Mathematical Methods for Physicists: A concise introduction - Site Map

VECTOR AND TENSOR ANALYSIS
(2) Addition (subtraction): The sum (di€erence) of two or more tensors of the
same type and rank is also a tensor of the same type and rank. Addition of
tensors is commutative and associative.
(3) Outer product of tensors: The product of two tensors is a tensor whose rank
is the sum of the ranks of the given two tensors. This product involves
ordinary multiplication of the components of the tensor and it is called
the outer product. For example, A B ˆ C
is the outer product
of A and B .
(4) Contraction: If a covariant and a contravariant index of a mixed tensor are
set equal, a summation over the equal indices is to be taken according to the
summation convention. The resulting tensor is a tensor of rank two less than
that of the original tensor. This process is called contraction. For example, if
we start with a fourth-order tensor T , one way of contracting it is to set
ˆ , which gives the second rank tensor T . We could contract it again
to get the scalar T .
(5) Inner product of tensors: The inner product of two tensors is produced by
contracting the outer product of the tensors. For example, given two tensors
A and B , the outer product is A B . Setting ˆ , we obtain the
inner product A B .
(6) Symmetric and antisymmetric tensors: A tensor is called symmetric with
respect to two contravariant or two covariant indices if its components
remain unchanged upon interchange of the indices:
A ˆ A ; A ˆ A :
A tensor is called anti-symmetric with respect to two contravariant or two
covariant indices if its components change sign upon interchange of the
indices:
A ˆA ; A ˆA :
Symmetry and anti-symmetry can be de®ned only for similar indices, not
when one index is up and the other is down.
Quotient law
A quantity Q ... ... with various up and down indexes may or may not be a tensor.
We can test whether it is a tensor or not by using the quotient law, which can be
stated as follows:
Suppose it is not known whether a quantity X is a tensor or not.
If an inner product of X with an arbitrary tensor is a tensor, then
X is also a tensor.
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