# Special form of the Divergence theorem

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?

• For $\phi$ a scalar field, those dots don't make sense. – Muphrid Jul 8 '14 at 15:49
• Apologies, does it make sense now? – user144895 Jul 8 '14 at 15:54

Apply the divergence theorem to the vector field $\phi\mathbf{i}$.
$$\int \phi \mathbf{i} \cdot \mathbf{n}\,dS=\int \phi n_x\,dS=\int \frac{\partial \phi}{\partial x}\,dV$$.
Repeat for $\phi\mathbf{j}$ and $\phi\mathbf{k}$. Multiply each result by the appropriate unit basis vector and sum to get
$$\int \phi \mathbf{n}\,dS=\int \nabla \phi \,dV$$.