Tensors and Relativity: Chapter 3
General properties of tensors
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Tensors have the following general properties:
- If
and
are M/N tensors, so are
,
and
for any
and
. Thus tensors of a given type form a vector space .
- If
is a M/N tensor and
is of type M'/N', then the outer product
is a tensor of type M+M'/N+N'.
- For any tensor of type M/N, one can construct a tensor of type M-1/N-1 by contracting an upper index with a lower index for example
is constructed from
[
or
]. Note however that there may be several ways of contracting and they give different tensors.
- Tensors can be symmetric on pairs of upper or lower indices for example
and
, or anti- symmetric . However symmetric or anti- symmetric pairs of indices with one index up and the other down cannot be defined since the relationship need not hold in all frames.