I'm having trouble understanding some vector calculus equations.
If $$ f(r) = \nabla_r g(r)$$ then $$ \Delta g(r) = \int f(r) \cdot \mathrm{d} r$$
and
if $$ f(r) = \frac{\mathrm{d} g(r)}{\mathrm{d} V_r} $$ then $$ \Delta g(r) = \int f(r) \, \mathrm{d} V_r $$
But the inverse for other operators is hard.
How would I rewrite these equations as $g = T[f ]$ for some operator $T$?
$$ f(r) = \nabla \times g(r) $$
$$ f(r) = \nabla \cdot g(r) $$
And for the Jacobian which doesn't have a standard notation but I like to use $\nabla \otimes$ for.
$$ f(r) = \nabla \otimes g(r) $$
And for the Laplacian operators
$$ f(r) = \nabla \times \nabla \times g(r)$$
$$ f(r) = \nabla \cdot \nabla g(r) $$
$$ f(r) = \nabla \nabla \cdot g(r) $$