Mathematical Tools for Physics

12—Tensors 359

Similarly for multilinear functionals, with as many arguments as you want.
Now apply the representation theorem for functionals to the subject of tensors. Start with a bilinear
functional:^02 T(~v 1 ,~v 2 )is a scalar. This function of two variables can be looked on as a function of one variable
by holding the other one temporarily fixed. Say~v 2 is held fixed, then^02 T(~v 1 ,~v 2 )defines a linear functional on the
variable~v 1. Apply the representation theorem now and the result is

0

2 T(~v^1 ,~v^2 ) =A~.~v^1.

The vectorA~however will depend (linearly) on the choice of~v 2. It defines a new function that I’ll call^11 T

A~=^11 T(~v 2 ). (8)

This defines a tensor^11 T, a vector valued function of a vector. The above paragraph shows that from a
bilinear functional you can construct a linear vector function and vice versa. With this close association between
the two concepts it is natural to extend the definition of a tensor to include bilinear functionals. To be precise, I
used a different name for the vector-valued function of one vector variable (^11 T) and for the scalar-valued function
of two vector variables (^02 T). This may be overly fussy, and it’s common practice to use the same symbol (T) for
both, with the hope that the context will make clear which one you actually mean.
Until I get tired of doing so however, I’ll follow this (unconventional) notation and indicate the number of
arguments by a preceding subscript, and the nature of the output by a preceding superscript. Eventually, I’ll drop
these indices, hoping that it will be clear from context which one I mean. Therankof the tensor is the sum of
these two indices.
The next extension of the definition follows naturally from the previous reformulation. A tensor ofnthrank
is ann-linear functional, or any one of the several types of functions that can be constructed from it by the
preceding argument. The meaning and significance of the last statement should become clear a little later. In
order to clarify the meaning of this terminology, some physical examples are in order. The tensor of inertia was
mentioned before:
~L=^11 I

(

~ω

)

.

The dielectric tensor relatedD~ andE~:

D~ =^11 ε

(

E~

)

.