Read an Excerpt
Tensor Calculus
By J. L. Synge, A. Schild Dover Publications, Inc.
Copyright © 1949 Canada
All rights reserved.
ISBN: 978-0-486-14139-8
CHAPTER 1
SPACES AND TENSORS
1.1. The generalized idea of a space. In dealing with two real variables (the pressure and volume of a gas, for example), it is a common practice to use a geometrical representation. The variables are represented by the Cartesian coordinates of a point in a plane. If we have to deal with three variables, a point in ordinary Euclidean space of three dimensions may be used. The advantages of such geometrical representation are too well known to require emphasis. The analytic aspect of the problem assists us with the geometry and vice versa.
When the number of variables exceeds three, the geometrical representation presents some difficulty, for we require a space of more than three dimensions. Although such a space need not be regarded as having an actual physical existence, it is an extremely valuable concept, because the language of geometry may be employed with reference to it. With due caution, we may even draw diagrams in this "space," or rather we may imagine multidimensional diagrams projected on to a two-dimensional sheet of paper; after all, this is what we do in the case of a diagram of a three-dimensional figure.
Suppose we are dealing with N real variables x1, x2, ..., xN. For reasons which will appear later, it is best to write the numerical labels as superscripts rather than as subscripts. This may seem to be a dangerous notation on account of possible confusion with powers, but this danger does not turn out to be serious.
We call a set of values of x1, x2, ..., xN a point. The variables x1, x2, ..., xN are called coordinates. The totality of points corresponding to all values of the coordinates within certain ranges constitute a space of N dimensions. Other words, such as hyperspace, manifold, or variety are also used to avoid confusion with the familiar meaning of the word "space." The ranges of the coordinates may be from - ∞ to + ∞, or they may be restricted. A space of N dimensions is referred to by a symbol such as VN.
Excellent examples of generalized spaces are given by dynamical systems consisting of particles and rigid bodies. Suppose we have a bar which can slide on a plane. Its position (or configuration) may be fixed by assigning the Cartesian coordinates x, y of one end and the angle θ which the bar makes with a fixed direction. Here the space of configurations is of three dimensions and the ranges of the coordinates are
- ∞ < x < + ∞, - ∞ < y < + ∞, 0 ≤ θ < 2π.
Exercise. How many dimensions has the configuration-space of a rigid body free to move in ordinary space? Assign coordinates and give their ranges.
It will be most convenient in our general developments to discuss a space with an unspecified number of dimensions N, where N ≥ 2. It is a remarkable feature of the tensor calculus that no essential simplification is obtained by taking a small value of N; a space of two million dimensions is as easy to discuss (in its general aspects) as a space of two dimensions. Nevertheless the cases N = 2, N = 3, and N = 4 are of particular interest: N = 2 gives us results in the intrinsic geometry of an ordinary surface; N = 3 gives us results in the geometry of ordinary space; N = 4 gives us results in the space-time of relativity.
The development of the geometry of VN is a game which must be played with adroitness. We take the familiar words of geometry and try to give them meanings in VN. But we must of course remember that N might be 3 and VN might be our familiar Euclidean space of three dimensions. Therefore, to avoid confusion, we must be careful to frame our definitions so that, in this particular case, these definitions agree with the familiar ones.
A curve is defined as the totality of points given by the equations
1.101.
xr = fr(u) (r = 1, 2, ..., N).
Here u is a parameter and fr are N functions.
Next we consider the totality of points given by
1.102.
xr = fr(u1, u2, ..., uM) (r = 1, 2, ..., N).
where the u's are parameters and M < N. This totality of points may be called VM, a subspace of VN. There are two cases of special interest, namely M = 2 and M = N - 1. Either of these might be called a surface, because if N = 3 they both coincide with the familiar concept of "surface." It seems, however, that VN-1 has the better right to be called a surface, because it has (for any N) the fundamental property of a surface in ordinary space, viz. it divides the neighbouring portion of space into two parts. To see this, we eliminate the parameters from 1.102. Since M = N - 1, the number of parameters is one less than the number of equations, and so elimination gives just one equation:
1.103.
F(x1, ..., xN) = 0.
The adjacent portion of VN is divided into two parts for which respectively F is positive and negative. VN-1 is often called a hypersurface in VN.
Other familiar geometrical ideas will be extended to VN as the occasion arises.
Exercise. The parametric equations of a hypersurface in VN are
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where a is a constant. Find the single equation of the hypersurface in the form 1.103, and determine whether the points (1/2a, 0, 0,.... 0), (0, 0, 0, ... 0, 2a) lie on the same or opposite sides of the hypersurface.
Exercise. Let U2 and W2 be two subspaces of VN. Show that if N = 3 they will in general intersect in a curve; if N = 4 they will in general intersect in a finite number of points; and if N > 4 they will not in general intersect at all.
1.2. Transformation of coordinates. Summation convention. It is a basic principle of tensor calculus that we should not tie ourselves down to any one system of coordinates. We seek statements which are true, not for one system of coordinates, but for all.
Let us suppose that in a VN there is a system of coordinates x1, x2, ..., xN. Let us write down equations
1.201.
x'r = fr(x1, ..., xN) (r = 1, 2, ..., N).
where the f's are single valued continuous differentiable functions for certain ranges of x1, x2, ..., xN. These equations assign to any point x1, x2, ..., xN a new set of coordinates x'1, x'2, ..., x'N. The Jacobian of the transformation is
1.202.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
or, in a briefer notation,
1.203.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
the ranges r, s = 1, 2, ..., N being understood. We shall suppose that the Jacobian does not vanish. Then, as is well known from the theory of implicit functions, the equations 1.201 may be solved to read
1.204.
xr = gr(x'1, x'2, ..., x'N) (r = 1, 2, ..., N).
