# multilineal lección six

### tensor fields on a surface and its covariant derivatives

the collection $T\Sigma$ of all the tangent spaces $T_p\Sigma$ is called the tangent bundle of the surface, i.e.

$T\Sigma=\bigsqcup_pT_p\Sigma$

A vector field in a surface is a mapping $X:\Sigma\to T\Sigma$ with the condition $p\mapsto X\in T_p\Sigma$ and since $\partial_1,\partial_2$ span $T_p\Sigma$ then

$X=X^s\partial_s$

This construction determines a contravariant tensor field of rank one, which is taken as the base to ask how other tensor fields -of any rank and any variance- vary…

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