Editing Covariant derivative (section)

===Tensor fields===
Once the covariant derivative is defined for fields of vectors and covectors it can be defined for arbitrary [[Tensor (intrinsic definition)|tensor]] fields by imposing the following identities for every pair of tensor fields <math> \varphi</math> and <math>\psi </math> in a neighborhood of the point {{mvar|p}}:
<math display="block">\nabla_\mathbf{v}\left(\varphi \otimes \psi\right)_p = \left(\nabla_\mathbf{v}\varphi\right)_p \otimes \psi(p) + \varphi(p) \otimes \left(\nabla_\mathbf{v}\psi\right)_p,</math>
and for <math>\varphi</math> and <math>\psi</math> of the same valence
<math display="block">\nabla_\mathbf{v}(\varphi + \psi)_p = (\nabla_\mathbf{v}\varphi)_p + (\nabla_\mathbf{v}\psi)_p.</math>
The covariant derivative of a tensor field along a vector field {{math|'''v'''}} is again a tensor field of the same type.

Explicitly, let {{mvar|T}} be a tensor field of type {{math|(''p'', ''q'')}}. Consider {{mvar|T}} to be a differentiable [[multilinear map]] of [[smooth function|smooth]] [[section (fiber bundle)|sections]] {{math|''α''{{isup|1}}, ''α''{{isup|2}}, ..., ''α''<sup>''q''</sup>}} of the cotangent bundle {{math|''T''{{isup|∗}}''M''}} and of sections {{math|''X''{{sub|1}}, ''X''{{sub|2}}, ..., ''X''<sub>''p''</sub>}} of the [[tangent bundle]] {{math|''TM''}}, written {{math|''T''(''α''{{isup|1}}, ''α''{{isup|2}}, ..., ''X''{{sub|1}}, ''X''{{sub|2}}, ...)}} into {{math|'''R'''}}. The covariant derivative of {{mvar|T}} along {{mvar|Y}} is given by the formula

<math display="block">\begin{align}
(\nabla_Y T)\left(\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) =
&{} \nabla_Y\left(T\left(\alpha_1,\alpha_2, \ldots, X_1, X_2, \ldots\right)\right) \\
&{}- T\left(\nabla_Y\alpha_1, \alpha_2, \ldots, X_1, X_2, \ldots\right) - T\left(\alpha_1, \nabla_Y\alpha_2, \ldots, X_1, X_2, \ldots\right) - \cdots \\
&{}- T\left(\alpha_1, \alpha_2, \ldots, \nabla_YX_1, X_2, \ldots\right) - T\left(\alpha_1, \alpha_2, \ldots, X_1, \nabla_Y X_2, \ldots\right) - \cdots
\end{align}</math>