The term 'tensor rank' is sporadically used in the mathematical literature to denote the minimum number of simple terms (i.e. tensor products of vectors) needed to express the tensor. This is analogous (and equivalent) to the usual and prevalent notion of [matrix rank](http://en.wikipedia.org/wiki/Rank_%28linear_algebra%29) in linear algebra; a good example of its use in the literature is [*Exploring Tensor Rank*](http://www.its.caltech.edu/~matilde/WeitzMa10Abstract.pdf) by Benjamin Weitz. Is this concept useful in physics?

Note that I am not asking about the total number of indices of the tensor, as in e.g. [this MathWorld article](http://mathworld.wolfram.com/TensorRank.html).

For example, in general relativity, we deal with tensor fields. For a given field, can the rank of a tensor at one point be different that the rank of the tensor at a different point? Is there any physical significance to this?