Tensor field

Assignment of a tensor continuously varying across a mathematical space

Intuitively, a vector field is best visualized as an "arrow" attached to each point of a region, with variable length and direction. One example of a vector field on a curved space is a weather map showing horizontal wind velocity at each point of the Earth's surface.

In general, we want to specify tensor fields in a coordinate-independent way: It should exist independently of latitude and longitude, or whatever particular "cartographic projection" we are using to introduce numerical coordinates.

to emphasize that the tangent bundle is the range space of the (1,0) tensor fields (i.e., vector fields) on the manifold M. This should not be confused with the very similar looking notation

in the latter case, we just have one tensor space, whereas in the former, we have a tensor space defined for each point in the manifold M.

It is worth noting that differential forms, used in defining integration on manifolds, are a type of tensor field.