quantity, even though one is ex- pressed by a certain triplex, as ordinary Cartesian coordinates, and the other by another triplex, as polar coordinates. The numerical method of defining the vector will be considered as incidental. 2. Notation. We shall represent vectors for the most part by Greek small letters. Occasionally, however, as in Electricity, it will be more convenient to use the standard symbols, which are generally Gothic type. As indicated on page 12 there is a great variety of notation, and only one principle seems to be used by most writers, namely that of using heavy type for vectors, whatever the style of type. In case the vector is from the origin to the point (x, y, z) it may be indicated by Px, y, z> while for the same point given by polar coordinates r, ,a lf! = /?„. The equivalence of two glissants implies sets of equalities reducible in every case to five independent equalities. The equivalence of two radials reduces to sets of six equalities. 4. Vector Fields. Closely allied to the notion of radial is that of vector field. A vector field is a system of vectors each associated with a point of space, or a point of a surface, or a point of a line or curve. The vector is a function of the position of the point which is itself usually given by a vector, as p. The vector function may be monodromic or polydromic. We will consider some of the usual vector fields. EXAMPLES (1) Radius Vector, p [L]. This will usually be indicated by p. In case it is a function of a single parameter, as t, the points defined will lie on a curve;* in case it is a function * We are discussing mainly ordinary functions, not the "pathologic type." VECTOR FIELDS 27 of two parameters, u, v, the points defined will lie on a surface. The term vector was first introduced by Hamilton in this sense. When we say that the field is p, we mean that at the point whose vector is p measured from the fixed origin, there is a field of velocity, or force, or other quantity, whose value at the point is p. (2) Velocity, a [XT 7-1 ]. Usually we will designate veloc- ity by c. In the case of a moving gas or cloud, each particle has at each point of its path a definite velocity, so that we can describe the entire configuration of the moving mass at any instant by stating what function a is of p, that is, for the point at the end of the radius vector p assign the velocity vector. The path of a moving particle will be called a trajectory. At each point of the path the velocity a is a tangent of the trajectory. If we lay off from a fixed point the vectors a which corre- spond to a given trajectory, their terminal points will lie on a locus called by Hamilton the hodograph of the trajectory. For instance, the hodographs of the orbits of the planets are circles, to a first approximation. If we multiply a by dt, which gives it the dimensions of length, namely an infinitesimal length along the tangent of the trajectory, the differential equation of the trajectory becomes dp = adt. The integral of this in terms of t gives the equation of the trajectory. (3) Acceleration. t[LT~ 2 ]. An acceleration field is simi- lar to a velocity field except in dimensions. The accelera- tion is the rate of change of the vector velocity at a point, consequently, if a point describes the hodograph of a trajec- tory so that its radius vector at a given time is the velocity in the trajectory at that time, the acceleration will be a 3 L\S VECTOR CALCULUS tangent to the hodograph, and its length will be the velocity of the moving point in the hodograph. We will use r to indicate acceleration. (4) Momentum Density. T [$QL~ 4 ]. This is a vector function of points in space and of some number which can be attached to the point, called density. In the case of a moving cloud, for instance, each point of the cloud will have a velocity and a density. The product of these two factors will be a vector whose direction is that of the velocity and whose length is the product of the length of the velocity vector and the density. However, momentum density may exist without matter and without motion. In electro- dynamic fields, such as could exist in the very simple case of a single point charge of electricity and a single magnet pole at a point, we also have at every point of space a momentum density vector. This may be ascribed to the hypothetical motion of a hypothetical ether, but the essen- tial feature is the existence of the field. If we calculate the integral of the projection of the momentum density on the tangent to a given curve from a point A to a point B, the value of the integral is the action of an infinitesimal volume, an action density, along that path from A to B. The integration over a given volume would give the total action for all the particles over their various paths. This would be a minimum for the paths actually described as compared with possible paths. Specific momentum is momentum density of a moving mass. (5) Momentum. Y [TOL -1 ]. The volume integral of momentum density or specific momentum is momentum. Action is the line-integral of momentum. (6) Force Density. F [^QL^T- 1 ]. If a field of momen- tum density is varying in time then at each point there is a vector which may be called force-density, the time derivative VECTOR FIELDS 29 of the momentum density. Such cases occur in fields due to moving electrons or in the action of a field of electric intensity upon electric density, or magnetic intensity on magnetic density. (7) Force. X [mL- 1 ? 7 - 1 ]. The unit of force has re- ceived a name, dyne. It is the volume integral of force density. The time integral of a field of force is momentum. In a stationary field of force the line integral of the field for a given path is the difference in energy between the points at the ends of the path, or what is commonly called work. In case the field is conservative the integral has the same value for all paths (which at least avoid certain singular points), and depends only on the end points, This takes place when the field is a gradient field of a force- function, or a potential function. If we project the force upon the velocity at each point where both fields exist, the time integral of the scalar quantity which is the product of the intensity of the force, the intensity of the velocity and the cosine of the angle between them, is the activity at the point. (8) Flux Density. 12 [UT~ 1 }. In the case of the flow of an entity through a surface the limiting value of the amount that flows normally across an infinitesimal area is a vector whose direction is that of the outward normal of the surface, and whose intensity is the limit. In the case of a flow not normal to the surface across which the flux is to be de- termined, we nevertheless define the flux density as above. The flux across any surface becomes then the surface integral of the projection of the flux density on the normal of the surface across which the flux is to be measured. Flux density is an example of a vector which depends upon an area, and is sometimes called a bivector. The notion of two vectors involved in the term bivector may 30 VECTOR CALCULUS be avoided by the term cycle, or the term feuille. It is also called an axial vector, in opposition to the ordinary vectors, called polar vectors. The term axial is applicable in the sense that it is the axis or normal of a portion of a surface. The portion (feuille, cycle) of the surface is traversed in the positive direction in going around its boundary, that is, with the surface on the left-hand. If the direction of the axial vector is reversed, we also traverse the area attached in the reverse direction, so that in this sense the axial vector may be regarded as invariant for such change while the polar vector would not be invariant. The distinction is not of much importance. The important idea is that of areal integration for the flux density or any other so-called axial vector, while the polar vector is sub- ject only to linear integration. We meet the distinction in the difference below between the induction vectors and the intensity vectors. (9) Energy Density Current. R [TOL -2 ? 7 - 2 ]. When an energy density has the idea of velocity attached to it, it becomes a vector with the given dimensions. In such case we consider it as of the nature of a flux density. (10) Energy Current. 2 [$QT~ 2 ]. If a vector of energy density current is multiplied by an area we arrive at an energy current. (11) Electric Density Current. J [SL^T- 1 ]. A number of moving electrons will determine an average density per square centimeter across the line of flow, and the product of this into a velocity will give an electric density current. To this must also be added the time rate of change of electric induction, which is of the same dimensions, and counts as an electric density current. (12) Electric Current. C [97 1-1 ]. The unit is the ampere = 3-10 9 e.s. units = 10 _1 e.m. units. This is the product of an electric density current by an area. VECTOR FIELDS 31 (13) Magnetic Density Current. G [$Ir 2 T- 1 }. Though there is usually no meaning to a moving mass of magnetism, nevertheless, the time rate of change of magnetic induction must be considered to be a current, similar to electric current density. (14) Magnetic Current. K [^T' 1 ]. The unit is the heavy side = 1 e.m. unit = 3 • 10 10 e.s. units. In the phenom- ena of magnetic leakage we have a real example of what may be called magnetic current. Both electric current and magnetic current may also be scalars. For instance, if the corresponding flux densities are integrated over a given surface the resulting scalar values would give the rate at which the electricity or the magnetism is passing through the surface per second. In such case the symbols should be changed to corresponding Roman capitals. (15) Electric Intensity. E fMr 1 ! 1 " 1 ]. When an electric charge is present in any portion of space, there is at each point of space a vector of a field called the field of electric intensity. The same situation happens when lines of magnetic induction are moving through space with a given velocity. The electric intensity will be perpendicular to both the line of magnetic induction and to the velocity it has, and equal to the product of their intensities by the sine of their angle. The electric intensity is of the nature of a polar vector and its flux, or surface integral over any surface has no meaning. Its line integral along any given path, however, is called the difference of voltage between the two points at the ends of the path, for that given path. The unit of voltage is the volt = J • 10~ 2 e.s. units = 10 8 e.m. units. The symbol for voltage is V [$T~ 1 ]. Its dimensions are the same as for scalar electric potential, or magnetic current. 32 VECTOR CALCULUS (16) Electric Induction. D [QL~ 2 ]. The unit is the line = 3-10 9 e.s. units — 10 -1 e.m. units. This vector usually has the same direction as electric intensity, but in non- isotropic media, such as crystals, the directions do not agree. It is a linear function of the intensity, however, ordinarily indicated by D = k(E) where k is the symbol for a linear operator which converts vectors into vectors, called here the permittivity, [0^> -1 Z _1 T], measurable in farads per centimeter. In isotropic media k is a mere numerical multiplier with the proper dimensions, which are essential to the formulae, and should not be neglected even when k = 1. The flux is measured in coulombs. (17) Magnetic Intensity. H [eL" 1 ? 7 " 1 ]. The field due to the poles of permanent magnets, or to a direct current traversing a wire, is a field of magnetic intensity. In case we have moving lines of electric induction, there is a field of magnetic intensity. It is of a polar character, and its flux through a surface has no meaning. The line integral between two points, however, is called the gilbertage between the points along the given path, the unit being the gilbert = 1 e.m. unit = 3 • 10 10 e.s. units. The symbol is N [GT- 1 ]' Its dimensions are the same as those of scalar magnetic potential, or electric current. (18) Magnetic Induction. B [$L~ 2 ]. The unit is the gauss = 1 e.m. unit = 3 • 10 10 e.s. units. The direction is usually the same as that of the intensity, but in any case is given by a linear vector operator so that we have B-m(H) where \x is the inductivity, [^>0 -1 Z _1 T], measurable in henrys per centimeter. The flux is measured in maxwells. VPPf VECTOR FIELDS 33 (19) Vector Potential of Electric Induction. T [eZ -1 ]. A vector field may be related to another vector field in a certain manner to be described later, such that the first can be called the vector potential of the other. (20) Vector Potential of Magnetic Induction. ^ [M -1 ]. This is derivable from a field of magnetic induction. This and the preceding are line-integrable. (21) Hertzian Vectors. 9, <£. These are line integrals of the preceding two, and are of a vector nature. 5. Vector Lines. If we start at a given point of a vector field and consider the vector of the field at that point to be the tangent to a curve passing through the point, the field will determine a set of curves called a congruence, since there will be a two-fold infinity of curves, which will at every point have the vector of the field as tangent. If the field is represented by a, a function of p, the vector to a point of the field, then the differential equation of these lines of the congruence will be dp = adt, where dt is a differential parameter. From this we can determine the equation of the lines of the congruence, in- volving an arbitrary vector, which, however, will not have more than two essential constants. For instance, if the field is given by a = p, then dp = pdt, and p = ae l , where a is a constant unit vector. The lines are, in this case, the rays emanating from the origin. The lines can be constructed approximately by starting at any given point, thence following the vector of the field for a small distance, from the point so reached following the new vector of the field a small distance, and so proceed- ing as far as necessary. This will trace approximately a vector line. Usually the curves are unique, for if the field is monodromic at all points, or at points in general, the 34 VECTOR CALCULUS curves must be uniquely determined as there will be at any point but one direction to follow. Two vector lines may evidently be tangent at some point, but in a monodromic field they cannot intersect, except at points where the in- tensity of the field is zero, for vectors of zero intensity are of indeterminate direction. Such points of intersection are singular points of the field, and their study is of high importance, not only mathematically but for applications. In the example above the origin is evidently a singular point, for at the origin a = 0, and its direction is indetermi- nate. 6. Vector Surfaces, Vector Tubes. In the vector field we may select a set of points that lie upon a given curve and from each point draw the vector line. All such vector lines will lie upon a surface called a vector surface, which in case the given curve is closed, forming a loop, is further particularized as a vector tube. It is evident that the vector lines are the characteristics of the differential equation dp = adt, which in rectangular coordinates would be equivalent to the equations dx _dy _ dz X ~ Y~ Z' In case these equations are combined so as to give a single exact equation, the integral will (since it must con- tain a single arbitrary constant) be the equation of a family of vector surfaces. The vector lines are the intersections of two such families of vector surfaces. The two families may be chosen of course in infinitely many different ways. Usually, however, as in Meteorology, those surfaces are chosen which have some significance. When a vector tube becomes infinitesimal its limit is a vector line. 7. Isogons. If we locate the points at which a has the VECTOR FIELDS 35 same direction, they determine a locus called an isogon for the field. For instance, we might locate on a weather map all the points which have the same direction of the wind. If isogons are constructed in any way it becomes a simple matter to draw the vector lines of the field. Machines for the use of meteorologists intended to mark the isogons have been invented and are in use.* As an instance con- sider the vector field a = (2x, 2y, — z). An isogon with the points at which a has the direction whose cosines are /, m, n is given by the equations 2x : 2y : — z = I : m : n or 2x = It, 2y = mt, z = — nt. It follows that the vector to any point of this isogon is given by p = t(l, m, n) - (0, 0, 3nt). That is to say, to draw the vector p to any point of the isogon we draw a ray from the origin in the direction given, then from its outer end draw a parallel to the Z direction backward three times the length of the Z projection of the segment of the ray. The points so determined will evi- dently lie on straight lines in the same plane as the ray and its projection on the XY plane, with a negative slope twice the positive slope of the ray. The tangents of the vector lines passing through the points of the isogon will then be parallel to the ray itself. The vector lines are drawn ap- proximately by drawing short segments along the isogon parallel to its corresponding ray, and selecting points such that these short segments will make continuous lines in *Sandstrdm: Annalen der Hydrographie und Maritimen Meteor- ologie (1909), no. 6, pp. 242 et.seq. Bjerknes: Dynamic Meteorology. See plates, p. 50. 36 VECTOR CALCULUS passing to adjacent isogons. The figure illustrates the method. All the vector lines are found by rotating the figure about the X axis 180°, and then rotating the figure so produced about the Z axis through all angles. Fig. 3. 8. Singularities. It is evident in the example preceding that there are in the figure two lines which are different from the other vector lines, namely, the Z axis and the line which is in the XY plane. Corresponding to the latter would be an infinity of lines in the XY plane passing through the origin. These lines are peculiar in that the other vector lines are asymptotic to them, while they are themselves vector lines of the field. A method of studying the vector lines in the entire extent of the plane in which they lie was used by Poincare. It consists in placing a sphere tangent VECTOR FIELDS 37 to the plane at the origin. Lines are then drawn from the center of the sphere to every point of the plane, thus giving two points on the sphere, one on the hemisphere next the plane and one diametrically opposite on the hemisphere away from the plane. The points at infinity in the plane correspond to the equator or great circle parallel to the plane. In this representation every algebraic curve in the plane gives a closed curve or cycle on the sphere. In the present case, the axes in the plane give two perpendicular great circles on the sphere, and the vector lines will be loops tangent to these great circles at points where they cross the equator. These loops will form in the four Junes of the sphere a system of closed curves which Poincare calls a topographical system. The equator evidently belongs to the system, being the limit of the loops as they grow nar- rower. The. two great circles corresponding to the axes also belong to the system, being the limits of the loops as they grow larger. If a point describes a vector line its projection on the sphere will describe a loop, and could never leave the lune in which the projection is situated. The points of tangency are called nodes', the points which represent the origin, and through which only the singular vector lines pass, are called fames. 9. Singular Points. The simplest singular lines depend upon the singular points and these are found comparatively simply. The singular points occur where o" = or a —• oo . Since we may multiply the components of a by any ex- pressions and still have the lines of the field the same, we may equally suppose that the components of a are reduced to as low terms as possible by the exclusion of common factors of all of them. We will consider first the singular 38 VECTOR CALCULUS points for fields in space, then those cases which have lines every point of which is a singular point, which will include the cases of plane fields, since these latter may be considered to represent the fields produced by moving the plane field parallel to itself. The classification given by Poincare is as follows. (1) Node. At a node there may be many directions in which vector lines leave the point. An example is a = p. At the origin, it is easy to see, a = 0, and it is not possible to start at the origin and follow any definite direction. In fact the vector lines are evidently the rays from the origin in all directions. There is no other singular point at a finite distance. If, however, we consider all the rays in any one plane, and for this plane construct the sphere of projection, we see that the lines correspond to great circles on the sphere which all pass through the origin and the point diametrically opposite to it. This ideal point may be considered to be another node, so that all the vector lines run from node to node, in this case. Every vector line which does not terminate in a node is a spiral or a cycle. (2) Faux. From a faux* there runs an infinity of vector lines which are all on one surface, and a single isolated vector line which intersects the surface at the faux. The surface is a singular surface since every vector line in it through the faux is a singular line. The singular surface is approached asymptotically by all the vector lines not singular. An example is given by a = (x, y, — z). The vector lines are to be found by drawing all equilateral hyperbolas in the four quadrants of the ZX plane, and then * Poincare uses the term col, meaning mountain pass, for which faux is Latin. / VECTOR FIELDS 39 rotating this set of lines about the Z axis. Evidently all rays in the XY plane from the origin are singular lines, as well as the Z axis. Where fauces occur the singular lines through them are asymptotes for the nonsingular lines. If Fig. 4. we consider any plane through the Z axis, the system of equilateral hyperbolas will project onto its sphere as cycles tangent on the equator to the great circles which repre- sent the singular lines in that plane. From this point of view we really should consider the two rays of the Z axis as separate from each other, so that the upper part of the Z axis and the singular ray perpendicular to it, running in the same general direction as the other vector lines, would con- stitute a vector line with a discontinuity of direction, or with an angle. Such a vector line to which the others are tangent at points at infinity only is a boundary line in the sense that on one side we have infinitely many vector lines which form cycles (in the sense defined) while on the other sides we have vector lines which belong to different sys- tems of cycles. 40 VECTOR CALCULUS A simple case of this example might arise in the inward flow of air over a level plane, with an ascending motion which increased as the air approached a given vertical line, becoming asymptotic to this vertical line. In fact, a small fire in the center of a circular tent open at the bottom for a small distance and at the vertex, would give a motion to the smoke closely approximating to that described. A singular line from a faux runs to a node or else is a spiral or part of a cycle which returns to the faux. An example that shows both preceding types is the field a = (x 2 + y 2 — 1, bxy — 5, mz). In the X Y plane the singular points are at infinity as follows : A at the negative end of the X axis, and B at the positive end, both fauces; C at the end of the ray whose direction is tan -1 2, in the first quadrant, D at the end of the ray of direction tan -1 2 in the third quadrant; E at the end of the VECTOR FIELDS 41 ray of direction tan -1 — 2 in the fourth quadrant; and F at the end of the ray of tan -1 — 2 in the second quadrant, these four being nodes. Vector lines run from E to D separated from the rest of the plane by an asymptotic division line from B to D; from C to D on the other side of this division line, separated from the third portion of the plane by an asymptotic division line from C to A ; and from C to F in the third portion of the plane. The figure shows the typical lines of the field. (3) Focus. At a focus the vector lines wind in asymp- totically, either like spirals wound towards the vertex of a spindle produced by rotating a curve about one of its tangents, one vector line passing through the focus, or they are like spirals wound around a cone towards the Fig. 6. vertex. As an example o- = (x+ y, y - x, z). The Z axis is a single singular line through the origin, which is a singular point, a focus in this case. The XY plane contains vector lines which are logarithmic spirals wound in towards the origin. The other vector lines are spirals 42 VECTOR CALCULUS wound on cones of revolution, their projections on XY being the logarithmic spirals. By changing z to az we would have different surfaces depending upon whether 1 < a. a< 1 or In case a spiral winds in onto a cycle, the successive turns approaching the cycle asymptotically, the cycle is called a limit cycle. In this example the line at infinity in the X Y plane, or the corresponding equator on its sphere, is a limit cycle. It is clear that the spirals on the cones wind outward also towards the lines at infinity as limit cycles. From this example it is plain that vector lines which are spiral may start asymptotically from a focus and be bounded by a limit cycle. The limit cycle thus divides the plane or the surface upon which they lie into two mutually exclusive regions. Vector lines may also start from a limit cycle and proceed to another limit cycle. As an example of vector lines of both kinds consider the field Fig. 7. a = ( r 2 _ 1, r 2 + lf mz)f where the first component is in the direction of a ray in the XY plane from the origin, the second perpendicular to VECTOR FIELDS 43 this in the XY plane, and the third is parallel to the Z axis. The vector lines in the singular plane, the XY plane, are spirals with the origin as a focus for one set, which wind around the focus negatively and have the unit circle as a limit cycle, while another set wind around the unit circle in the opposite direction, having the line at infinity as a limit cycle. The polar equation of the first set is r~ l — r An example with all the preceding kinds of singularities is the field Fig. 8. a = ( [r 2 - l)(r - 9)], (r 2 - 2r cos 9 - 8), mz) with directions for the components as in the preceding example. The singular points are the origin, a focus; the point A (r = 3, = + cos -1 §), a node; the point B (r = 3, 6 = — cos -1 J), a faux. The line at infinity is a limit cycle, as well as the circle r = 1, which is also a vector line. The circle r = 3 is a vector line which is a cycle, 4 44 VECTOR CALCULUS starting at the faux, passing through the node and returning to the faux. The vector lines are of three types, the first being spirals that wind asymptotically around the focus, out to the unit circle as limit cycle; the second start at the node A and wind in on the unit circle as limit cycle; the third start at the node A and wind out to the line at in- finity as unit cycle. The second set dip down towards the faux. The exceptional vector lines are the line at infinity, the unit circle, both being limit cycles; the circle of radius 3; a vector line which on the one side starts at the faux B winding in on the unit circle, and on the other side starts at the faux B winding outward to the line at infinity as limit cycle. The last two are asymptotic division lines of the regions. The figure exhibits the typical curves. (4) Faux-Focus. This type of singular point has passing through it a singular surface which contains an infinity of spirals having the point as focus, while an isolated vector line passes through the point and the surface. No other surfaces through the vector lines approach the point. An instance is the field a- = (x, y, — z). The Z axis is the isolated singular line, while the XY plane is the singular plane. In it there is an infinity of spirals with the origin as focus and the line at infinity as limit cycle. All other vector lines lie on the surfaces rz = const. These do not approach the origin. (5) Center. At a center there is a vector line passing through the singular point, and not passing through this singular line there is a singular surface, with a set of loops or cycles surrounding the center, and shrinking upon it. There is also a set of surfaces surrounding the isolated singular line like a set of sheaths, on each of which there are vector lines winding around helically on it with a decreasing VECTOR FIELDS 45 Fig. 9. pitch as they approach the singular surface, which they therefore approach asymptotically. As an instance we have the field a = (y, - x, z). The Z axis is the singular isolated vector line, the XY plane the singular surface, circles concentric to the origin the singular vector lines in it, and the other vector lines lie on circular cylinders about the Z axis, approaching the XY plane asymptotically. The method of determining the character of a singular point will be considered later in connection with the study of the linear vector operator. A singular point at infinity is either a node or a faux. 10. Singular Lines. Singularities may not occur alone but may be distributed on lines every point of which is a singular point. This will evidently occur when cr = gives three surfaces which intersect in a single line. The dif- ferent types may be arrived at by considering the line of singularities to be straight, and the surfaces of the vector lines with the points of the singular line as singularities to be planes, -for the whole problem of the character of the singularities is a problem of analysis situs, and the deforma- tion will not change the character. The types are then as follows : (1) Line of Nodes. Every point of the singular line is a node. A simple example is a = (x, y, 0). The vector lines are all rays passing through the Z axis and parallel to the XY plane. 46 VECTOR CALCULUS (2) Line of Fauces. There are two singular vector lines through each point of the singular line. As an instance a = (x, — y, 0). The lines through the Z axis parallel to the X and the Y axes are singular, all other vector lines lying on hyperbolic cylinders. (3) Line of Foci. The points of the singular line are approached asymptotically by spirals. As an instance the isogon will have a singular point. It does not follow, however, that all the singular points of the isogons will appear as singular points such as are described above for the vector lines. When the differential equation of the isogons is reduced to the standard form dp = rdu we shall see later that r will be a linear vector function of a, and that a linear vector function may have zero directions, so that €3 = a 3 iai + a 32 a 2 + «33«3, then p becomes P = (a n x + a n y + a n z)ai + (a u x + a 22 y + a 32 z)a 2 + (a n x -f a 2z y + a 33 z)a 3 . It is evident then that if we transform the e's by a non- singular linear homogeneous transformation, the coeffi- cients of the new basis hypernumbers, a, are the transforms of the original coefficients under the contragredient trans- formation. Inasmuch as the transformation is linear, the transform of a sum will be the sum of the transforms of the terms of the original sum. The transformation as a geometrical process is equivalent to changing the axes. This process evidently gives us a new triple, but must be considered not to give us a new hypernumber nor a new vector. Indeed, a vector cannot be defined by a triple of numbers alone. There is also either explicitly stated or else implicitly understood to be a basis, or on the geometric side a definite set of axes such that the triple gives the components of the vector along these axes. It is evident that the success of any system of vector calculus must then depend upon the choice of modes of combination which are not affected by the change from one basis to another. This is the case with addition as we have defined it. We assume that we may express any vector or hypernumber in terms of any basis we like, and usually the basis will not appear. If the transformation is such as to leave the angles be- ADDITION OF VECTORS 55 tween ei, e 2 , e 3 the same as those between a\, a 2 , a 3 , the second trihedral being substantially the same as the first rotated into a new position, with the lengths in each case remaining units, then the transformation is called orthog- onal. We may define an orthogonal transformation algebra- ically as one such that if followed by the contragredient transformation the original basis is restored. 4. Differential of a Vector. If we consider two points at a small distance apart, the vector to one being p, to the other p', and the vector from the first to the second, Ap = p' — p, where Ap = As-e, e being a unit vector in the direction of the difference, we may then let one point ap- proach the other so that in the limit e takes a definite posi- tion, say a, and we may write ds for As, and call the result the differential of p for the given range over which the p f runs. In the hypernumbers we likewise arrive at a hyper- number dp = dxei -f- dye?, + dzez, where now ds is a linear homogeneous irrational function of dx, dy, dz, which = V (dx 2 + dy 2 + dz 2 ) in case e ly e 2 , e 3 form a trirectangular system of units. The quotient dpjdt is the velocity at the point if t repre- sents the time. The unit vector a: is the unit tangent for a curve. We generally represent the principal normal and the binormal by jS, 7 respectively. When p is given as dependent on a single variable parameter, as t for instance, then the ends of p may describe a curve. We may have in the algebraic form the coordinates of p alone dependent upon the parameter, or we may have both the coordinates and the basis dependent upon t. For instance, we may ex- press p in terms of ei, e 2 , e 3 which are not dependent upon t but represent fixed directions geometrically, or we may express p in terms of three hypernumbers as w, r, J* which 56 VECTOR CALCULUS themselves vary with t, such as the moving axes of a system in space. In relativity theories the latter method of repre- sentation plays an important part. 5. Integral of a Vector. If we add together n vectors and divide the result by n we have the mean of the n vectors, which may be denoted by p. If we select an infinite number of vectors and find the limit of their sum after multiplication by dt, the differential of the parameter by which they are expressed, such limit is called the integral of the vector expressed in terms of t, and if we give t two definite values in the integral and subtract one result from the other, the difference is the integral of the vector from the first value of t to the second. More generally, if we multiply a series of vectors, infinite in number, by a corre- sponding series of differentials, and find the limit of the sum of the results, such limit, when it exists, is called the integral of the series. In integration, as in differentiation, the usual difficulties met in analysis may appear, but as they are properly difficulties due to the numerical system and not to the hypernumbers, we will suppose that the reader is familiar with the methods of handling them. The mean in the case of a vector which has an infinite sequence of values is the quotient of the integral taken on some set of differentials, divided by the integral of the set of differentials itself. The examples will illustrate the use of the mean. EXAMPLES (1) The centroid of an arc, an area, or a volume is found by integrating the vector p itself multiplied by the dif- ferential of the arc, ds, or of the surface, du, or of the volume dv. The integral is then divided by the length of the arc, the area of the surface, or the volume. That is - Sheets ffpdu • fffpdv m P — — , or - — or — b— a A V ADDITION OF VECTORS 57 (2) An example of average velocity \s found in the following (Bjerknes, Dynamic Meteorology, Part II, page 14) obser- vations of a small balloon. 2 = Ht. in Meters Az Direction Velocity (w/sec.) Products 77 680 960 1240 1530 1810 2090 2430 2730 3040 3400 3710 4030 4400 603 280 280 290 280 280 340 300 310 360 310 320 370 S. 50° E. S. 57° E. S. 36° E. S. 28° W. S. 2°W. S. 2°W. S. 35° W. S. 53° W. S. 69° W. S. 55° W. S. 53° W. S. 58° W. S. 37° W. 3.4 4.0 5.3 1.5 1.8 2.0 1.5 1.8 1.8 3.0 2.8 4.4 10.2 2050 1120 1484 435 504 560 510 540 558 1080 868 1408 3773 To average the velocities we notice that on the assump- tion that the upward velocity was uniform the distances vertically can be used to measure the time. We therefore multiply each velocity by the difference of elevations corresponding, the products being set in the last column. These numbers are then taken as the lengths of the vectors whose directions are given by the third column. The sum of these is found graphically, and divided by the total difference of distance upward, that is, 4323. In the same manner we can find graphically the averages for each 1000 meters of ascent. We may now make a new table in order to find other important data, as follows : Height . Pressure (ra-bars) Dens, (ton/w 3 ) Veloc. . Spec. Mo- mentum (ton/ra 2 sec.) 4000 3000 2000 1000 75 622 705 797 899 1003 0.00083 0.00092 0.00102 0.00112 3.8 1.6 • 2.4 3.7 0.0032 0.0015 0.0025 0.0041 ;,s VECTOR CALCULUS We now find the average velocity between the 1000 m-bar, the 900 ra-bar, the 800 m-bar, the 700 ra-bar, and the 600 ra-bar. The direction is commonly indicated by the in- tegers from to 63 inclusive, the entire circle being divided into 64 parts, each of 5f°. East is 0, North is 16, NW. is 24, etc. The following table is found. Pressure Height Spec. Vol. (m 3 /Ton) Direction Veloc. Spec. Mo- mentum 600 700 800 ' 900 1000 1002.6 4274 3057 1970 989 99 76 1217 1087 981 890 890 8 7 20 25 25 5.2 1.7 2.4 3.7 3.4 0.0043 0.0016 0.0024 0.0042 0.0040 Of course, specific momenta should be averaged like veloc- ities but usually owing to the rough measurements it is sufficient to find specific momenta from the average velocities. ADDITION OF VECTORS 59 EXERCISES 1. Average as above the following observations taken at places mentioned (Bjerknes, p. 20), July 25, 1907, at 7 a.m. Greenwich time. Isobar Dyn. Ht. Az Direction Veloc. 100 200 300 400 500 600 700 800 900 1000 1001.2 16374 11947 9320 7301 5648 4240 3020 1938 975 107 98 4427 2627 2019 1653 1408 1220 1082 963 867 9 10 18 19 8 5 4 4 36 35 4.7 3.2 3.4 3.3 2.6 2.5 2.5 1.4 4.5 4.5 Uccle, Lat. 50° 48' Long. 4° 22' 100 200 300 400 500 600 700 800 900 955.9 16238 11817 9240 7248 5626 4244 3038 1955 977 471 4421 2577 1992 1622 1382 1206 1083 978 506 3 6 7 2 3 2 62 4 30 10.0 6.5 7.6 10.2 6.7 6.8 5.3 0.6 2.1 Zurich, Lat. 47° 23' Long. 8° 33' 200 300 400 500 600 700 800 900 1000 11890 9241 7240 5643 4196 2991 1927 981 118 17 2649 2001 1597 1447 1205 1064 946 863 101 59 57 58 55 49 41 38 56 55 9.2 10.5 8.8 8.0 2.9 2.9 1.9 4.3 3.4 Hamburg, Lat. 53° 33' Long. 9° 59' GO VECTOR CALCULUS 2. If the direction of the wind is registered every hour how is the average direction found? Find the average for the following observa- tions. Station Pikes Peak Vienna Mauritius Cordoba S Orkneys Elev 4308 m. 26 m. 15 m. 437 m. 25 m Summer Winter Dec-Feb. Winter Summer Time Vel. Az. Vel. Az. Vel. Az. Vel. Az. Vel. Az. a.m 0.84 1.34 1.46 1.05 0.43 0.66 1.03 100° 83 71 57 12 279 262 0.47 0.56 0.42 0.33 0.22 0.17 0.36 62 61 59 57 46 303 257 1.00 1.30 1.30 1.00 1.10 1.80 2.40 100 97 98 119 241 312 326 0.94 1.06 1.44 2.03 2.17 0.50 2.78 115 111 121 132 136 252 314 0.52 0.52 0.51 0.51 0.52 0.53 0.54 70 2 51 4 30 6 5 8 343 10 ?m 12 noon 255 14 1.09 256 0.58 242 1.60 332 3.56 315 0.54 245 16 0.95 253 0.64 232 1.30 304 3.36 305 0.54 42 18 0.74 247 0.47 223 0.20 10 1.75 299 0.53 35 20 0.49 47 0.14 186 0.90 101 0.72 44 0.53 50 22 0.36 153 0.25 72 1.00 102 0.89 128 0.52 60 Bigelow, Atmospheric Circulation, etc., pp. 313-315. 3. The following table gives the mean magnetic deflecting vectors, in four zones, the intensity measured in 10~ 6 dynes,

