I was reading a book about Sobolev Spaces and to prove Grene's Theorem for weak derivatives they have used the following statement of Green's Theorem:
Let $\omega$ be an bounded open subset of $\mathbb R^n$. Let $u,v$ be infinitely many times differentiable functions on $\omega$. Then
$
\int_{\omega}u\displaystyle \frac{\partial v}{\partial x_i}dx=-\int_{\omega}\displaystyle \frac{\partial u}{\partial x_i}v dx+\int_{\partial\omega}uv\nu_id\sigma{(x)} \\$
The book says it is the classic Green's Theorem. I have searched on Google to find this form of Green's Theorem but I could not. Can any one provide some reference to understand this form?
Best Answer
First, use the product rule $u\frac{\partial v}{\partial x^i}=-\frac{\partial u}{\partial x^i}v + \frac{\partial (uv)}{\partial x^i}$. Now, integrate over the region $\omega$, and for the second term, use the Green's theorem/Divergence theorem/Gauss' divergence theorem, as stated in Proof of Stokes' theorem for differentiable manifolds, to transform it into an integral over the boundary.
If you're more familiar with the divergence theorem stated as $\int_{\Omega}\text{div}(\mathbf{F})\,dV=\int_{\partial \Omega}\mathbf{F}\cdot\mathbf{n}\,d\sigma$, then the version in the link follows from this version by applying this to the vector field $\mathbf{F}= u\,\mathbf{e}_i= (0,\dots, u,\dots, 0)$, with the function $u$ in the $i^{th}$ spot.