---
tag: WX302T93TW
kind: concept
name: Tensor field
msc: [53A45, 58A10]
sources:
  - url: https://en.wikipedia.org/wiki/Tensor_field
same_as:
  wikidata: Q1758924
  wikipedia: Tensor field
---

## Summary

A tensor field on a [[smooth manifold|F45VQ47GPV]] assigns to each point a tensor built from the tangent space and its dual.
The assignment varies smoothly with the point.
Vector fields, differential forms, Riemannian metrics and curvature tensors are all tensor fields.
Tensor fields are the basic objects of differential geometry, because their meaning does not depend on the choice of coordinates.

## Explanations

### L1 human

At each point of a curved space there is a flat space of directions, the tangent space.
A tensor at a point is a multilinear gadget that eats some directions and some covectors and returns a number.
A tensor field places one such gadget at every point, smoothly.
For example, the dot product $dx \otimes dx + dy \otimes dy$ at every point of the plane is a tensor field that measures lengths of arrows.
A wind map that puts an arrow at each point is a simpler tensor field, of type $(1,0)$.