Differentiation of 1.201 gives
1.205.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Thus the transformation of the differentials of the coordinates is a linear homogeneous transformation, the coefficients being functions of position in VN. We shall return to this transformation presently, but first let us introduce two notational conventions which will save us an enormous amount of writing.
Range Convention. When a small Latin suffix (superscript or subscript) occurs unrepeated in a term, it is understood to take all the values 1, 2, ..., N, where N is the number of dimensions of the space.
Summation Convention. When a small Latin suffix is repeated in a term, summation with respect to that suffix is understood, the range of summation being 1, 2, ..., N.
It will be noticed that the reference is to small Latin suffixes only. Some other range (to be specified later) will be understood for small Greek suffixes, while if the suffix is a capital letter no range or summation will be understood.
To see the economy of this notation, we observe that 1.205 is completely expressed by writing
1.206.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Repeated suffixes are often referred to as "dummies" since, due to the implied summation, any such pair may be replaced by any other pair of repeated suffixes without changing the expression. We have, for example,
arsb8 = arkbk.
This device of changing dummies is often employed as a useful manipulative trick for simplifying expressions.
In order to avoid confusion we make it a general rule that the same suffix must never be repeated more than twice in any single term or product. If this cannot be avoided, the summation convention should be suspended and all sums should be indicated explicitly.
Exercise. Show that
(arst + astr + asrt)xrxsxt = 3arstxrxsxt.
Exercise. If φ = arsxrxs, show that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Simplify these expressions in the case where ars = asr.
Let us introduce a symbol δrs called the Kronecker delta; it is defined by
1.207.
δrs = 1 if r = s,
δrs = 0 if r ≠ s.
Exercise. Prove the relations
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
It is evident that [partial derivative]xr/[partial derivative]xs = δrs, or equivalently
1.208.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
From this we may derive an identity which will be useful later. Partial differentiation with respect to xp gives (since the Kronecker delta is constant)
1.209.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
If we multiply across by [partial derivative]x'q/[partial derivative]xr, we get
1.210.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
We may of course interchange primed and unprimed symbols in the above equations.
Exercise. If Ars are the elements of a determinant A, and Brs the elements of a determinant B, show that the element of the product determinant AB is ArnBns. Hence show that the product of the two Jacobians
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
is unity.
1.3. Contravariant vectors and tensors. Invariants. Consider a point P with coordinates xr and a neighbouring point Q with coordinates xr + dxr. These two points define an infinitesimal displacement or vector [??] PQ; for the given coordinate system this vector is described by the quantities dxr, which may be called the components of this vector in the given coordinate system. The vector dxr is not to be regarded as "free," but as associated with (or attached to) the point P with coordinates xr.
Let us still think of the same two points, but use a different coordinate system x'r. In this coordinate system the components of the vector [??] are dx'r; these quantities are connected with the components in the coordinate system xr by the equation
1.301.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
as in 1.206. If we keep the point P fixed, but vary Q in the neighbourhood of P, the coefficients [partial derivative]x'r /[partial derivative]xs remain constant. In fact, under these conditions, the transformation 1.301 is a linear homogeneous (or affine) transformation.
The vector is to be considered as having an absolute meaning, but the numbers which describe it depend on the coordinate system employed. The infinitesimal displacement is the prototype of a class of geometrical objects which are called contravariant vectors. The word "contravariant" is used to distinguish these objects from "covariant" vectors, which will be introduced in 1.4. The definition of a contravariant vector is as follows:
A set of quantities Tr, associated with a point P, are said to be the components of a contravariant vector if they transform, on change of coordinates, according to the equation
1.302.
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
where the partial derivatives are evaluated at P. Thus an infinitesimal displacement is a particular example of a contravariant vector. It should be noted that there is no general restriction that the components of a contravariant vector should be infinitesimal. For a curve, given by the equations 1.101, the derivatives dxr/du are the components of a finite contravariant vector. It is called a tangent vector to the curve.
Any infinitesimal contravariant vector Tr may be represented geometrically by an infinitesimal displacement. We have merely to write
1.303.
dxr = Tr.
If we use a different coordinate system x'r, and write
1.304.
dx'r = T'r,
we get an infinitesimal displacement. The whole point of the argument is that these two equations define the same displacement, provided Tr are the components of a contravariant vector. If Tr and T'r were two sets of quantities connected by a transformation which was not of the form 1.302, but something different, say, T'r = Ts]partial derivative] xs/[partial derivative][x'r, then the connection between the dxr of 1.303 and the dx'r of 1.304 would not be the transformation 1.301 which connects the components of a single infinitesimal displacement in the two coordinate systems. In that case, dxr and dx'r would not represent the same infinitesimal displacement in VN.
This point has been stressed because it is very useful to have geometrical representations of geometrical objects, in order that we may use the intuitions we have developed in ordinary geometry. But it is not always easy to do this. Although we can do it for an infinitesimal contravariant vector, we cannot do it so completely for a finite contravariant vector. This may appear strange to the physicist who is accustomed to represent a finite vector by a finite directed segment in space. This representation does not work in the general type of space we have in mind at present.
(Continues...)
Excerpted from Tensor Calculus by J. L. Synge, A. Schild. Copyright © 1949 Canada. Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
