Mathematical Tools for Physics

12—Tensors 381

variable).

For each vector space, you can discuss the tensors that act on that space and so, by picking one such tensor
for each point of the manifold a tensor field is defined.

A physical example of a tensor field (of second rank) is stress in a solid. This will typically vary from
point to point. But at each point a second rank tensor is given by the relation between infinitesimal area vectors
and internal force vectors at that point. Similarly, the dielectric tensor in an inhomogeneous medium will vary
with position and will therefore be expressed as a tensor field. Of course even in a homogeneous medium the
dielectric tensor would be a tensor field relatingD~ andE~ at the same point. It would however be a constant
tensor field. Like a uniform electric field, the tensors at different points could be thought of as “parallel” to each
other (whatever that means).

12.6 Coordinate Systems
In order to obtain a handle on this subject and in order to be able to do computations, it is necessary to put a
coordinate system on the manifold. From this coordinate system there will come a natural way to define the basis
vectors at each point (and so reciprocal basis vectors too).

There is no need to restrict the discussion to rectangular or even to orthogonal coordinate systems. A
coordinate system is a means of identifying different points of the manifold by different sets of numbers. This
is done by specifying a set of functions: x^1 ,x^2 ,x^3 , which are the coordinates. (There will be more in more
dimensions of course.) These functions are real valued functions of points in the manifold. The coordinate axes
are defined as in the drawing by

2

x

x

x = constant

x = constant

x = constant

x = constant^2

3

3

1

1

Specify the equationsx^2 =constant andx^3 =constant for thex^1 coordinate axis. For example in rectan-
gular coordinatesx^1 =x,x^2 =y,x^3 =z, and thex-axis is the liney= 0, andz= 0.