# Vector Calculus Theorems - Duality Between Differentiation & Integration?

I'm trying to go over some vector analysis using forms & kind of noticed what look like random vector identities are more appropriately thought of, to me at least, as differential analogues of the classical integral theorems in the way Maxwell's equations can be cast in differential & integral form. However I'm missing a theorem:

Integral Gradient Theorem: $\smallint_{\vec{a}}^{\vec{b}} \nabla f \cdot d \vec{r} = f(\vec{b}) - f(\vec{a})$

Integral Curl Theorem: $\smallint_S (\nabla \times \vec{F}) \cdot \hat{n} dS = \smallint_{\partial S} \vec{F} \cdot d \vec{r}$

Integral Divergence Theorem: $\smallint_V \nabla \cdot \vec{F} dV = \smallint_{\partial V} \vec{F} \cdot \hat{n} dS$

Derivative Curl Theorem: $\nabla \times (\nabla \phi) = 0$

Derivative Divergence Theorem: $\nabla \cdot (\nabla \times \vec{F}) = 0$

Any ideas as to what I should put in there?

Also, could one bluff an appropriate differential & integral theorem for the scalar or vector laplacians?

To justify this, note that the integral theorems you write down are all specializations of Stokes' theorem, of which there should be $n$ in $n$ dimensions. The derivative theorems you write down are all specializations of the fact that the de Rham differential squares to zero, of which there should be $n$ in $n$ dimensions, except one of them just says that the composition $0 \to \Omega^0(M) \to \Omega^1(M)$ is zero and this is the least interesting one.