. BA,CU = y cks OCA. Construct A ACX directly similar to A A UB. 18. Find the condition that the three lines perpendicular to the three vectors pa, qa, ra at their extremities be concurrent. We have p + xkp = q + ykq = r + zkr. Taking conjugates q — xkp = p — ykq = r— zkr. Eliminate x, y, z from the four equations. 19. If a ray at angle is reflected in a mirror at angle a the reflected ray is in the direction whose angle is 2 a — /3. Study a chain of mirrors. Show that the final direction is independent of some of the angles. 20. Show that if the normal to a line is a and a point P is distant y from the line, and from P as a source of light a ray is reflected from the line, its initial direction being — qa, then the reflected ray has for equation — 2ya + tqa = p. For further study along these lines, see Laisant: Theorie et Application des Equipollences. 11. Alternating Currents. We will notice an application of these hypernumbers to the theory of alternating currents and electromotive forces, due to C. P. Steinmetz. If an alternating current is given by the equation I = Io cos 2wf(t - h), the graph of the current in terms of t is a circle whose diameter is 7 making an angle with the position for t = of 2wfti. The angle is called the phase angle of the current. If two such currents of the same frequency are superim- 72 VECTOR CALCULUS posed on the same circuit, say we may set 7 = 7 cos 2irf(t - ti), F = Jo' cos 2tt/(* - fcO, sex 7 cos 2vfh + h' cos 2tt/V = 7 " cos 2wfh, 7 sin 2tt/k)(g + Cirk), I 2 = (r + Lak)/(g + Cwk), so that m is [X -1 ] while / is ohms/mile, the solution of the equations is E = E cosh ms + ll sinh ms, I = Iq cosh ms + 1~ 1 Eq sinh ms, where E and 7 are the initial values, that is, where s = 0. If we set Eq = ZqIq and then set Z = Z cosh h, I = Z sinh h we have E = Z cosh (ms + h)I , I = l~ l Z sinh (ms + h)I , E = I coth (ms + h)I, E = sech h cosh (ms + h)E , I = csch h sinh (ms + h)I . To find where the wattless current of the initial station has become the power current we set I = kl , that is, sinh (ms -f- h) = k sinh h. VECTORS IN A PLANE 75 The value of s must be real. EXAMPLES (1) Let r = 2 ohms/mile, L = 0.02 henrys/mile, C = 0.0000005 farads/mile, g = 0, to = 2000, coL = 40 ohms/mile, conductor reactance, r + Look = 2 + 40/c ohms/mile impedance = 40.5 87 .i5 o . uC = 0.001 mhos/mile dielectric susceptance. g + Coik = 0.001 k mhos/mile dielectric admit- tance = 0.001 90 °. (g + Cuk)~ l = 1000/j" 1 = 1000 27 o° ohms/mile dielectric impedance. m 2 = 0.0405i 77 .i5°, m = 0.2001 88 .58°, P = 40500_.2.85°, I = 201.25_i.43°. Let the values at the receiver (s = 0) be E = 1000 o volts, 7 = o. Then we have E = 1000 cosh s0.2001 8 8.58°, for s = 100 E = 1000 cosh 20.01 88 . 58 = 625.9 45 .92°, I = 2.77 27 o, for s = 8 E = 50.01i26.ot, for s = 16 E = 1001i 80 .3°, for s = 15.7 E = 1000i 8 o°, a reversal of phase, for s = 7.85 E = 90 o. At points distant 31.4 miles the values are the same. If we assume that at the receiver end a current is to be maintained with Jo = 50 40 ° with E = 1000 o, E = 1000 cosh s0.2001 88 . 58 ° + 10062 38 . 5 7° sinh s0.2001 88 . 5 8°, I = 50 4 o° cosh sm + 5i. 4 3° sinh sm. At s = 100 E = 10730n355°. MacMahon, Hyperbolic Functions. 76 VECTOR CALCULUS (2) Let E - 10000, 7 - 65i 3 . 5 ° r = 1, g = 0.00002 Ceo = 0.00020 period 221.5 miles, o>L = 4. (3) The product P = EI represents the power of the alternating current, with the understanding that the fre- quency is doubled. The real or scalar part is the effective power, the imaginary part the wattless or reactive power. The value of TP is the total apparent power. The cos z P is the power factor, and sin / P is the induction factor. The torque, which is the product of the magnetic flux by the armature magnetomotive force times the sine of their angle is proportional to TIP, where E is the generated electromotive force, and/ is the secondary current. In fact, the torque is TI'EI-p/2irf where p is the number of poles (pairs) of the motor. 12. Divergence and Curl. In a general vector field the lines have relations to one another, besides having the peculiarities of the singularities of the field. The most important of these relations depend upon the way the lines approach one another, and the shape and position of a moving cross-section of a vector tube. There is also at each point of the field an intensity of the field as well as a direction, and this will change from point to point. Divergence of Plane Lines. If we examine the drawing of the field of a vector distribution in a plane, we may easily measure the rate of approach of neighboring lines. Starting from two points, one on each line, at the intersec- tion of the normal at a point of the first line and the second line, we follow the two lines measuring the distance apart on a normal from the first. The rate of increase of this normal distance divided by the normal distance and the distance traveled from the initial point is the divergence of the lines, or as we shall say briefly the geometric divergence of the field. It is easily seen that in this case of a plane VECTORS IN A PLANE 77 field it is merely the curvature of the curves orthogonal to the curves of the field. For instance, in the figure, the tangent to a curve of the field is a, the normal at the same point /5. The neighboring curve goes through C. The differential of the normal, which is the difference of BD and AC, divided by AC, or BD, is the rate of divergence of the second curve from the first for the distance AB. Hence, if we also divide by AB we will have the rate of angular turn of the tangent a in moving to the neigh- boring curve, the one from C. This rate of angular turn of the tangent of the field is the same as the rate of turn of the normal of the orthogonal system, and is thus the curva- ture of the normal system. Curl of Plane Lines. If we find the curvature of the original lines of the field we have a quantity of much im- portance, which may be called the geometric curl. This must be taken plus when the normal to the field on the convex side of the curve makes a positive right angle with the tangent, and negative when it makes a negative right angle with the tangent. Curl is really a vector, but for the case of a plane field the direction would be perpendicular to the plane for the curl at every point, and we may con- sider only its intensity. Divergence of Field. Since the field has an intensity as well as a direction, let the vector characterizing the field be cr = Ta-a. Then the rate of change of TV in the direc- tion of a, the tangent, is represented by d a T = kVu. We also have for whatever fields belong to the two sets of orthog- onal lines for u curves, a = rVv, for the v curves, a' = sVu, or also we may write Vv = tot, Vu = tp, a = Ta-ct. 14. Nabla. The symbol V is called nabla, and evidently may be written in the form ad/dx + Pd/dy for vectors in a plane. We will see later that for vectors in space it may be written ad/dx + Pd/dy + yd/dz, where a, ft y are the usual unit vectors of three mutually perpendicular directions. However, this form of this very important differential operator is not at all a necessary form. In fact, if a and fi are any two perpendicular unit vectors in a plane, and dr, ds are the corresponding differential dis- tances in these two directions, then we have V = ad/dr + pd/ds. VECTORS IN A PLANE 81 For instance, if functions are given in terms of r, 6, the usual polar coordinates, then V = Upd/dr + kUpd/rdd. The proof that for any orthogonal set of curves a similar form is possible, is left to the student. In general, V is defined as follows : V is a linear differentiating vector operator connected with the variable vector p as follows: Consider first, a scalar function of p, say F(p). Differentiate this by giving p any arbitrary differential dp. The result is linear in dp, and may be looked upon as the product of the length of dp and the projection upon the direction of dp of a certain vector for each direction dp. If now these vectors so projected can be reduced to a single vector, this is by definition VF. For instance, if F is the distance from the origin, then the differential of F in any direction is the projection of dr in a radial direction upon the direc- tion of differentiation. Hence, V7p = Up. In the case of plane vectors, VF will lie in the plane. In case the differential of F is polydromic, we define VF as a poly- dromic vector, which amounts to saying that a given set of vectors will each furnish its own differential value of dF. In some particular regions, or at certain points, the value of J7F may become indefinite in direction because the differentials in all directions vanish. Of course, functions can be defined which would require careful investigation as to their differentiability, but we shall not be concerned with such in this work, and for their adequate treatment reference is made to the standard works on analysis. We must consider next the meaning of V as applied to vectors. It is evident that if V is to be a linear and there- fore distributive operator, then such an expression as Va must have the same meaning as VXa. + V Y(3 + VZy if a = Xa + F/3 -r Zy, where a, 0, y are any independent constant vectors. This serves then as the definition of 82 VECTOR CALCULUS Vo-, the only remaining necessary part of the definition is the vector part which defines the product of two vectors. This will be considered as we proceed. 15. Nabla as a Complex Number. We will consider now p to represent the complex number x + yk, or r e , and that all our expressions are complex numbers. The proper expression for V becomes then V = d/dx + kd/dy = Upd/dr + kUpd/rdd. In general for the plane, let p depend upon two parameters u, v, and let dp = p\du -f- p 2 dv. If a is a function of p (generally not analytic in the usual sense) and thus dependent on u, v, we will have da = dcr/du-du + da/dv-dv = R-dpV -a. If we multiply dp by kpi, which is perpendicular to pi, the real part of both sides will be equal and we have, since kpi is perpendicular to pi, Rkpidp — dvRkpip 2 , and similarly Rkpidp = duRkpipi = — duRkpip 2 since the imaginary part of pip 2 equals — the imaginary part of p 2 pi- Substituting in da we have A, = «.*,(-,*£- £+#-£) This gives the rate of change of the density at a point moving with the fluid. Hence if it is incompressible, the velocity is solenoidal, RV& = 0. 88 VECTOR CALCULUS This may also be written curl (— ka) = 0, hence — ka = V?, and " - 05' + 00", D"£' = B'B + C ,, (7 , , D'D = C'C. Hence the biradial of the sum is OD/OA, where the scalar part is the ratio of OD" to OA. This is clearly the sum of the scalar parts of q and r, and S( q + f ) = Sq+ Sr. The vector part of the quaternion for OD/OA is the ratio of D"D to OA in magnitude, and the unit part is repre- sented by a unit normal perpendicular to OD" and D"D. But D"D = B'B + C'C, and the ratio of D"D to OA equals the sum of the ratios of B'B and C'C to OA. If then we draw, in a plane through which is perpendicular to OA, the vector Vq along the representative unit normal of the plane OAB, and of a length to represent the numerical ratio of B'B to OA, and likewise Vr to represent the ratio of C'C to OA laid off along the representative unit normal 98 VECTOR CALCULUS to the plane OAC, because D"D is parallel to this plane, as well as B'B and C"C, the representative unit vector of q+ r will lie in the plane, and will be in length the vector sum of Vq and Vr, that is V(q + r) as shown. It follows at once since the addition of scalars is associa- tive, and the addition of vectors is associative, and the two parts of a quaternion have no necessary precedence, that the addition of quaternions is associative. 4. Product of Quaternions. To define the product of quaternions we likewise utilize the biradials. In this case however we bring the initial vector of the multiplier to coincide with the terminal line of the multiplicand, and define the product biradial as the biradial whose initial vector is the initial vector of the multiplicand, and the terminal vector is the terminal vector of the multiplier. In the figure, the product of the biradials OB/OA, and Fig. 13. OC/OB, is, writing the multiplier first, OC/OB- OB/OA = OC/OA. It is clear that the tensor of the product is the product of the tensors, so that T-qr= TqTr. It follows that U-qr = UqUr. It is evident from the figure that the angle of the product will be the face angle of the trihedral, AOC, or on a unit sphere would be represented by the side of the spherical VECTORS IN SPACE 99 triangle corresponding. It is clear too that the reversal of the order of the multiplication will change the plane of the product biradial, usually, and therefore will give a quaternion with a different unit vector, though all the other numbers dependent upon the product will remain the same. However we can prove that multiplication of quaternions is associative. In this proof we may leave out the tensors and handle only the versors. The proof is due to Hamilton. To represent the biradials, since the vectors are all taken as unit vectors, we draw only an arc on the unit sphere, from one point to the other, of the two ends of the two unit vectors of the biradial. Thus we represent the biradial of q by CA, or, since the biradial may be rotated in its plane about the vertex, equally by ED. The others in- volved are shown. The product qr is represented by FD, from the definition, or equally by LM. What we have to prove is that the product p • qr is the same as the product pq-r, that is, we must prove that the arcs KG and LN are on the same great circle and of equal length and direction. Fig. 14. Since FE = KH, ED - CA, HG = CB, LM = FD, the points L, C, G, D are on a spherical conic, whose cyclic planes are those of AB, FE, and hence KG passes through L, and with LM intercepts on AB an arc equal to AB. That is, it passes through N, or KG and LN are arcs of the 100 VECTOR CALCULUS same great circle, and they are equal, for G and L are points in the spherical conic. 5. Trirectangular Biradials. A particular pair of bira- dials which lead to an interesting product is a pair of which the vectors of each biradial are perpendicular unit vectors, and the initial vector of one is the terminal of the other, for in such case, the product is a biradial of the same kind. In fact the three lines of the three biradials form a tri- rectangular trihedral. If the quaternions of the three o Fig. 15. are i, j, k, then we see easily that the quaternion of the biradial OC/OB is represented completely by the unit vector marked i, the quaternion of OA/OC by j, and of OB/OA by k. The products are very interesting, for we have ij = k, jk = i, hi = j, and if we place the equal biradials in the figure we also have ji = — k, kj = — i, ik = — j. Furthermore, we also can see easily that, utilizing the common notation of powers, V- = - 1, ? - - 1, V - ■■- 1. Since it is evidently possible to resolve the vector part of any quaternion, when it is laid off on the unit vector of its plane as a length, into three components along the direc- tions of i, j, k, and since the sum of the vector parts of VECTORS IN SPACE 10] quaternions has been shown to be the vector part of the sum, it follows that any quaternion can be resolved into the parts q = w -\- xi -\- yj -\- zk. These hypernumbers can easily be made the base of the whole system of quaternions, and it is one of the many methods of deriving them. Hamilton started from these. The account of his invention is contained in a letter to a friend, which should be consulted. (Philosophical Maga- zine, 1844, vol. 104, ser. 3, vol. 25, p. 489.) 6. Product of Vectors. It becomes evident at once if we consider the product of two vector parts of quaternions, or two quaternions whose scalar parts are zero, that we may consider this product, a quaternion, as the product of the vector lines which represent the vector parts of the quaternion factors. From this point of view we ignore the biradials completely, and look upon every geometric vector as the representative of the vector part of a set of quaternions with different scalars, among which one has zero scalar. From the biradial definition we have VqVr= S-VqVr+ V-VqVr equal to the quaternion whose biradial consists of two vectors in the same plane as the vector normals of the Fig. 16. 