# Tensor field

A tensor field (also imprecise tensor) is examined in the mathematical sub-area of differential geometry, in particular in tensor analysis. It is a function that assigns a tensor to each point of an underlying space in a special way .

## definition

Let be a smooth manifold and an (r, s) - tensor bundle . An (r, s) -tensor field is a smooth cut in the tensor bundle . The number of tensor fields is denoted by. This set is a module on the algebra of smooth functions . ${\ displaystyle M}$${\ displaystyle T_ {s} ^ {r} (M)}$${\ displaystyle T_ {s} ^ {r} (M)}$${\ displaystyle \ Gamma ^ {\ infty} (T_ {s} ^ {r} (M))}$${\ displaystyle C ^ {\ infty} (M) = \ Gamma ^ {\ infty} (T_ {0} ^ {0} (M))}$

## Examples

Let M be a differentiable manifold, then a tensor field on M is a mapping that assigns a tensor to every point.