tldr: A scalar field assigns a single value (real or complex) to every set of coordinates (typically space and time) in the coordinate space. A vector field assigns a vector to each set of coordinates. A tensor field assigns a tensor (matrix with special transformation properties) to each set of coordinates.

tldr: A scalar field assigns a single value (real or complex) to every set of coordinates (typically space and time) in the coordinate space. A vector field assigns a vector to each set of coordinates. A tensor field assigns a tensor (matrix with special transformation properties) to each set of coordinates.

As a recovering differential geometer I applaud the effort and have doubts about its effectiveness.