102 VECTOR CALCULUS biradials of Vq, Vr and perpendicular to them respectively. In the figure the biradial of Vr is OAB, and of Vq is OBC, and of VqVr is OAC. If then we represent the vectors by Greek letters whether meant to be considered as lines or as vector quaternions, a = Vq, /3 = Vr, then the quaternion which is the product of a(3 has for its angle the angle be- tween /3 and a + 180°, and for its normal the direction OB. If we take UVa(3 in the opposite direction to OB, and of unit length, so as to be a positive normal for the biradial a /3 in that order, then we shall have, letting 6 be the angle from a to /3, a(3 = TaTj3(- cos + UVafi sin 0). We can write at once then the fundamental formulae S-a& = - TaTfi cos 6, V-a$ = TaTp-sm 6- UVaP. From this form it is clear also that any quaternion can be expressed as the product of two vectors, the angle of the two being the supplement of that of the quaternion, the product of their lengths being the tensor of the quater- nion, and their plane having the unit vector of the quater- nion as positive normal. If now we consider the two vectors a and to be resolved in the forms a = ai-\- bj + ck, (3 — li + mj + nk, where i, j, k have the significance of three mutually tri- rectangular unit vectors, as above, then since Ta Tfi cos 6 = al-\- bm-\- en, and since the vector Ta T(3 sin 6 • UVa(3 is (bn — cm)i + (cl — an)j + (am — bl)k, we have a/3 = — (al + bm + en) + (bn — cm)i + (cl — ari)j -\- (am — bl)k. VECTORS IN SPACE 103 But if we multiply out the two expressions for a and distributively, the nine terms reduce to precisely these. Hence we have shown that the multiplication of vectors, and therefore of quaternions in general, is distributive when they are expressed in terms of these trirectangular systems. It is easy to see however that this leads at once to the general distributivity of all multiplications of sums. 7. Laws of Quaternions. We see then that the addition and multiplication of quaternions is associative, that addition is commutative, and that multiplication is dis- tributive over addition. Multiplication is usually not commutative. We have yet to define division, but if we now consider a biradial as not being geometric but as being a quaternion quotient of two vectors, we find that P/a differs from a(3 only in having its scalar of opposite sign, and its tensor is T(3/Ta instead of TaTfi. It is to be noticed that while we arrived at the hyper- numbers called quaternions by the use of biradials, they could have been found some other way, and in fact were so first found by Hamilton, whose original papers should be consulted. Further the use of vectors as certain kinds of quaternions is exactly analogous, or may be considered to be an extension of, the method of using complex numbers instead of vectors in a plane. In the plane the vectors are the product of some unit vector chosen for all the plane, by the complex number. In space a vector is the product of a unit vector (which would have to be drawn in the fourth dimension to be a complete extension of the plane) by the hypernumber we call a vector. However, the use of the unit in the plane was seldom required, and likewise in space we need never refer to the unit 1, from which t^e vectors of space are derived. On the other hand, just as in the plane all complex numbers can be found as the ratios 104 VECTOR CALCULUS of vectors in the plane in an infinity of ways, so all quater- nions can be found as the ratios of vectors in space. All vectors are thus as quaternions the ratios of perpendicular vectors in space. And multiplication is always of vectors as quaternions and not as geometric entities. In the common vector systems other than Quaternions, the scalar part of the quaternion product, usually with the opposite sign, and the vector part of the quaternion product, are looked upon as products formed directly from geometric con- siderations. In such case the vector product is usually defined to be a vector in the geometric sense, perpendicular to the two given vectors. Therefore it is a function of the two vectors and is not a number or hypernumber at all. In these systems, the scalar is a common number, and of course the sum of a number and a geometric vector is an impossibility. It seems clear that the only defensible logical ground for these different investigations is that of the hypernumber. It is to be noticed too that Quaternions is peculiarly applicable to space of three dimensions, because of the duality existing between planes and their normals. In a space of four dimensions, for instance, a plane, that is a linear extension dependent upon two parameters, has a similar figure of two dimensions as normal. Hence, corre- sponding to a biradial we should not have a vector. To reach the extension of quaternions it would be necessary to define triradials, and the hypernumbers corresponding to them. Quaternions however can be applied to four dimensional space in a different manner, and leads to a very simple geometric algebra for four-dimensional space. The products of quaternions however are in that case not sufficient to express all the necessary geometrical entities, and recourse must be had to other functions of quaternions. VECTORS IN SPACE 105 In three-dimensional space, however, all the necessary ex- pressions that arise in geometry or physics are easily found. And quaternions has the great advantage over other systems that it is associative, and that division is one of its processes. In fact it is the most complex system of numbers in which we always have from PQ = the conclusion P = 0, or Q = 0.* 8. Formulae. It is clear that if we reverse the order of the product ce/3 we have 0a = Soft - Vafi. This is called the conjugate of the quaternion a(3, and written K-a(3. We see that SKq = Sq= KSq, VKq = - Vq = KVq. Further, since qr = SqSr + SqVr + SrVq + VqVr, we have K-qr= SqSr - SqVr - SrVq + VrVq = KrKq. From this important formula many others flow. We have at once K-qi- • -q n = Kq n > • >Kqi. And for vectors Koli- • -0L n = {—) n a n - • •«!. Since Sq = i(q+Kq), Vq=\{q-Kq), we have therefore S-OLl" 'Qt2n = i(<*l" * -«2n + «2n ' * 'Oil), S-ai- • -C^n-l = i(tti* • 'tt2n~l — «2n-l' * 'Oil), V'CXi- ' 'OL 2n = !(«!« * ,Q; 2n ~ « 2n ' " '«t), F'Qfi- • ■a 2 n-l = %(' 2 + x' 2 +y' 2 + s' 2 ) = (ww f — ao' — 2/2/' — zz') 2 + (wa;' + w'x + 2/2' — S/'s) 2 + (wy' + to'y + zx' — z'x) 2 + (W + w'z + #2/' — ^'2/) 2 - This formula expresses the sum of four squares as the product of the sums of four squares. It was first given by Euler. The problem of expressing the sum of three squares as the product of sums of three or four squares and the sum of eight squares as the product of sums of eight squares has also been considered. 108 VECTOR CALCULUS 9. Rotations. We see from the adjacent figure that we have for the product qrq- 1 a quaternion of tensor and angle the same as that of r. But the plane of the product is produced by rotating the plane of r about the axis of q through an angle double the angle of q. In case r is a vector /3 we have as the product a vector fi f which is to be found by rotating conically the vector (3 about the axis of q through double the angle of q. It is obvious that operators* of the type qQq~ l , r()r -1 , which are called rotators, follow the same laws of multiplica- tion as quaternions, since g(r()r _1 )<7 -1 = qrQ[qr]~ l . A gaussian operator is a rotator multiplied by a numerical multiplier, and is called a mutation. The sum of two mutations is not a mutation. As a simple case of rotator we see that if q reduces to a vector a we have as the result of after 1 = /3' the vector which is the reflection of /3 in a. The reflection of /3 in the plane normal to a is evidently — a$or l . EXAMPLES (1) Successive reflection in two plane mirrors is equivalent * QOq' 1 represents a positive orthogonal substitution. VECTORS IN SPACE 109 to a rotation about their line of intersection of double their angle. (2) Successive reflection in a series of mirrors all per- pendicular to a common plane, 2h in number, making angles in succession (exterior) of ), pV = i(a"» + p*-«). 11. If T P = Ta = Tp = 1 and S-afip = 0, S-U(p-a)U(p -P) = ±iV[2(l -Sap)]. 12. If a, P, 7 and a h Pi, 71 are two sets of trirectangular unit vectors such that if a = Py, a x = Piy, then we may find angles called Eulerian angles such that a 2 = a COS yp + P sin \J/, P 2 = — a sin i£ + P COS ^, 73 = 7 cos 6 + <*2 sin 0, a 3 = — 7 sin -f « 2 cos 0, 71 = 73, «i = «3 cos ^ + /?2 sin + /3 2 cos - 2S 7 (r? - ft : 2Sy(i - a) : S(£ + a)(i, - ft. The six vectors are not independent. Vq is easily found and thence Sq from qa = £q. (11). If (p - a)" 1 + (p - ft" 1 - (P ~ 7)" 1 ~ (P ~ 5)- 1 = 0, then if we let ifi' ~ aT 1 = 1 * (TO - 5)" 1 - 5] - [(a - 6)" 1 - 5]) = (p — 8)(p — a) _1 (« — 5), etc., where p', a', 0', 7' are the vectors from D, the extremity of 5, to the inverses with respect to D, of the extremities of p, a, ft 7, then (p' - a')" 1 + (p' - ft)" 1 - (p' - 7T 1 = 0. Prove that 1 - ft _ y - ft _ P ' - y _ r y - /n i/2 p whence p' and p. (R. Russell.) (12). If (q - a)" 1 + (q - 6)" 1 - (q - c)" 1 - (q - d)~' = 0, we set (q - d)(q' - d)= (a- d){a' - d) = (b - d)(b' - d) = (c - d){c' - d) - 1, VECTORS IN SPACE 125 thence (q - d)-> -(q- a)' 1 = (4 - d)-\a - d)(q - a)' 1 (q - d)~i [(a~d)/(q-a)+(b- d)l(q- b)- (c - d)/(q- e)] - (?' - a')' 1 + (?' - &T 1 ~ W ~ cT 1 and we have q' from (V - cW - C) = (g' - 6')/(g' - «0 = (q' - c')l(a' - c') - [(V - c')Ka' - c')]K (R. Russell.) 13. Characteristic Equation. If we write q = Sq + Vq and square both sides we have q 2 = S 2 q + (Vq) 2 + 2Sq-Vq whence g 2 - 2qSq + S 2 q - V 2 q = 0. This equation is called the characteristic equation of q. The coefficients 2Sq and S 2 q - V 2 q = T 2 q are the invariants of q; they are the same, that is to say, if q is subjected to the rotation r()r -1 . They are also the same if Kq is substituted for q. Hence they will not define q but only any one of a class of quaternions which may be derived from each other by the group of all rotations of the form rQr~ l or by taking the conjugate. The equation has two roots in general, Sq + Tqyl - 1 and Sq - Tq^ - 1. Since these involve the V — 1 it leads us to the algebra of biquaternions which we do not enter here, but a few re- marks will be necessary to place the subject properly. Since the invariants do not determine q we observe that we must also have UVq in order to have the other two parameters involved. 126 VECTOR CALCULUS If we look upon UVq as known then we may write the roots of the characteristic equation in the number field of quaternions as Sq + TVqUVq and Sq — TVqUVq or q and Kq. If we set q -f- r for q and expand, afterwards drop all the terms that arise from the identical equations of q and r separately, we have left the characteristic equation of two quaternions, which will reduce to the first form when they are made to be equal. This equation is qr+rq-2Sq-r- 2Sr-Vq + 2SqSr - 2SVqVr = 0. We might indeed start with this equation and develop the whole algebra from it. We may write it qr-\- rq- 2qSr - 2rSq + 4Sq-Sr + S-qr + S-rq = which involves only the scalars of q, r, qr, and rq. 14. Biquaternions. We should notice that if the param- eters involved in q can be imaginary or complex then division is no longer unique in certain cases. Thus if Q 2 =q 2 we have as possible solutions Q = ± q and also Q = ± V (- l)UVq-q. If q 2 = and Vq = then TVq = and we have Vq = x(i + j V — 1) where X is any scalar and i, j are any two perpendicular unit vectors. CHAPTER VII APPLICATIONS 1. The Scalar of Two Vectors 1. Notations. The scalar of the product of two vectors is defined independently by writers on vector algebra, as a product. In such cases the definition is usually given for the negative of the scalar since this is generally essentially positive. A table of current notations is given. If a and (3 define two fields, we shall call S*cfi the virial of the two fields. S-a(3 = — a X /3 Grassman, Resal, Somoff, Peano, Bura- li-Forti, Marcolongo, Timerding. — Cfft Gibbs, Wilson, Jaumann, Jung, Fischer. — a/3 Heaviside, Silberstein, Foppl, Ferraris, Heun, Bucherer. — (aft) Bucherer, Gans, Lorentz, Abraham, Henrici. — a|/3 Grassman, Jahnke, Fehr, Hyde. Cos a/3 Macfarlane. [a/3] Caspary. For most of these authors, the scalar of two vectors, though called a product, is really a function of the two vectors which satisfies certain formal laws. While it is evident that any one may arbitrarily choose to call any function of one or more vectors their product, it does not seem desirable to do so. For Gibbs, however, the scalar is defined to be a function of the dyad of the two vectors, which dyad is a real product. The dyad or dyadic of Gibbs, as well as the vectors of most writers on vector analysis, are not considered to be numbers or hypernumbers. 127 128 VECTOR CALCULUS They are looked upon as geometric or physical entities, from which by various modes of "combination" or de- termination other geometric entities are found, called products. The essence of the Hamiltonian point of view, however, is the definition by means of geometric entities of a system of hypernumbers subject to one mode of multiplica- tion, which gives hypernumbers as products. Functions of these products are considered when useful, but are called functions. 2. Planes and Spheres. It is evident that the condition for orthogonality will yield several useful equations, and of these we will consider a few. The plane through a point A, whose vector is a, per- pendicular to a line whose direction is 8 has for its equation, since p — a is any vector in the plane, S-d(p-a) = 0. If we set p = 8Sa/d we have the equation satisfied and as this vector is parallel to 5 it is the perpendicular from the origin to the plane. The perpendicular from a point B is b~ l S{a - 0)5. If a sphere has center D and radius T(3 where /? and — (3 are the vectors from the center to the extremities of a diameter, then the equation of the sphere is given by the equation S(p - 3 + fi)(p - d - P) = 0, orp 2 - 2S8 P + 5 2 - /3 2 = 0. The plane through the intersection of the two spheres p 2 - 2£5ip + ci = = p 2 - 2S8 2P + c 2 is 2S(5i — 5 2 )p = ci — c 2 . The form of this equation shows that it represents a plane APPLICATIONS 129 perpendicular to the center line of the spheres. The point where it crosses this line is X18] + x 2 8 2 P = i » Xi + x 2 whence solving, we find p = v(h + 8 2 )-\V8,8 2 + i(cj - <*)>. 3. Virial. If (3 is the representative of a force in direction and magnitude then its projection on the direction a is a~ 1 Sa^ f and perpendicular to this direction crWafi. If a is in the line of action of the force, the projection is fit If a is a direction not in the line of action then the projection gives the component of the force in the direction a. If a is the vector to the point of application of the force then Sa(3 is the virial of the force with respect to a, a term intro- duced by Clausius. It is the work that would be done by the force in moving the point of application through the vector distance a. If a fe an infinitesimal distance say, 8a, then — S8a(3 is the virtual work of a small virtual dis- placement. The total virtual work would be 8V = — 2S8a n (3 n for all the forces. 4. Circulation. In case a particle is in a vector field (of force, or velocity, or otherwise) and it is subjected to successive displacements 8p along an assigned path from A to B, we may form the negative scalar of the vector intensity of the field and the displacement. If the vector intensity varies from point to point the displacements must be infinitesimal. The sum of these products, if there is a finite number, or the definite integral which is the limit of the sum in the infinitesimal case, is of great importance. If a point is moving with a velocity a [cm./sec] in a field of force of /3 dynes, the activity of the field on the point is 130 VECTOR CALCULUS — S-(3S- E(J + D) and the magnetic activity — $H(G + B). EXERCISES 1. An insect has to crawl up the inside of a hemispherical bowl, the coefficient of friction being 1/3, how high can it get? 2. The force of gravity may be expressed in the form a = — mgk. Show that the circulation from A to B is the product of the weight by the vertical difference of level of A and B. 3. If the force of attraction of the earth is 4r. What is the amount of energy enclosed in a sphere of radius 3 cm. and center at a distance from the origin of 10 cm.? 11. In problem 9, if the inductivity is 1760 and the magnetic in- tensity is H = p~% how much energy is enclosed in a box 2 cm. each way, whose center is 10 cm. from the point and one face perpendicular to the line joining the point and the center? 12. If the current in a wire 1 mm. in diameter is 10 amperes and the drop in voltage is 0.001 per cm., what is the activity? APPLICATIONS 133 13. If there is a leakage of 10 heavisides through a magnetic area of 4 cm. 2 , and the magnetic field is 5 gilberts/cm., what is the activity? 14. Through a circular spot in the bottom of a tank which is kept level full of water there is a leakage of 100 cc. per second, the spot having an area of 20 cm. 2 . If the only force acting is gravity what is the activity? 15. If an electric wave front from the sun has in its plane surface an electric intensity of 10 volts per cm., and a magnetic intensity of 0033 gilberts per cm., and if for the free ether or for air y. = 1 and k = £-10~ 20 , what is the energy per cc. at the wave front? (The average energy is half this maximum energy and is according to Langley 4.3 -10 -5 ergs per cc. per sec.) 16. If a charge of e coulombs is at a point A and a magnetic point at B has m maxwells, what is the energy per cc. at P, any point in space, the medium being air? 8. Geometric Loci in Scalar Equations. (1). The equation of the sphere may be written in each of the forms a/p = Kp[a, S(p - a)/(p + a) = 0, S2a/(p + a) = 1, S2p/(p + <*) - 1, T(Sp/a + Vp/a) = 1, Tip - ca) m T(cp - a), S{p - a) (a - »08 - 7)(Y - B)(S - p) - 0, a 2 Sfiyp + j3 2 Syap + y 2 Sa(3p = p 2 Sa(3y (p-aO 2 (p-/3) 2 (p-7) 2 (P-5) 2 (p - a) 2 (a - /3) 2 (a - y) 2 (a - 5) 2 (p-/?) 2 (/? -«) 2 (/5-t) 2 (0-S) 2 (P-T) 2 (Y-«) 2 (Y-0) 2 (7-5) (p - 5) 2 (5 - a) 2 (5 - /3) 2 (5 - 7) 2 Interpret each form. (2). The equation of the ellipsoid may be written in the forms S 2 p/a - V 2 p/(3 = 1, where a is not parallel to ft T(p/y + Kpjb) = T(p/8 + tfp/7), rOup + pX)=x 2 -/* 2 . 134 VECTOR CALCULUS The planes a p cut the ellipsoid in circular sections on Tp = Tfi. These are the cyclic planes. Tfi is the mean semi-axis, Ufi the axis of the cylinder of revolution circumscribing the ellip- soid, a is normal to the plane of the ellipse of contact of the cylinder and the ellipsoid. In the second form let r 1 - -£, 7- 1 = - £> t 2 = n 2 - TJ, then the semi-axes are a=rX+7>, 6= ^~ TfX * > c=T\-T». T(\ - n) (3). The hyperboloid of two sheets is S 2 p/a + F 2 p//3 = 1. (4). The hyperboloid of one sheet is S 2 p/a + V 2 p/(3 = — 1. (5) . The elliptic paraboloid of revolution is SplP+V 2 p/(3 = 0. (6). The elliptic paraboloid is Sp/a + V 2 p/(3 = 0. (7). The hyperbolic paraboloid is Sp/a Sp/fi = Sp/y. (8). The torus is T(± bUarWap - p) = a, 2bTVap = ± (Tp 2 +b 2 - a 2 ), 4b 2 S 2 ap = 4b 2 T 2 p - (T 2 p + b 2 - a 2 ) 2 , Aa 2 T 2 p - 4b 2 S 2 ap = (T 2 p - b 2 + a 2 ) 2 , SU(p - «V (a 2 - b 2 ))l(p + cW (a 2 - b 2 )) = ± b/a, p = ± bJJoTWar + at/Y, r any vector. (9). Any surface is given by p = sl-sv>vl) 2 = o. \ a a a a) 136 VECTOR CALCULUS For further examples consult Joly : Manual of Quater- nions. 2. The Vector of Two Vectors Notations, If a and /3 are two fields, we shall call V-a(3 the torque of the two fields. Va(3 = Va(3 Hamilton, Tait, Joly, Heaviside, Foppl, Ferraris, Carvallo. cqS Grassman, Jahnke, Fehr. aX Gibbs, Wilson, Fischer, Jaumann, Jung. [a, /3] Lorentz, Gans, Bucherer, Abraham, Timer- ding. [a | /?] Caspary . a A j3 Burali-Forti, Marcolongo, Jung. aj8 Heun. Sin a/3 Macfarlane. Iaccb Peano. 1. Lines. The condition that two lines be parallel is that Vafi = 0. Therefore the equation of the line through the origin in the direction a is Vap — 0. The line through parallel to a is Va(p — fi) = or Vap = Va(3 = y. The perpendicular from 5 on the line Vap = 7 is — a~ l Vab + a~ l y. The line of intersection of the planes, S\p = a, S^p = b, is VpV\fx = a/x — 6X. If we have lines Vpa — y and Vp& = 8 then a vector from a point on the first to a point on the second is 5/3" 1 — 7a -1 + #/3 — ya. If now the lines in- tersect then we can choose x and y so that this vector will vanish, corresponding to the two coincident points, and thus S{bp~ l - ya~ l )$a = = S8a + Syp. APPLICATIONS 137 If we resolve the vector joining the two points parallel and perpendicular to Vaft we have* 5/3 -1 — ya~ l + xfi — ya = • (Va^S • VaP(bpr l - yoT 1 + zp - ya) = -(VaP)-\S5(x+ Spy) L a Fa/3 Fa/3 J L Va0 P Vap] - «-* f- SaPS ^~ + a 2 S JL 1 Vap Vap] Hence the vector perpendicular from the first line to the second is - (Vafl-KStct + Spy) and vectors to the intersections of this perpendicular with the first and second lines are respectively and ya x — a 1 \ 8 ' — ^— L Va(3 J * Note that (Va0)- l V(Vu0)(z0 - ya) = xp - ya (y« j S)- 1 F-7a/3(5 J 3- 1 - ya~ l ) = (Vc0)- l (- a'^Sfiya - (r l S<*&) Va ,(-^S^ + p-S^) 10 138 VECTOR CALCULUS The projections of the vectors a, y on any three rectangular axes give the Pluecker coordinates of the line. For applica- tions to linear complexes, etc., see Joly: Manual, p. 40, Guiot: Le Calcul Vectoriel et ses applications. 