Understanding tensor fields

Question

This is just a simple question: Is every vector field a first-order tensor field?

I understand the definition of vector field, but I have problems understanding first-order tensor fields.

Answer 1

Before going to tensor fields start with tensors! A "tensor field" is simply a function that assigns a tensor to every point. You don't say what definition of tensor you use but there are some definitions that specifically define a vector to be a first order tensor. More generally, a tensor is any function of the coordinates that transforms homogeneously as the coordinate system changes. That is, if tensor A= 0 in one coordinate system then A= 0 in all coordinate systems. Since that is true of a vector, yes, a vector is a tensor and so a vector field is a tensor field.