/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
.
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 + /3 2 cos 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
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
SVav VUVpVap = ^P-,
V(Vap)-i=0, V (g)=0.
EXERCISE
Show that (Fa/3 - 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 , i) 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 ,
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\(pfxp].
Adding to this result S\pu> -m%ipp, we have
S\pv((p 3 p — mnp 2 p + m + bfx + cX.
Since
( ,
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,
so that
Sa(3y-x 4 ) = ra 3 4 .
4. Show that for the function + c, where c is a scalar multiplier,
mi( + c ) = w s(«p) + cw 2 ( ) = aHiv)', tifilPi) = ^( 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 \ /3, m 3 (d) =
o,
" 2 (a 2 — aSa),
12. 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 )(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
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
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General Library
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Berkelev
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;