2. Congruence. The differential equation of a curve or set of curves forming a congruence whose tangents have given directions cr, that is, the vector lines of a vector field a 2 a, a s a, • • • , then the moments are F(aift + a 2 ft + a 3 ft + • • •)« [dyne cm.]. If we set «ift + 02ft + 03ft + • • • = ftai + a 2 + a z + • • •)/ then /3 is the vector to the mean point of application, which, in case the forces are the attractions of the earth upon a set of weighted points, is called the center of gravity. If ai + #2 + a 3 + • • • = 0, we cannot make this substitution. APPLICATIONS 139 4. Couple. A couple consists of two forces of equal magnitude, opposite directions and different lines of action. In such case the mean point becomes illusory and the sum of the moments for any point from which vectors to points on the lines of action of the forces are a h a 2 respectively, is V{a x - a 2 )P. But a\ — a 2 is a vector from one line of action to the other, and this sum of the moments is called the moment of the couple. It is evidently unchanged if the tensor of /? is increased and that of a\ — a 2 decreased in the same ratio, or vice versa. 5. Moment of Momentum. If the velocity of a moving mass m is a cm./sec, then the momentum of the mass is defined to be ma gr. cm./sec. The vector to the mass being p, the moment of momentum of the mass is defined to be Vpma = mVpa [gm. cm. 2 /sec.]. 6. Electric Intensity. If a medium is moving in a mag- netic field of density B gausses, with a velocity a cm./sec, then there will be set up in the medium an electromotive intensity E of value E=Fo-B-10~ 8 [volts/centimeter]. For any path the volts will be - fSd P E= + fSdpBa-10- 8 . If this be integrated around any complete circuit we shall arrive at the difference in electromotive force at the ends of the circuit. 7. Magnetic Intensity. If a magnetic medium is moving in an induction field of D lines, with a velocity a, then there will be produced in the medium at every point a magnetic 140 VECTOR CALCULUS intensity field H = OAwVDa [gilberts/cm.]. For any path the gilbertage will be OAirf SdpaD. 8. Moving Electric Field. If an electric field of induc- tion, of value D lines, is moving with a velocity a, then there will be produced in the medium at the point a mag- netic field of intensity H gilberts/cm. where H m OAirVaD. For a moving electron with charge e, this will be — (eUp/4:irTp 2 ). For a continuous stream of electrons along a path we would have the point being the origin. 9. Moving Magnetic Field. If a magnetic field of in- duction of value B gausses is moving with a velocity cr, it will produce at any given point in space an electric intensity E = V - BolO -8 volts per centimeter. 10. Torque. If a particle of length dp is in a field of intensity — TV> T\p, that is, practically — t along X and + t in all directions perpendicular to X. CHAPTER VIII DIFFERENTIALS AND INTEGRALS 1. DlFFEKENTIATION AS TO A SCALAR PARAMETER 1. Differential of p. If the vector p depends upon the scalar parameter t, say p = ati ~t o an involutory substitution. If ac = b 2 , k becomes co ex- cept when also the numerator = 0. [Joly, Manual, Chap. VII, art. 48.] In general the equation of the tangent of any curve is IT = p + Xp'. We may also find the derivatives of functions of p, when p = (p(t), by substituting the value of p in the expression and differentiating as before. Thus let p = a cos 6 + P sin 6 where Ta # Tp. 148 VECTOR CALCULUS Then Tp = V [- a 2 cos 2 6 - 2Sap sin 6 cos 6 - 2 sin 2 6], We may then find the stationary values of Tp in the manner usual for any function. Thus differentiating after squaring a 2 sin 26 - 2Sa(3 cos 26 - fi 2 sin 26 = 0, tan 26 - 2Sap/(a 2 - /3 2 ). 2. Frenet-Serret Formulae. Since the arc is essentially the natural parameter of a curve we will suppose now that p is expressed in terms of s, and accents will mean only differentiation as to s. Then both p and p + dsp' are points upon the curve. The derivative of the latter gives p' + dsp", which is also a unit vector since the parameter is s. Thus the change in a unit vector along the tangent is dsp", and since this vector is a chord of a unit circle its limiting direction is perpendicular to p', and its quotient by ds has a length whose limit is the rate of change of the angle in the osculating plane of the tangent and a fixed direction in that plane which turns with the plane. That is to say, p" in direction is along the principal normal of the curve on the concave side, and in magnitude is the curmture of the curve, which we shall indicate by the notation Unit tangent is a = p', Unit normal is |9 = Up", curvature is Ci = Tp", Unit binormal is y = Va(3, so that Ciy = Vp'p". The rate of angular turn of the osculating plane per centi- meter of arc is found by differentiating the unit normal of the plane. Thus we have Ti = cf 2 hW - Fp'p"-c 2 ].' DIFFERENTIALS 149 But d 2 = T 2 p" = - Sp"p" and therefore Cl c 2 = - S P "p f ". Substituting for c 2 we have 71 - cr 3 [- Sp"p"Vp'p f " + SpV'Wl = cr z [Vp , Vp"Vp ,,, p"] = crWaVc 1 (3Vp'"c 1 p = crW-aPVp"^ = cr l VyVp'"p = cr l pSyp"' = - «lft where «i is written for the negative tensor of 71 and is the tortuosity. It may also be written in the form Again since /? = ya we have at once the relations j3i = 7i« + 7«i = «i7 ~~ C\a. Thus we have proved Frenet's formulae for any curve «i = erf, ft = ai7 — ci«, 71 = — a$. It is obvious now that we may express derivatives of any order in terms of a, ft y, and Oi, Ci, and the derivatives of ai and Ci. For instance we have Pi = OL, p 2 = fci, Pa = ftci + Pc 2 = fe + (701 — aci)ci, Pi = 0c 3 + 2{yai — aci)c 2 + (ya 2 — ac 2 )ci - ^( ai 2 + Cl 2 ) Cl . The vector w = aai + 7C1 is useful, for if 77 represents in turn each one of the vectors a, /3, 7, then 771 = Fa^ It is the vector along the rectifying line through the point. The centre of absolute curvature k is given by K = p - lip" m p + Pld. 150 VECTOR CALCULUS The centre of spherical curvature is given by a = k + yd/da • c{~ 1 = k — yc 2 /aiCi 2 . The polar line is the line through K in the direction of 7. It is the ultimate intersection of the normal planes. 3. Developable s. If we desire to study certain de- velopables belonging to the curve, a developable being the locus of intersections of a succession of planes, we proceed thus. The equation of a plane being S(w — p)rj = 0, where t is the vector to a variable point of the plane, and p is a point on the curve, while rj is any vector belonging to the curve, then the consecutive plane is S(t - p)f) + ds'd/dsS(w - p)r) = 0. The intersection of this and the preceding plane is the line whose equation is 7r = p + (— r)Sar) + t)lVr}r}i. This line lies wholly upon the developable. If we find a secOnd consecutive plane the intersection of all three is a point upon the cuspidal edge of the developable, which is also the locus of tangents of the cuspidal edge. This vector is tv = p + (VwySar} + 2Vr)7]iSar)i + Vr}7}iS^rjCi)/ST]r}ir]2' By substituting respectively for 77, a, ft 7, we arrive at the polar developable, the rectifying developable, the tangent- line developable. EXAMPLE Perform the substitutions mentioned. 4. Trajectories. If a curve be looked upon as the path of a moving point, that is, as a trajectory, then the param- eter becomes the time. In this case we find that (if p = dp/dt, etc.) the velocity is p = av, the acceleration is DIFFERENTIALS 151 p = ficiv 2 + av. The first term is the acceleration normal to the curve, the centrifugal force, the second term is the tangential acceleration. In case a particle is forced to describe a curve, the pressure upon the curve is given by (3civ 2 . There will be a second acceleration, p = a(v — wi 2 ) + (3(2cii + c 2 v) + yaiCiV. The last term represents a tendency per gram to draw the particle out of the osculating plane, that is, to rotate the plane of the orbit. 5. Expansion for p. If we take a point on the curve as origin, we may express p in the form p = sa + %cis 2 (3 — %s*(ci 2 a — c 2 /3 — cmy) — ^ 4 (3c 2 cia: ~~ £I C3 ~~ c * — Clttl2 l ~~ T[2c 2 ai + da 2 ]) EXERCISES 1. Every curve whose two curvatures are always in a constant ratio is a cylindrical helix. 2. The straight line is the only real curve of zero curvature every- where. 3. If the principal normals of a curve are everywhere parallel to a fixed plane it is a cylindrical helix. 4. The curve for which Ci = 1/ms, ai = 1/ns, is a helix on a circular cone, which cuts the elements of the cone under a constant angle. 5. The principal normal to a curve is normal to the locus of the centers of curvature at points where Ci is a maximum or minimum. 6. Show that if a curve lies upon a sphere, then cr 1 = A cos a + B sin a = C cos (a + e), A, B, C, e are constants. The converse is also true. 7. The binormals of a curve do not generate the tangent surface of a curve. 8. Find the conditions that the unit vectors of the moving trihedral afiy of a given curve remain at fixed angles to the unit vectors of the moving trihedral of another given curve. Two Parameters 6. Surfaces. If the variable vector p depends upon two arbitrary parameters it will terminate upon a surface of 152 VECTOR CALCULUS some kind. For instance if p = i + 2dudvS(piw 2 + Pivv\) + dv 2 Sp 2 w 2 = 0. We may also write the equation in the form dp + xv + ydv = = pidu -\- p 2 dv + xv + yv\du + yv 2 dv. Multiply by (pi + yv\){p 2 + yv 2 ) and take the scalar part of the product, giving S(pi + yvi)(pi + P2#> = o = y 2 Svviv 2 + 2ySv{piv 2 + ^ip 2 ) + ^ 2 . The ultimate intersection of the two normals is given by t = p + dp + yv + y<&>, that is by yv. Hence we solve for yTv, giving two values R and R f which are the principal radii of curvature at the point. The product and the sum of the roots are re- spectively RR' = yy'Tv 2 - Tv%- Sw 1 v 2 ), R + R' = — 2TvSv(piv 2 -\- vip 2 )/Swiv 2 . The reciprocal of the first, and one-half the second divided by the first, that is, — Spvivt/v 4 and Sv(piv 2 + vip 2 )/Tv*, are the absolute curvature and the mean curvature of the surface at the point. The equation of the lines of curvature may be also written vSdpvdv = = V-VdpVvdv = VdpV(dv/vv) = VdpdUv. Hence the direction of dUv is that of a line of curvature, when du and dv are chosen so that dp follows the line of curvature. That is, along a line of curvature the change li 154 VECTOR CALCULUS in the direction of the unit normal is parallel to the line of curvature. When the mean curvature vanishes the surface is a minimal surface, the kind of surface that a soapfilm will take when it extends from one curve to another and the pressures on the two sides are equal. The pressure indeed is the product of the surface tension and twice the mean curvature, so that if the resultant pressure is zero, the mean curvature must vanish. If the radii are equal, as in a sphere, then the resultant pressure will be twice the surface tension divided by the radius, for each surface of the film, giving difference of pressure and air pressure = 4 times surface tension/radius. The difference of pressure is thus for a sphere of 4 cm. radius equal to the surface tension, that is, 27.45 dynes per cm. When a surface is developable the absolute curvature is zero, and conversely. Surfaces are said to have positive or negative curvature according as the absolute curvature is positive or negative. EXERCISES 1. The differential equation of spheres is Vp(p - a) = 0. 2. The differential equations of cylinders and cones are respectively Sva = 0, Sv(p - a) = 0. 3. The differential equation of a surface of revolution is Sapv = 0. 4. Why is the center of spherical curvature of a spherical curve not of necessity the center of the sphere? 5. Show how to find the vector to an umbilicus (the radii of curvature are equal at an umbilicus). 6. The differential equation of surfaces generated by lines that are perpendicular to the fixed line a is SVavU(p - a)l = P — OL J or Sadp[(p + a)- 1 - (p - a)- 1 ] = 0. Now in a meridian section a is constant so that Vdp[(p + a)" 1 - (p - a)" 1 ] - and dp is for such section tangent to a sphere through A and —A. EXERCISES 1. The potential due to a mass m at the distance Tp is m/Tp in DIFFERENTIALS 159 gravitation units. Find the differential of the potential in any direc- tion, and determine in what directions it is zero. 2. The magnetic force at the point P due to an infinite straight wire carrying a current a is H = — 2h/Vap. Find the differential of this and determine in what direction, if any, it is zero. For Vdpa = 0, dH = 0; for dp = dsVa^Vap, dH = - Hds/TVr) = - 2V(q-dq- l -r) = 2V(Vdq/q)r - 2q-V 'V{q- l dq-a)q~ l dUVq= V'Vdq/Vq-UVq, dzq= S[dqKUVq-q)]. We define when 7a = 1 a x = cos • irx/2 + sin • 7nc/2 • a = catf • %tx; thus d-a* = tt/2-o:^ 1 ^. If Ta # 1, then d-a x = dz[log 7W* + tt/2 -a x+1 /Ta\, 3. Extremals. For a stationary value of /(p) in the vicinity of a point p we have ay(p) = 0. If /(p) is to be stationary and at the same time the terminal point of p is to remain on some surface, or in general if p is to be subject *Tait, Quaternions, 3d ed., p. 97. DIFFERENTIALS 161 to certain conditioning equations, we must also have, if there is one equation, q(p) = 0, dq(p) = 0, and if there are two equations, g(p) = and h(p) = 0, then also dg(p) = 0, dh(p) = 0. Whether in all these different cases /(p) attains a maximum of numerical value or a minimum, or otherwise, we will consider later. EXERCISES 1. g( p ) = (p — a) 2 -f- « 2 = 0, find stationary values of Tp = f(j>). Differentiating both expressions, Sdp(p —a) = = Sdpp, for all values of dp. Hence we must have dp parallel to V • tp where t is arbitrary, and hence Srp(p — a) = 0, for all values of r. Therefore we must have Vp(p — a) = 0, or Yap = 0, or p = ya. Substituting and solving for y, y = 1 ± a/Tcc, p = a ± aUa. . 2. g( p ) = ( p — «) 2 + a 2 = 0. Find stationary values of &/3p. Sdp(p - a) = = *S/3ap, whence dpP.WjS, £'T,3(p - a) = 0, 7/3(p - a) = 0. p - a =y0, y = a/T(3, p = a ± at//3. 3. ^( p ) = (p — a )2 -f a 2 = 0, &G>) = *S/3p = 0, find stationary values of Tp. Sdp( P - a) = = Spdp = Spdp, whence S-p0(p — a) « - £pa/S, and since £p0 = 0, p = yV-fiVafi. p = V0VaP(l ± V[a 2 - S*a0)/TVal3). 4. #(p) = p 2 — SapSpp + a 2 = 0. Find stationary values of Tp. £d P p = o = £dp(p - «S/8p - jSflap), p = x(aS$p + /8/Sap) = (a£/3p + pSap)/(Sa(3 ± Ta0), whence Sap = TaSpU/3, = SpU(3(Ua ± U0)/(SUaU0 T 1). Substituting in the first equation, we find SpUp, thence p. 5. Sfip = c, >Sap = c', find stationary values of Tp. Sd P p = Sadp = Spdp = 0, p = xa + y$ and z£a/3 -f 2//3 2 = c, xa 2 + ySafi = c', whence x and y. 162 VECTOR CALCULUS 6. Find stationary values of Sap when (p — a) 2 -f- a 2 = 0. Sctdp = = Sdp(p - a); hence p = ya = a ± aJ7a and Sap = a 2 ± aTa. 7. Find stationary values for Sap when p 2 — SppSyp + a 2 = 0. Sadp = = £d P (p - )857P - ySfip), P =xa+ fiSyp + ySfip, etc. 8. Find stationary values of TV8p when (p - a) 2 + a 2 = 0. 9. Find stationary values of SaUp when (p - a) 2 + a 2 = 0. 10. Find stationary values of SaUpSpUp when Syp + c = 0. 4. Nabla. The rate of variation in a given direction of a function of p is found by taking dp in the given direction. Since df(p) is linear in dp it may always be written in the form where $ is a linear quaternion, vector, or scalar function of dp. In case / is a scalar function, $ takes the form — Sdpv, where v is a function of p, which is usually independent of dp. In case v is independent of the direction of dp, we call / a continuous, generally differentiable, function. Functions may be easily constructed for which v varies with the direction of dp. If when v is independent of dp we take differentials in three directions which are not in the same plane, we have pS - dipd 2 pd 3 p = V'dipd 2 p-Sd 3 pp + V '• d 2 pd 3 p • Sdipp + V - d 3 pdipSd 2 pp = — V 'd 1 pd 2 p'd 3 f '— Vd 2 pd 3 p-dif — V-d 3 pdip-d 2 f. DIFFERENTIALS 163 It is evident that if we divide through by Sdipdipdzp, the different terms will be differential coefficients. The entire expression may be looked upon as a differential operation upon/, which we will designate by V. Thus we have v= V/ = _ ( Vdipdip - d z + V- d 2 pd s p • di + V- d^pdip ■ d 2 ) ,, , S • dipdipdzp We may then write df( P ) = - SdpVfip). If the three differentials are in three mutually rectangular directions, say i, j, k, then V = id/dx + jd/dy + kd/dz. It is easy to find V/ for any scalar function which is gener- ally differentiate from the equation for df(p) above, that is, df(p) = — SdpVf. For instance, VSap = - a, Vp 2 = - 2p, VTp = Up, V(Tp) n = nTp n - l Up = nTp n ~ 2 -p, V TVap = TJVap-a, VSaUp = - p-WUpa, V • SapSpp = - pSd$ - Vap(3, V-log TVap = -^~, vap VT(p - a)-' = - U(p - a)lT\p- a), VSaUpS(3Up = p-WpVap^P, Vlog Tp= U P /Tp= -p~\ 1 V(ZpA*) = - p~ 1 UVpa = pUVap 5. Gradient. If we consider the level surfaces of /(p), /(p) = C, then we have generally for dp on such surface or tangent to it S dp p = = df(p) where p. is the normal of the 164 VECTOR CALCULUS surface. Since Sdp\7f — and since the two expressions hold for all values of dp in a plane M = *V/, or since the tensor of p. is arbitrary, we may say V/(p) is the normal to the level surface of /(p) at p. It is called the gradient of /(p), and by many authors, particularly in books on electricity and magnetism, is written grad. p. The gradient is sometimes defined to be only the tensor of V/, and sometimes is taken as — V/. Care must be exercised to ascertain the usage of each author. Since the rate of change of /(p) in the direction a is — &*V/(p), it follows that the rate is a maximum for the direction that coincides with UVf, hence the gradient V/(p) gives the maximum rate of change off(p) in direction and size. That is, TVf is the maximum rate of change of /(p) and UVfis the direction in which the point P must be moved in order that /(p) shall have its maximum rate of change. 6. Nabla Products. The operator V is sometimes called the Hamiltonian and it may be applied to vectors as well as to scalars, yielding very important expressions. These we shall have occasion to study at length farther on. It will be sufficient here to notice the effect of applying V and its combinations to various expressions. It is to be ob- served that VQ may be found from dq, by writing dq = $-dp, then VQ = i$i + j$j + k$k. For examples we have Vp = {Vdipdzp-dzp + Vd 2 pdsp-dip + Vdzpdip • d 2 p) I '(— Sdipd 2 pd 3 p) = - 3 since the vector part of the expression vanishes. DIFFERENTIALS 165 Vp _1 = — (Vdipd 2 p-p~ 1 d 3 pp~ 1 + •••)/(— Sd 1 pd 2 pd 3 p) Since - - P" 2 . dUp = V^ • Up, dTp = - SUpdp. Hence VUp = 2iV--Up= -~, VTp= Up. p Tp Expressions of the form 2F(i, i, Q) are often written F($ > r> Q)> a notation due to McAulay. Vap = a, VfaSfap + cx 2 S/3 2 p + mSfop) = - 0m + /5 2 a 2 + 1830:3), VFap = 2a, VVap(3 = &xft VSapVfip = - Sapp + 3/3£a

