Theorem for Divergence

The integral of a derivative (divergence) over a volume equals the value of the function at the boundary (surface).

$${\int }_V(\nabla \cdot A)d\tau ={\oint }_{S}A\cdot da$$

The divergence of a vector field (medium) is interpreted as its rate of expansion or compression in the defined space. It is the flux per unit volume flux density.

$$\text{div}F=\nabla \cdot F=\frac{\partial M}{\partial x}+\frac{\partial N}{\partial y}+\frac{\partial P}{\partial z}$$

Deriving the Identity by Decomposition by Partial Integration

$$\int \_Vf(\nabla \cdot A)d\tau =-\int \_VA\cdot (\nabla f)d\tau +\oint \_SfA\cdot da$$

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We begin with a common divergence integral over a volume $V$. Apply the del identity $\nabla \cdot (fA)=f\nabla A+A\nabla f$.

$$\int \_V\nabla \cdot (fA)d\tau =\int \_Vf(\nabla \cdot A)d\tau +\int \_VA\cdot (\nabla f)d\tau$$

We can use the divergence theorem above to rewrite the left hand side as a surface integral along a closed crease

$${\oint }_S(fA)\cdot da={\int }_{V}f(\nabla \cdot A)d\tau +\int \_VA\cdot (\nabla f)d\tau$$

By subtraction we arrive at the identity that is stated above