Mathematical Tools for Physics

12—Tensors 371

Many of the common tensors such as the tensor of inertia and the dielectric tensor are symmetric. The
magnetic field tensor in contrast, is antisymmetric. The basis of this symmetry in the case of the dielectric tensor
is in the relation for the energy density in an electric field,

∫~

E.dD~.* Apply an electric field in thexdirection,
then follow it by adding a field in theydirection; undo the field in thexdirection and then undo the field in the
ydirection. The condition that the energy density returns to zero is the condition that the dielectric tensor is
symmetric.
All of the above discussions concerning the symmetry properties of tensors were phrased in terms of second
rank tensors. The extensions to tensors of higher rank are quite easy. For example in the case of a third rank
tensor viewed as a 3-linear functional, it would be called completely symmetric if

T(~u, ~v, ~w) =T(~v, ~u, ~w) =T(~u, ~w, ~v) =etc.

for all permutations of~u,~v,~w, and for all values of these vectors. Similarly, if any interchange of two arguments
changed the value by a sign,

T(~u, ~v, ~w) =−T(~v, ~u, ~w) = +T(~v, ~w, ~u) =etc.

then theT is completely antisymmetric. It is possible to have a mixed symmetry, where there is for example
symmetry on interchange of the arguments in the first and second place and antisymmetry between the second
and third.

Alternating Tensor
A curious (and very useful) result about antisymmetric tensors is that in three dimensions there is, up to a factor,
only one totally antisymmetric third rank tensor; it is called the “alternating tensor.” So, if you take any two such
tensors,^03 Λand^03 Λ′, then one must be a multiple of the other. (The same holds true for thenthrank totally
antisymmetric tensor inndimensions.)

• This can be proved by considering the energy in a plane parallel plate capacitor, which is, by definition of
  potential,

∫

V dq. The Potential differenceV is the magnitude of theE~field times the distance between the the

capacitor plates. [V =Ed.] (E~ is perpendicular to the plates by∇×E~ = 0.) The normal component ofD~

related toqby∇.D~ =ρ. [AD~.ˆn=q.] Combining these, and dividing by the volume gives the energy density
as

∫ ~

E.dD~.