VUVpVap = ^P-, V(Vap)-i=0, V (g)=0. EXERCISE Show that (Fa/3 -y -+- y0y<£>a + Vy = S8( ) + 2ai Fischer. da Projection of directional derivative perpendicular to the direction V-trhi'SV'a, Tait, Joly. —— * Fischer. da Gradient of a scalar V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo, Bucherer. grad, Lorentz, Gans, Abraham, Burali-Forti, Marcolongo, Peano, Jaumann, Jung. . — grad, Jahnke, Fehr. [Fischer's multiplication follows Gibbs, d/dr d p. , being after the operand, the whole being dr read from right to left; e.g., Fischer's Vfl is equiv. to — vSV.] Gradient of a vector V, Tait, Joly, Gibbs, Wilson, Jaumann, Jung, Carvallo. grad, Jaumann, Jung. -=- > Fischer. dr 7. Directional Derivative. One of the most important operators in which V occurs is— SaV, which gives, the DIFFERENTIALS 167 rate of variation of a function in the direction of the unit vector a. The operation is called directional differentiating. SaV'Sfo = - SaP, SaV-p 2 = - 2Sap, SaVTp - SaUp, SaVTp- 1 = - Sap/Tp* = UY^p- 2 , SaVTVap= 0, SaV-Up= - ^~ • An iteration of this operator upon Tp~ l gives the series of rational spherical and solid harmonics as follows : - SaVTp- 1 = - Sap/Tp* = UYiTff*, Sl3VSaVTp- 1 = (3SapS(3p+ Tp 2 Sa(3)Tp- 5 = 2\Y 2 Tp~\ SyVSWSaVTp- 1 = - (3.5SapS(3pSyp + 32S(3ySapTp 2 )Tp- 7 = 3\Y 3 Tp~\ For an n axial harmonic we apply n operators, giving Y n = S.(- l) 8 (2n - 2s)!/[2 n -*nl(n - s)l\ES n - 28 aUpS s a 1 a 2 , ^ s^ n/2. The summation runs over n — 2s factors of the type SaiUpSoi2Up • - • and s factors of the type SajCtjSotnar - - , each subscript occurring but once in a given term. The expressions Y are the surface harmonics, and the expressions arising from the differentiation are the solid harmonics of negative order. When multiplied by Tp 2n+1 we have corresponding solid harmonics of positive order. The use of harmonics will be considered later. 8. Circuital Derivative. Another important operator is Va\7 called the circuital derivative. It gives the areal density of the circulation, that is to say, if we integrate the function combined with dp in any linear way, around an infinitesimal loop, the limit of the ratio of this to the area of the loop is the circuital derivative, a being the normal to the area. We give a few of its formulae. We may also 168 VECTOR CALCULUS find it from the differential, for if dQ = $dp, Fa V • Q VaV • Tp - VaUp, FaV • Tp n = nTp n ~ 2 Vap t VaV - Up = (Sap 2 - pSap)/Tp\ VaV-SQp = F/3a, Fa V • V(3p = a(3+ S-aP, FaV -ft> = 2Sa(3, FaV • 7Tft> - - V-apUVpp, FaV -p - - 2a, Fa V • (aiSftp + a 2 »S/3 2 p + a 3 S/3 3 p) - Sa(« A + "A + a 3 fo) + FaiFa/3i + Fa 2 Fa/3 2 + Fa 3 Fa/3 3 . 9. Solutions of VQ = 0, V 2 Q =0. In a preceding formula we saw that V(Vap)~ l = 0. We can easily find a number of such vectors, for if we apply Sa V to any vector of this kind we shall arrive at a new vector of the same kind. The two operators V and Sa V • are commutative in their operation. For instance we have d(Vap)~ l = - (VapyWadp-iVap)- 1 ; hence T = ^V-(Fap)- 1 = {Vap)- l V$a>{Vap)- 1 is a new vector which gives Vr = 0. The series can easily be extended indefinitely. Another series is the one de- rived from Up/T 2 p. This vector is equal to p/T 3 p, and its differential is (-p 2 dp+SSdpp.p)/T% The new vector for which the gradient vanishes is then (-ap 2 +3Sap-p)/7V The latter case however is easily seen to arise from the vector V Tp~ l , and hence is the first step in the process of using V twice, and it is evident that S7 2 Tp~ l = 0. So also the first case above is the first step in applying V 2 to log TVap-a~ l so that V 2 (log TVap-a) = 0. Functions of p that satisfy this partial differential equation are called DIFFERENTIALS 169 harmonic functions. That is,/(p) is harmonic if V 2 /(p) = 0. Indeed if we start with any harmonic scalar function of p and apply V we shall have a vector whose gradient van- ishes, and it will be the beginning of a series of such vectors produced by applying &*iV, Sa 2 V, • • • to it. However we may also apply the same operators to the original harmonic function deriving a series of harmonics. From these can be produced a series of vectors of the type in question. V 2 • F(p) is called the concentration of F(p) . The concentra- tion vanishes for a harmonic function. EXERCISES Show that the following are harmonic functions of p: 1. Tp- 1 tan" 1 Sap/Spp, where a and /? are perpendicular unit vectors, 2. Tf* log tan ^ Z £ 3. where and tan -1 Sap/S/3p Sa(3 = a 2 = £2 = _ 1# 4. logtan^ z - • £j CL 10. Harmonics. We may note that if u, v are two scalar functions of p, then V -uv = u Vfl + v\7u and thus V 2 -uv = u\7h + vV 2 u + 2SVuVv. Hence the product of two harmonics is not necessarily harmonic, unless the gradient of each is perpendicular to the gradient of the other. Also if u is harmonic, then \7 2 -uv = u\7 2 v + 2SVu\7v. 12 170 VECTOR CALCULUS If u is harmonic and of degree n homogeneously in p, then w/7p 2n+I is a harmonic of degree — (n + 1). For V 2 (fp 2n+1)-1 . V[ _ ( 2n+ l) p r p -2n-3] = - (2n+ l)(2n)Tp~ 2n - 3 and SVuVTp- 2 "- 1 = - (2n+ l)Tp- 2n -*SVup = (2n+ l)(2n)uTp- 2n -*; hence V 2 -u/Tp 2n+1 - 0. In this case w is a solid harmonic of degree n and uTp~ 2n ~ l is a solid harmonic of degree — n — 1. Also uTp" 11 is a corresponding surface harmonic. The converse is true. EXAMPLES OF HARMONICS Degree n = 0;

