Tensors can be of type (n, m), denoting n covariant and m contravariant indicies. Rank is a concept that comes from matrix rank and is basically the number of "simple" terms it takes to write out a tensor. Sometimes, however, I recall seeing or hearing things like "the metric tensor is a rank 2 tensor" and also "the metric is a covariant 2-tensor or type 2 tensor" I assume the two concepts, that of "type" and "rank" are unrelated, but I want another perspective. Also, in GR mostly we deal with tensor fields as well as tensors. At different points the rank (as in matrix rank) may be different. Is this true?