# Corallary of Divergence Theorem. Can we do this always? $\oint_S T\cdot\hat n\,dS=\int_V \nabla T\,dV$

Divergence Theorem $$\oint_S \vec F\cdot\hat n\,dS=\int_V \nabla \cdot \vec F\,dV$$

We want to prove $$\oint_S T\cdot\hat n\,dS=\int_V \nabla T\,dV.$$

In the proof we take $$\vec F= T\hat c$$ and we got $$\nabla \cdot \vec F=\hat c\cdot\nabla T$$

Question: How can we be sure that $$\vec F= T\hat c$$ is always true ? What is the mathematical reasoning?

• What does the proof say that $\hat{c}$ is? Doesn't it say something like "Let $\hat{c}$ be a constant vector field and set $\vec{F} = T \hat{c}$? – md2perpe Nov 8 '18 at 20:55