Vector Integration: Fundamental theorem use

[WWCY]

Homework Statement

Could someone illustrate why
$$\int_{V} \nabla \cdot (f\vec{A}) \ dv = \int_{V} f( \nabla \cdot \vec{A} ) \ dv + \int_{V} \vec{A} \cdot (\nabla f ) \ dv = \oint f\vec{A} \cdot \ d\vec{a}$$
?

Homework Equations

The Attempt at a Solution

I understand that the integrand can be split by using vector product rules to give two integrals, but I don't see how the divergence theorem,
$$\int_{V} (\nabla \cdot \vec{A}) \ dv = \oint \vec{A} \cdot d\vec{a}$$
gets me from step 2 to 3.

Assistance is greatly appreciated!

 

[kuruman]

The divergence theorem takes you directly from step 1 to step 3. Define
$$\vec B= f\vec{A}$$ Then
$$\int_{V} (\vec{\nabla} \cdot \vec{B}) \ dv = \oint \vec{B} \cdot d\vec{a}=\oint f\vec{A} \cdot d\vec{a}$$

 

[WWCY]

The divergence theorem takes you directly from step 1 to step 3. Define
$$\vec B= f\vec{A}$$ Then
$$\int_{V} (\vec{\nabla} \cdot \vec{B}) \ dv = \oint \vec{B} \cdot d\vec{a}=\oint f\vec{A} \cdot d\vec{a}$$

I see it now, thank you very much!