Special form of the Divergence theorem

Question

It states in my notes relating to the derivation of the euler equations that a 'special form' of the divergence theorem is:

$\iint{\phi\hat{n}}\space{dS}=\iiint\nabla\phi\space{dV}$

with $\phi$ a scalar field. I thought the divergence theorem only related to vector fields?

Answer 1

Apply the divergence theorem to the vector field $\phi\mathbf{i}$.

Then

$$\int \phi \mathbf{i} \cdot \mathbf{n}\,dS=\int \phi n_x\,dS=\int \frac{\partial \phi}{\partial x}\,dV$$.

Repeat for $\phi\mathbf{j}$ and $\phi\mathbf{k}$. Multiply each result by the appropriate unit basis vector and sum to get

$$\int \phi \mathbf{n}\,dS=\int \nabla \phi \,dV$$.