Identifying Tensor Fields
Just as for vector fields, we need a good condition to identify tensor fields in the wild. And the condition we will use is similar: if is a smooth tensor field of type , then for any coordinate patch in the domain of , we should be able to write out
for some smooth functions on . Conversely, this formula defines a smooth tensor field on .
Indeed, we can find these coefficient functions by evaluation:
for using this definition, if we plug these coordinate vector fields and coordinate covector fields into either the left or the right side of the expression above we will get the same answer. Any vector or covector fields on can be written as a linear combination of these coordinate fields with smooth functions as coefficients, and the multilinear properties of tensors will ensure that both sides get the same value no matter what fields we evaluate them on.
Similarly, if is a differential -form and is a coordinate patch within its domain, then we can write
for some smooth functions on . The proof in this case is similar, following from the definition
In this case we can pick the indices to be strictly increasing because of the antisymmetry of the tensors.