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Illustration Divergence Integral Theorem (Divergence + Flux)

Divergence theorem
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The Divergence Integral Theorem states that the divergence of a vector field \(\boldsymbol{F}\) in a considered volume \(V\) corresponds to the flow of the vector field through the surface of that volume:\[ \int_{V} \left(\nabla \cdot \boldsymbol{F}\right) \, \text{d}v ~=~ \oint_{A}\boldsymbol{F} \cdot \text{d}{\boldsymbol a} \]

Here the vector field \( \boldsymbol{F} \) was divided into the parallel and perpendicular parts to the considered surface element to show that due to the scalar product only the field part \( \boldsymbol{F}_{||} \) parallel to the surface element contributes to the integral.