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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?

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

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

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