Prove an integral identity

Question

I want to prove the identity 17 but I don't know how to start. I believe I should pull apart the n on the left side so I can use identity 14 and then follow. So my question is how can I pull apart that n ? Or there is a better way to prove this identity?Integral identities

Answer 1

HINT:

Write

$$\oint_S \vec A (\hat n\cdot \vec B)\,dS=\sum_i \hat x_i \oint_S A_i \vec B \cdot \hat n\,dS $$

Then, apply the divergence theorem and use

$$\nabla \cdot(A_i \vec B)=A_i \nabla \cdot \vec B+\vec B \cdot \nabla A_i$$