I know that in physics the divergence of a vector over a volume is equal to the dot product of the vector by the normal to the surface, over the surface, ie:
$$\int_{\Omega} \nabla \cdotp \vec{u}\ d\Omega = \int_{\Gamma} u\ \cdotp n\ d\Gamma$$ In my case my vector is the gradient of a scalar function $f(\alpha)$ depending of the scalar $\alpha$ and i'm in the following case: $$\int_{\Omega} \alpha\nabla \cdotp \nabla f(\alpha)\ d\Omega$$ I've a scalar inside on my integrale, infront of my divergence. Can i write: $$\int_{\Omega} \alpha\nabla \cdotp \nabla f(\alpha)\ d\Omega = \int_{\Gamma} \alpha \nabla f(\alpha)\cdotp n\ d\Omega $$ or is this false ? Note that $\alpha$ is variable through $\Omega$. Thanks.