= Tensoriality
{title2=$C^\infty(M)\text{-linearity}$}
A multilinear operation on <vector fields> is tensorial when it is linear over <smooth functions> in every argument. Its value at a point then depends only on the argument vectors at that point. To see this, use a <smooth cutoff function> to reduce to local fields, expand them in a local frame, and apply linearity over <smooth functions> to their coefficients. A smoothly valued tensorial operation therefore defines a <tensor field>. The curvature of an <affine connection> is tensorial, whereas a <covariant derivative> differentiates a scalar coefficient in its second argument and is not tensorial there.
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