# Tensor Derivations

Define $(\mathcal{D}_\mathcal{U}B)_p = \mathcal{D}(fB)_p$. This is well defined since if $g$ is another bump function then

$\mathcal{D}(fgB)_p = \mathcal{D}(f)_p(gB)(p) + f(p)\mathcal{D}(gB)_p = \mathcal{D}(gB)_p$

and reversing $f$ and $g$ gives

$\mathcal{D}(gfB)_p = \mathcal{D}(g)_p(fB)(p) + g(p)\mathcal{D}(fB)_p = \mathcal{D}(fB)_p$