pp where Sc& = 0, a 2 = /3 2 = - 1; ^ = log cot ^/ -a 2 = - 1; a S-a(3UpSapS(3p/V 2 -a(3p; Sa(3UpS(a + 0)pS(a - /3)p/F 2 a/3p. The gradients of these as well as the result of any opera- tion Sy V are solid harmonics of degree — 1, hence multiply- ing the results by Tp[n = 1, 2n — 1 = 1] gives harmonics again of degree 0. These will be, of course, rational harmonics but not integral. Taking the gradient again or operating by $71 V any number of times will give harmonics of higher negative degree. Multiplying any one of degree — n by Tp 2n ~ 1 will give a solid harmonic of degree n — 1. Degree n = — 1. Any harmonic of degree divided by Tp, for example, 1/Tp, ip/Tp, f/Tp, Saf3UpSaUpS(3p/V 2 a(3p, • • • , DIFFERENTIALS 171 Degree n = — 2. SaUp/p 2 , &*ft>, ^Softa + 7p • • • . Other degrees may easily be found. 11. Rational Integral Harmonics. The most interesting harmonics from the point of view of application are the rational integral harmonics. For a given degree n there are 2n + 1 independent rational integral harmonics. If these are divided by Tp n we have the spherical harmonics of order n. When these are set equal to a constant the level surfaces will be cones and the intersections of these with a unit sphere give the lines of level of the spherical harmonics of the given order. A list of these follow for certain orders. Drawings are found in Maxwell's Electricity and Magnetism. Rational integral harmonics, Degree 1. Sap, S(3p, Syp, a, ft, y a trirectangular unit system. Degree 2. SapS(3p, SfoSyp, SypSap, 3S 2 ap + p 2 , S 2 ap - s 2 p P . These correspond to the operators 7p 5 [£ 2 7V, SyVSaV, SyVSPV, S(a + 0) VS(a - 0) V, SaVSQV] on Tp'K Degree 3. Representing Sap by — x, Sfip by — y, Syp by — z, SaV by — D x , S(3V by — D y , SyV by — D z we have 2z 3 — 3x 2 z — 3y 2 z, 4:Z 2 x — x* — y 2 x, A.z 2 y — x 2 y — y 3 , x 2 z — y 2 z, xyz, x z — 3xy 2 , 3x 2 y — y 3 corresponding to 7)3 7)3 7)3 7) 3 _ 7) 3 7) 3 _ Q7) 3 ^ zzz ) -lszzx , Lf zzy , ^xxz > J^xyz , ^xxx > OU X yy , 7) 3 _ Q7) 3 ■LSyyy j OJ^xxy • 172 VECTOR CALCULUS Degree 4. 3z 4 + 3y 4 + 8z 4 + 6*y - 24z 2 z 2 - 24yV, *z(4z 2 - Sx 2 - 3y 2 ), yz(4z 2 - 3^ - 3i/ 2 ) (^ _ y 2 )(6z 2 — x 2 — y 2 ), xy(6z 2 — x 2 — y 2 ), xz(x> - Sy 2 ), yzQx 2 - y 2 ), x* + y* - My 2 , xyix 2 - y 2 ) 7) 4 7) 4 7) 4 is zzzz ) *-* zzzx y -L/zzzy D 4 - ■LS ZZXX . T) 4 7) 4 7) 4 _ OT) 4 *s zzyy j M* zzxy i J^xxxz *->±s X yyz j 7) 4 1J yyyz _ Q7) 4 7) 417) 4_ ft/) 4 oiyxxyz ) Uxxxx T ^ yyyy ^^xxyy y D 4 — 7) 4 J^xxxy ^xyyy • The curves of the intersections of these cones with the unit sphere are inside of zero-lines as follows : Degree 1. Equator, standard meridian, longitude 90°. Degree 2. Latitudes ± sin -1 JV 3, equator and standard meridian, equator and longitude 90°, longitude ± 45°. Standard meridian and 90°. Degree 3. Latitudes 0°, db sin -1 V 0.6, latitudes ± sin -1 V 0.2 and standard meridian, latitudes ± sin -1 V 0.2 and 90° longitude, equator, longitude ± 45°, equator, longitudes 0°, 90°. Longitudes ± 30°, 90°, longitudes ± 60°, 0°. 12. Variable System of Trirectangular Unit Vectors. We will consider next a field which contains at every point a system of three lines which are mutually perpendicular. That is, the lines in one direction are given by a, say, at the same point another set by ft and the third set by y. Each is a given function of p, subject to the conditions a/3 = 7, /?7 = a, ya = /?, a 2 = (3 2 = y 2 = — 1. For example, in the ordinary congruence, a being the unit tangent at any point of one line of the congruence, then the normal and the binormal are determined and would be ]S and 7. However /? and 7 may be other perpendicular DIFFERENTIALS 173 lines in the plane normal to a. If we follow the vector line for /3 after we leave the point we shall get a determinate curve, provided we consider a to be its normal. We may however draw any surface through the point which has a for its normal and then on the surface draw any curve through the point. All such curves can serve as ft curves but a might not be their principal normal. It can happen therefore that the j8 curves and the y curves may start out from the point on different surfaces. However a, (3, and y are definite functions of the position of the point P, with the condition that they are unit vectors and mutually perpendicular. If we go to a new position infinitesimally close, a becomes a + da, ft becomes fi + dp, and y becomes y + dy. The new vectors are unit vectors and mutually perpendicular, hence we have at once S-ada = S-pdp = S>ydy = 0, Sadp = - S(3da, n . Spdy = - Sydp, Syda = - Sady. {L) These equations are used frequently in making reductions. We have likewise since a 2 = — 1, Va-a = - VW, V/3-/3 = - VW, (2) vy-t = - v'rr'j where the accent on the V indicates that it operates only on the accented symbols following. Similarly we have Va-j8 + V(3-a= - V'a0' - V'j&x', etc. (3) We notice also that S-a(SQV)a = 0, S-a(SQV)0 = - S-p(S()V)a, etc. (4) We now operate on the equation y = afi with V, and 174 VECTOR CALCULUS remember that for any two vectors X/x we have X/x = — juX + 2 Fischer. dr Dyad of the gradient. Gradient of the divergence — VSV, Tait, Joly. VV-, Gibbs, Wilson. V; V, Jaumann, Jung. grad div, Buroli-Forti, Marcolongo. DIFFERENTIALS 181 Planar dyad of the gradient. Vortex of the vortex VVVV(), Tait, Joly. V*V, Jaumann, Jung. rot 2 , Lorentz, Bucheoer, Gons, Abraham. curl 2 , Heaviside, Foppl, Ferraris. rot rot, Burali-Forti, Marcolongo. 13. Vector Potential, Solenoidal Field. If £ = VVv, then we say that a is a vector potential of £. Obviously £v£ = SV 2 id ; But this is the circulation about a small rectangle in the 13 ISC. VECTOR CALCULUS plane normal to /?. Hence the component of VVcr in the direction is — (3 Lim J'Sdpff/aresi of loop in plane normal to /?. Likewise the other term reduces to a similar form and the component of V\7S7) = for every v. DIFFERENTIALS 191 This may be written in the form Q'VV'l ) .- identically. EXERCISES 1. Vadp is exact only when a = a a constant vector. For VaV\7 v = for every v, that is S\(vSS7p- — VSav) = for every X, v, and for X perpendicular to v therefore SXS/Sav = 0, or Sdav = for every v perpendicular to the dp that produces da. Again if X = v, SV* + SvVSav = 0, for every v. Therefore S\/a = and Sv\7 Sav = 0, or Sdav = for every dp in the direction of v. Hence da = for every dp and a = a a constant. 2. Examine the expressions S^, V(Vap)dp, F.&. Integrating Factor If an expression becomes ezactf &?/ multiplication by a scalar function of p, let the multiplier be m. Then mQ(W) = 0, where V operates on m and Q, or QWm() + mQVV() = 0, where V operates on m only in the first term and on Q only in the second. This gives for Sadp SaVmi ) + mS( )Vo- = 0, or VaVm + mVV()) = 0,

