Mathematical Methods for Physicists: A concise introduction - Site Map

BASIC OPERATIONS WITH TENSORS
two contravariant vectors
T ˆ @x
@x 0 @x
@x 0 T 0 ;
…1:97†
is a contravariant tensor of rank two. We may also form a covariant tensor of
rank two from two covariant vectors, which transforms according to the formula
T ˆ @x 0
@x @x 0
@x T 0 :
…1:98†
Similarly, we can form a mixed tensor T
follows:
of order two that transforms as
T ˆ @x @x 0
@x 0 @x T 0 :
…1:99†
We may continue this process and multiply more than two vectors together,
taking care that their indexes are all di€erent. In this way we can construct tensors
of higher rank. The total number of free indexes of a tensor is its rank (or order).
In a Cartesian coordinate system, the distinction between the contravariant and
the covariant tensors vanishes. This can be illustrated with the velocity and
gradient vectors. Velocity and acceleration are contravariant vectors, they are
represented in terms of components in the directions of coordinate increase; the
gradient vector is a covariant vector and it is represented in terms of components
in the directions orthogonal to the constant coordinate surfaces. In a Cartesian
coordinate system, the coordinate direction x coincides with the direction orthogonal
to the constant-x surface, hence the distinction between the covariant and
the contravariant vectors vanishes. In fact, this is the essential di€erence between
contravariant and covariant tensors: a covariant tensor is represented by components
in directions orthogonal to like constant coordinate surface, and a
contravariant tensor is represented by components in the directions of coordinate
increase.
If two tensors have the same contravariant rank and the same covariant rank,
we say that they are of the same type.
Basic operations with tensors
(1) Equality: Two tensors are said to be equal if and only if they have the same
covariant rank and the same contravariant rank, and every component of
one is equal to the corresponding component of the other:
A ˆ B :
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