i)g(dipt, d 2 pi, d 3 pi). For instance, if we consider /(p) = a, we have for ffadp along the straight line p = fi + #7, dp = cfo-7 and Lim "Zadx-y from # = # to x = Xi is 0:7(21 — Xo), hence ^P = «(Pi - Po). The same function along the ellipse p = /3 cos + 7 sin 0, where dp = (— /? sin 6 + 7 cos 0)d0 has the limit (a/3 cos 6 -\- ay sin 0) between = O , 6 = 0i, that is, again a(pi — p ). EXAMPLES (1). j£« £dp/p = log TWpo, for any path. ( 2 )- Su ~ q~ l dqq~ l = qr 1 — g _1 , for any path. (3). The magnetic force at the origin due to an infinite straight current of direction a and intensity / amperes is H = 0.2-I-Va/p, where p is the vector perpendicular from the origin to the line. In case then we have a ribbon whose right cross-section by a plane through the origin is any curve, we have the magnetic force due to the ribbon, expressible as a definite integral, H = 0.2IfVaTdp/p. For instance, for a segment of a straight line p = a(3 -\- xy, /3, 7 unit vectors Tdp = dx, H = 0.27 / '(ay - xt3)dx/(a 2 + x 2 ) = - 0.2/0 -log (a 2 + * 2 2 )/(a 2 + *i 2 ) -f- 0.2 -I-yitsoT 1 x 2 /a — tan -1 xj/a), = 0.27/3 -log OA/OB + O.27J. L AOB. 198 VECTOR CALCULUS (4). Apply the preceding to the case of a skin current in a rectangular conductor of long enough length to be prac- tically infinite, for inside points, and for outside points. (5). Let the cross-section in (4) be a circle p — b3 — a(3 cos 6 — ay sin 0. Study the particular case when b = and the origin is the center. (6). The area of a plane curve when the origin is in the plane is \TfVpdp. If the curve is not closed this is the area of the sector made by drawing vectors to the ends of the curve. If we calculate the same integral \fVpdp for a curve not in the plane, or for an origin not in the plane of a curve we will call the result the areal axis of the path, or circuit. This term is due to Koenigs (Jour, de Math., (4) 5 (1889), 323). The projection of this vector on the normal to any plane, gives the projection of the circuit on the plane. (7). If a cone is immersed in a uniform pressure field (hydrostatic) then the resultant pressure upon its surface is "~ 2^Vpdp-P, where p is taken around the directrix curve. (8). According to the Newtonian law show that the at- traction of a straight segment from A to B on a unit point at is in the direction of the bisector of the angle AOB, and its intensity is 2/x sin ^AOB/c, where c is the perpen- dicular from to the line. (9). From the preceding results find the attraction of an infinite straight wire, thence of an infinite ribbon, and an infinite prism. (10). Find the attraction of a cylinder, thence of a solid cylinder. 19. Integration by Parts. We may integrate by parts INTEGRALS 199 just as in ordinary problems of calculus. For example, f y s V-adpSp P = iVa(B8P8 - ySfa) + \VaVPf*V pdp, which is found by integrating by parts and then adding to both sides J* y V -adpSpp. The integral is thus reduced to an areal integral. In case y and 5 are equal, we have an integral around a loop, indicated by J?. EXAMPLES (1). SfdpVcxp = HdVaS - yVay) - \Vaf*Vpdp + iSafjVpdp. (2). f y *V.VadpV(3p = ilaSPSy'Vpdp + pS-afy'Vpdp - 5 Sap 5 + y Softy]. (3). f y *S'VadpV(3p = i(Sa8S(35 - Say Spy - 8 2 SaP -y 2 Sap- S-a(3f y s Vpdp). (4). JfV-adpVPp = U*Spf y s Vpdp + pSafjVpdp - dSa(38 + y Softy + Sa5S(38 - Say Spy - 8 2 SaP + y 2 SaP - SaPffVpdp). (5). f y s SapSpdp = USadSpd - SaySPy -S-Voftf'Vpdpl (6). ffdpSap = itfSad - ySay + V-affVpdp]. (7). f y s Va P Spdp = HVadSpb - VaySPy - SoftffVpdp + PSaJfVpdp]. (8). f y s Vap-dp = i[Va6'B - Vayy + af y 8 Vpdp + SaffVpdp]. (9). fjapdp = h[a(8 2 - y 2 ) + 2af*Vpdp]. As an example of this formula take the scalar, and notice that the magnetic induction around a wire carrying a 200 VECTOR CALCULUS current of value Ta amperes, for a circular path a B - - 2p.Vap/a 2 . Therefore - fO^Sapdp/a 2 = - SfdpB = - OSfia^SafVpdp = ATafia~ 2 wr 2 . For fj, = 1, r = a, this is OAwC. This gives the induction in gausses per turn. (10). SfSdpw - i[S8cp8 - SySV $& \fSad18pd2dp = SVvodv. 206 VECTOR CALCULUS If now we can dissect any volume into elements in which the function has no singularities and sum the entire figure, then pass to the limit as usual, we have the important theorem ffSV = — pSvV = — pVVv + 2a and SSfvdv = ifffpVVvdv - \ffpVUvodA. Let $Uv = pSpUva, then V = pSpVv + Fpo-, whence fffVpa dv = - fffp&Va dv + ffpSpUvadA. Let $17V = - pVpVUixr, then $V = - pFpFVo" + 3PV, hence SSSVpadv = ifffpVpWadv- \ffpVpVUvadA. Let $E7V = SprUiHT, then 3>V = SprV j> 'SdAU r i>\? 'w/T 'p over the surface. A simple case is — SUv\/w = const. = C. Then 4ttWo = CffdAITp. The integration of this and of the forms arising from a different assump- tion as to the normal component of V^ can be effected by the use of fundamental functions proper to the problem and determined by the boundary conditions, such as Fourier's series, spherical harmonics, and the like. One very simple case is that of the sphere. If we take 212 VECTOR CALCULUS the origin at the center of the sphere we have to find the integral ,f,fdA/T( P - P o) where po is the vector to the point. Now the solid angle subtended by po is given by the integral — r~ l ffdASpU{p — po)/T*(p — p ) and equals 4t or 0, according as the point is inside or outside of the sphere. This integral, however, breaks up easily into two over the surface, the integrands being r-^T-Kp - po) - SpoU( P - P0 )/T*(p - po), but the last term gives or — 47rr 2 /7 T p , as the point is inside or outside of the sphere. Hence the other term gives ffdAlT{p - po) - 47rr or 4Trr 2 /Tp as the point is inside or outside. We find then in this case that w m Cr 2 /Tpo. If in place of the law above for — SUvS7w, it is equal to C/T 2 (p — p ) we find that ffdAIT\ P - po) = 47rr/(r' + p 2 ) or 47^/(7^0 - r^po). Inside _ r r ,A S(p ~ pp)(p + po) - ffdA TKp - po) ' dA = 27rr 2 sin Odd =- d[a 2 + r 2 - x 2 ] = —xdx, a a „ po(p — po) = ax cos 4/ _ a 2 + x 2 — r 2 T*(p - po) " x 3 2x* ffdAS^f^=^f r+a ' a+r ( a ^ + l)dx = T 2 (p — po) aJr-a,a-r \ X 2 J or 47TT 2 a The differentiation of these integrals by using Vp as operator under the sign leads to some vector integrals over the surface of the sphere. 2. Show that we have ££UvdAIT(p - po) = |ttpo or |7rr 3 /^ 3 Po-po for inside or outside points of a sphere. INTEGRALS 213 3. Find ffdAUu/T 3 ( P - Po ) for the sphere. 4. Prove f fdAT^{p-fi)T-\p-oc) =4 7 rr/[(r 2 -a 2 )7 7 ( / S-«)] or = ^r 2 J[a(r 2 -a 2 )T(r 2 a- 1 +0)]. 5. Consider the case in which the value of w is zero on a surface not at infinity but surrounding the first given surface. We have an example in two concentric spheres which form a condenser. On the inner sphere let w be const. = Wi, on the outer let w = 0, on the inner let — SUpVw = 0, inside, = E h outside, on the outer let — SUv\/w = E 2 on the inside, = Oon the outside. 6. If w is considered with regard to one of its level surfaces, it is constant on the surface, and the integral — £ f SdAU vU p\T 2 pio becomes for any inside point 4:irw, hence we have 4irw - A.™ = fffdv\7 2 wlT P - £ £SdAUuVw/T P . If then w is harmonic inside the level surface, it is constant at all points and 47r(w - to) m - £fSdAUv\7wlTp. But since w is constant as we approach the surface, V^o =0, and V(w — Wo) = 0, so that X7w = 0. Hence w = w. If w vanishes at oo and is everywhere harmonic it equals zero. 7. If two functions Wi, w 2 are harmonic without a given surface, vanish at » , and have on the surface values which are constantly in the ratio X : 1, X a constant, then W\ = \W2. 8. If the surface Si is a level for both the functions u and w, as also the surface S 2 inside Si, and if between Si and $2, u and w are harmonic, then (U — Ui)(w 2 — Wi) = (W — Wi)(ll2 — Ui). For if w = f(p)> we can proceed by the method of integral equations to arrive at the integral. However the integral is express- ible in the form of a definite integral, as well as a series, w = l/4:w[fSSdvV 2 w/Tp - ffSUviVw/Tp + wUp/T 2 p)dAl The first of these integrals is called the potential and written Pot. Thus for any function of p whatever we have Vot q, = fffqdvlT(p- p Q ) where p describes the volume and p is the point for which Pot qo is desired. Let Vo be used to indicate operation as to po, then we have Vo Pot g = VoffSqdv/T(p - p ) = fff[dvU(p - p )/r 2 (p - Po )]q - -SSfV[qlT(p- p )]dv + SffdWq/T(p- po) = Pot Vg - ffdAUvqlT(p - Po ). If we operate by Vo again, we have Vo 2 Pot q = Pot V 2 ? - ffdA[Uv\7qlT(p - po) + V'Uvq/T'(p - po)]. But the expression on the right is 4x^0, hence we have the INTEGRALS 215 important theorem Vo 2 Pot q = 4:irq . That is, the concentration of a potential is 4x times the function of which we have the potential. In the case of a material distribution of attracting matter, this is Poisson's equation, stating that the concentration of the potential of the density is 4r times the density; that is, given a distribution of attracting masses, they have a potential at any given point, and the concentration of this potential at that point is the density at the point -5- 4-7T. The gradient of Pot q was called by Gibbs the Newtonian of g , when the function q is a scalar, and if q is a vector, then the curl of its potential is called the Laplacian, and the convergence of its potential is called the Maxwellian of q . Thus New q = Vo Pot P, Lap - po)]. 3. We may, therefore, break up (in an infinity of ways) any vector into two parts, one solenoidal and the other lamellar. Thus, let a = 7T + t where £v r ■ 0, and Wir = 0, then Sv + S, an equation to determine g, which we shall write g z - mig 2 + m 2 g - m 3 = 0, called the /a<6n< equation of #>, where we have set Wl = (S\jA)/S\fJLl>, rri2 = (S\(pii(pp -+- S(

S\(pfxp]. Adding to this result S\pu> -m%ipp, we have S\pv((p 3 p — mnp 2 p + m, we have the Hamilton cubic for

+ bfx + cX. Since (m = 9iV> We notice that if c # 0, we can choose v' = v — (cjg 2 )\, whence ipv' = giv' and we could therefore take c = 0. Hence g 3 - g\2gi + g 2 ) + ^(2fir^ 2 + g Y 2 ) - g?g 2 = 0,

, 221 VECTOR CALCULUS and the general equation is (

(*> — c/x) = g{y — c/jl) = gv' , and the general equation i* - g) 3 = 0,

= gi(x\ + 2/M + •*) + a(a*M + 0*0 + (6 - a)yy, ) + fo^,

and all vectors /x in the direction v. In the first case we see that there is at least one vector p such that {

(

X = g{K + \x, /3 = a, satisfies the cubic and no reduced equation, there are three vectors (of which fi and 7 are not unique) such that 7 = PY, so that &Vp = gSap + Sft>, Wp = gS(3p + S 7 p, #7 so that Sa(3y-x% Whence we have also X + lA = *>'x' + ^', m 3 = ^ = ^V- EXERCISES 1. If

4 ) = ra 3 4 . 4. Show that for the function + c ) = w s(«p) + cw 2 () = aHiv)', tifilPi) = ^() + m 3 (0) + mi[*V(4) + 0V(*>)]. 11. x can have the three forms : t (ff* + 9t)*Sa + (g 3 + 0i)0S0 + (oi + g 2 )ySj; II. fo + o 2 )a£5 + fo + fln)/aSg + 2^x7^7 + apSa; III. 2g - (a/3 + c 7 )>S£ - bySfT The operator x is the rotor dyadic of Jaumann. 12. The forms of \f/ for the three types are I. g 2 g g aSa + g 3 gi&Sl3 + gig 2 ySy; II. gig 2 aSZ + g 2 gi0S8 + 0i 2 7#t" - agtfSZ; III. o 2 - [O03 + (ab - gc)y]Sa - bgySfi. 238 VECTOR CALCULUS 13. An operator called the deviator is defined by Schouten,* and is for the three forms as follows: I. (l9i - 9* - gs)aSZ + (Itfi - 0* - gi)0S8 + (lg* - g x - g%)ySy'; II. (- fci - <7*)(«S5 + fiSfi) + (§0» - 2^)7^ + apSZ; III. (o£ + Cy)Sa + bySd. It is V

z(

we find that S(

— 2$ (p\(m 3 — ra 3 + Wi(^» — mi(^')). But it is easy to see that this expression vanishes identically, for the first two terms cancel, and if

\' = F-a(), ,p 2 » = ^2n+l = a 2n7 a () > 13. For any two operators is a similitude when for every unit vector a, T^a = c, a constant. Show that the necessary and sufficient condition is = to, the gradient is 2e(07V0). 9. For any

(). 16. Change of Variable. Let F be a function of p, and p a function of three parameters u, v, w. Let A = ad/du + f3d/dv + yd/dw, where a, /3, y form a right-handed system of unit vectors. Then we have the following formulae to pass from expres- sions in terms of p to differential expressions in terms of the parameters. AF = - AiS Pl VF t FA' A" = |FAi'A 2 "£Fpip 2 Fv'V", SA'A"A'" - - i(a), '(a), ( ), Hamilton, Tait, Joly, Shaw. 'a f a-4>, , Gibbs, Wilson, Jaumann, Jung. Reciprocal dyadic 4>~ l , Hamilton, Tait, Joly, Gibbs, Wilson, Burali-Forti, Marcolongo, Shaw. q~ l , Timerding. I6I" 1 , filie. THE LINEAR VECTOR FUNCTION 249 The adjunct dyadic \j/ = m(f)'~ l , Hamilton, Tait, Joly, Shaw. WO2, Gibbs, Wilson, Macfarlane. R{a), Burali-Forti, Marcolongo. x((f>, (f>), Shaw. D4>~ 1 , Jaumann, Jung. The transverse or conjugate dyadic ', Hamilton, Tait, Joly. 0, Taber, Shaw. c , Gibbs, Wilson, Jaumann, Jung, Macfarlane. K(ct), Burali-Forti, Marcolongo. \b / , Elie. The planar dyadic X = Wi — (f> r , Hamilton, Tait, Joly. 4>J — c , Gibbs, Wilson. — /, Jaumann, Jung. CK(a), Burali-Forti, Marcolongo. x(0), Shaw. Self-transverse or symmetric part of dyadic o t Hamilton, Tait, Shaw. $, Joly. f , Gibbs, Wilson. [], Jaumann, Jung. D(a), Burali-Forti, Marcolongo. \ b /, Elie. \ b° / , Elie. In this case expressed in terms of the axes. Skew part of dyadic \{4> — ') = V-e( ), Hamilton, Tait, Joly, Shaw. ", Gibbs, Wilson. II, Jaumann, Jung. Va A , Burali-Forti, Marcolongo. 17 i 250 VECTOR CALCULUS \ b / , £lie. Sin , Macfarlane. Mixed functions of dyadic X«>, 0), Shaw. \l 0, Gibbs, Wilson. R{(f>, 0), Burali-Forti, Marcolongo. Vector of dyadic e, Hamilton, Tait, Joly. <£ x , Gibbs, Wilson. (f> r 8 , — <}>/, Jaumann, Jung. Va, Burali-Forti, Marcolongo. E, Carvallo. R = Te, filie. c(<£), Shaw. Negative vector of adjunct dyadic e, Hamilton, Tait, Joly. 0-0 x , Gibbs, Wilson. - r 8 , Jaumann, Jung. olVol, Burali-Forti, Marcolongo. «x(> )> Shaw. Square of pure strain factor of dyadic 4>', Hamilton, Tait, Joly. c , Gibbs, Wilson. {(f)} 2 , Jaumann, Jung. aKa, Burali-Forti, Marcolongo. [6], filie. ', Shaw. Dyadic function of negative vector of adjunct 2 e, Hamilton, Tait, Joly, Shaw. 2 -4> x , Wilson, Gibbs. THE LINEAR VECTOR FUNCTION 251 2 -0/, Jaumann, Jung. a 2 Va, Burali-Forti, Marcolongo. K 2 , Elie. Scalar invariants of dyadic. Coefficients of characteristic equation m" ', ra', m, Hamilton, Tait, Joly, Carvallo. 1%, h, h, Burali-Forti, Marcolongo, Elie. F, G, H, Timerding. S , (2) s , 03, Gibbs, Wilson, mi, ra 2 , ra 3 , Shaw. fc, ] 4> 8 *, >■ • • • 03, Jaumann, Jung. - w, J cos ••• 03, Macfarlane. (Mer scalar invariants ™>i(o 2 ), mi(00'), 2(rai 2 — m 2 ), rai(00')> wi[x(0, *)> 0L Shaw. [0 8 ] 2 «, {0j s 2 > [01/, •'* -j Jaumann, Jung. • • •, • • •, • • ., : 0, 0* : ft Gibbs, Wilson. Elie uses ifi for $e0e. Notations for Derivatives of Dyadic In these V operates on unless the subscript n indicates otherwise. Gradient of dyadic V0, Tait, Joly, Shaw. Dyadic of gradient. Specific force of field 0V, Tait, Joly, Shaw, grad a, Burali-Forti, Marcolongo. -3 — , Fischer. dr 252 VECTOR CALCULUS Transverse dyadic of gradient 0'V, Tait, Joly. grad Ka, Burali-Forti, Marcolongo. —r^-y Fischer. V -, Jaumann, Jung. Divergence of dyadic - SV( ), Tait, Joly, Shaw. X grad Ka, Burali-Forti, Marcolongo. Vortex of dyadic VV4>( ), Tait, Joly, Shaw. Rot a, Burali-Forti. V X 0, Jaumann, Jung. Directional derivatives of dyadic - S( ) V • 0. Sa' 1 V ■ a. ScT 1 V -Va(), Tait, Joly, Shaw. S(a, ( )), Burali-Forti. P , IX*, F i sch e r . da da Burali-Forti, Marcolongo. (»<>)<»• Gradient of bilinear function ju„(Vn, «), Tait, Joly, Shaw. <£(/z)a, Burali-Forti. Bilinear gradient function ju(Vn, u n ), Tait, Joly, Shaw. \//(n, u), Burali-Forti. Planar derivative of dyadic n VVn( ), Tait, Joly, Shaw. X-^> Fischer, CHAPTER X DEFORMABLE BODIES Strain 1. When a body has its points displaced so that if the vector to a point P is p, we must express the vector to the new position of P, say P', by some function of p, cpp, then we say that the body has been strained. We do not at first need to consider the path of transition of P to P'. If cp is a linear vector function, then we say that the strain is a linear homogeneous strain. We have to put a few restrictions upon the generality of )(1 + where v is the unit normal, \x a given vector. That is to say, we have for the transition of the surface [S()V-a] = »Sv. Whence [SVcr] = Spix, [W DEFORMABLE BODIES 267 in the case in which the time t is not involved; and for a moving surface in which / is a function of t as well as of p, we would have [-SOV- wherein we can interchange the subscripts, and where S = - a/z[« the quotient of a simple longitudinal tension by the stretch produced, and called Young's modulus. Also we set s = X/(2X + 2/x), Poisson's ratio, 278 VECTOR CALCULUS the ratio of the lateral contraction to the longitudinal stretch. It is clear that if any two of the three moduli are known, the other may be found. We have X = E/[(l + *)(1 - 2*), M - \Ej(X + *), k - IE/(1 - 2s). In terms of E and s we have t»i(S)'« po -m* E (4). If | < s, k < 0, and the material would expand under pressure. If s < — 1, W would not be positive. (5). If Cauchy's relations hold, s = \ and X = /x. For numerical values of the moduli see texts such as Love, Elasticity. 31. Bodies that are not isotropic are called aelotropic. For discussion of the cases and definitions of the moduli, see texts on elasticity. 32. There is still the problem of finding a from cp after the latter has been found from S. This problem we can solve as follows: i-p)VvM = f > « constants. Substituting a and 6, and constructing -1 . The wave-surface, or surface of ray- velocity, is the envelope of Sp/o) = 1, or Spp = — 1, where /x = — w _1 . The condition is that given by the equations of the two other surfaces. It is the reciprocal of the index surface with respect to the unit sphere p 2 = — 1, or the envelope of the plane wave-fronts in unit time after passing the origin, or the wave of the vibration propagated from the origin in unit time. The vectors p that satisfy its equation are in magnitude and direction the ray- velocities. When there is an energy function, this ray-velocity is found easily, as follows: The wave-surface is the result of eliminating between 0(/x, p, a) = ca, Q(dp, p, a) + 0(ju, dp, a) + 0(ju, /x, da) = cdcr, Sup = - 1-Spdfi= 0. From the second equation 2SdfxG(% where the function 0=-S()V-p', 0'=-VV(), 0o = K-sovy-wo), 2 e = FVp'. This statement of the motion in terms of the coordinates of 296 VECTOR CALCULUS any point and the time is the statement in terms of Eulers variables. Since near po, p = po + po'dt, we have the former function

<%'. These vector lines are called the vortex lines of the fluid. Occasionally the vortex lines may be closed, but as a rule the solutions of such a differential equation as the above do not form closed lines, in which case they may terminate on the walls of the containing vessel, or they may wind about indefinitely. The integral of this equation will usually contain t, and the vortices then vary with the time, but in a stationary motion they will depend only upon the point under consideration. 14. The equations of motion may be expressed in terms of the vortex as follows, since we have and thus Vp'VVp' ' = Sp'V-p'-iVp' 2 ,. Sp'V-p' =2Vp'e + iVp'\ aVp = i - dp' Idt + JVp /2 + 2Vp'e. 15. When now £ = \/u{p, t), and c = f(p), we set P = fadp, giving VP = aVp, and thence VP = Vu - dp' Idt + JVp /2 - 2Vep'. 302 VECTOR CALCULUS Or, if we set II = u-\- Jp' 2 — P, we have dp'/dt + 2Ve P ' = VII. Operate on this with V-V(), and since VV dp'/dt = 2de/dt, and WVep' = SeV -p' - eSVp' - Sp'V-e, de/dt — Sp'V-e = de/dt, SVp' by the continuity equation is equal to c~ l dc/dt = — a~ l da/dt, we have d(ae)/dt = - S(ae)V-p' = 6(ae). This equation is due to Helmholtz. If we remember the Lagrangian variables, it is clear that 6 is a function of the initial vector p and of t, hence the integral of this equation will take the form ae = e fm 'a,e Q = e' ~ s ^^' dt a e = ^(t)a e . But the operator is proved below to be equal to

VECTOR CALCULUS Farad 32, 73 Faux 37, 38 Faux-focus 44 Feuille 30 Feuillets 2 Field 13 Flow 142 Flux 29, 130, 142 Flux density 29 Focus 41 Force 29 Force density 28, 141 Force function 18 Franklin 90 Free vector 8, 25 Frenet-vSerret formulae 148 Functions of dyadic 238 Function of flow 88 Functions of quaternions. ... 121 Gas defined 87 Gauss 4/ Gauss (magnetic unit). . . .32, 130 Gaussian operator 108 General equation of dyadic . . 220 Geometric curl 76 Geometric divergence 76 Geometric loci 133 Geometric vector 1 Geometry of lines 2 Gibbs 2, 11, 215 Gilbert 32, 130, 143 Glissant 26 Gradient 16, 163 Gram 15 Grassmann 2, 3, 9 Green's Theorem 205 Groups 8 Guiot 138 Hamilton 2, 3, 4,65,95 Harmonics 84, 169 Heaviside 31 Henry (electric unit) 32, 73 Hertzian vectors 33 Hitchcock 49 Hodograph 27 Hypernumber 3, 94 Imaginary 65 Impedance 73 Inductance. 73 Inductivity 32 Integral of vector 56 Integrating factor 191 Integration by parts 198 Interior multiplication 10 Invariant line 219 Irrotational 88 Isobaric 15,288 Isogons 34 Isohydric 15 Isopycnic 15, 288 Isosteric 15, 288 Isothermal 15 Joly 138, 147 Joule 14 Joule-second 14 Kinematic compatibility .... 266 Kirchoff's laws .' 73 Koenig 198, 205 Laisant 71 Lamellae 15 Lamellar field 84, 181 Laplace's equation 214 Latent equation 220 Laws of quaternions 103 Leibniz 3 Level 15 Line (electric unit) 32, 130 Lineal multiplication 9 Linear associative algebra ... 3 Linear vector function 218 Line of centers 46 Line of convergence 47 Line of divergence 47 Line of fauces 46 Line of foci 46 Line of nodes 45 Lines as levels 80 Liquid defined 87 MacMahon 75 Magnetic current 31 Magnetic density current 31 Magnetic induction 32 Magnetic intensity 32, 139 Mass 15 Matrix unity 65 Maxwell 13 McAulay 3 Mobius 8 Modulus 66 Moment 138 Moment of momentum 139 INDEX 313 Momentum 28 Momentum density 28 Momentum of field 141 Monodromic 14 Monogenic 89 Moving electric field 140 Moving magnetic field 140 Multenions 3 Multiple 6 Mutation 108 Nabla as complex number. . . 82 Nabla in plane 80 Nabla in space 162 Neutral point 47 Node 37,38 Node of isogons 48 Non-degenerate equations . . . 225 Norm 66 Notations One vector 12 Scalar 127 Two vectors 136 Derivative of vectors 165 Divergence, vortex, deriva- tive dyads 179 Dyadics 248 Ohm (electric unit) 73 Orthogonal dyadic 241 Orthogonal transformation . . 55 Peirce, Benjamin 3 Peirce, B. O 85 Permittance 73 Permittivity 32 Phase angle 71 Plane fields.. 84 Poincare 36, 46 Polar vector 30 Polydromic 14 Potential ,. .. 15, 17 Progressive multiplication ... 10 Power 76 Poynting vector 141 Pressure 142 Product of quaternions 98 Product of several quater- nions 113 Product of vectors 101 Quantum 14 Quaternions 2, 3, 6, 7, 95 Radial 26 Radius vector 26 Ratio of vectors 62 Reactance 73 Real 65 Reflections \ 108 Refraction . . 112 Regressive multiplication. ... 10 Relative derivative 18 Right versor 96 Rotations 108 Rotatory deviation 175 Saint Venant's equations. . . . 260 Sandstrom 35, 49 Saussure 2 Scalar 13 Scalar invariants 220, 239 Scalar of q 96 Schouten 7 Science of extension 2 Self transverse 234 Servois 4 Shear 256 Similitude 242 Singularities of vector lines . . 244 Singular lines 45 Solenoidal field 84, 181 Solid angles 117 Solution of equations 123 Solution of differential equa- tions 195 Solution of linear equation. . . 229 Specific momentum 28 Spherical astronomy 110 Squirt 90 Steinmetz 68, 71 Stoke's theorem 200 Strain 253 Strength of source or sink ... 90 Stress 143,269 Study . . . 2 Sum of quaternions 96 Surfaces 151 Symmetric multiplication ... 9 Tensor 65 Tensor of q 96 Torque 140 Tortuosity 149 Trajectories 150 Transport 130,298 Transverse dyadic 231 Triplex 25 314 VECTOR CALCULUS Triquaternions 3 Trirectangular biradials 100 Unit tube 18 Vacuity 220 Vanishing invariants 240 Variable trihedral 172 Vector 1 Vector calculus 1, 25 Vector field 23, 26 Vector lines 33 Vector of q 96 Vector potential 33, 93, 181 Vector surfaces 34 Vector tubes 34 Velocity 27 Velocity potential 18 Versor 65 Versor of q 96 Virial 129 Volt 31, 130, 143 Vortex 92, 187, 187 Vorticity 247,304 Waterspouts 50 Watt 15 Weber 14 Wessel 4 Whirl 90 Zero roots of linear equations. 230 foist r Ot— C/ p^V A^y 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. ■ JSJ — ^o. JM181S8 - 1MIH r- — l^MM o^ p - DEC 9 1968 52 "HI ttulb'68-3PM LOAN DEPT. ^ ^ REC'DLD JUN ^ & & * 2 172-9PMK8 LD 21A-60m-2,'67 (H241slO)476B LD 21A-50to-11 '62 (D3279sl0)476B jMCWI General Library University of California Berkeley General Library University of California Berkelev