How to take care of boundary term of integration by parts?

Question

Let $f:[a,b]\to \mathbb{R}$ be a smooth positive function and let $\Omega = \{(x,y)\in\mathbb{R}^2\big|x\in[a,b],0\le y\le f(x)\}$. Consider a smooth function $F:\mathbb{R}^2\to\mathbb{R}^2$. Use integration by parts to show that $$\int_{\Omega}\nabla\cdot Fd\mathbf{x}=\int_{\partial\Omega}F(s)\cdot\mathbf{n}(s)ds$$

I know that on left hand: $$\int_{\Omega}\nabla\cdot Fd\mathbf{x} = \int_{\Omega}(\frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y})dxdy$$ And $$\int_{\Omega}\frac{\partial F_2}{\partial y}dxdy=\int_a^b\int_0^{f(x)}\frac{\partial F_2}{\partial y}dydx = \int_a^b(F_2(x,f(x))-F_2(x,0))dx$$ And I want to do the same to $\int_{\Omega}\frac{\partial F_1}{\partial x}dxdy$, but I just don't know how to take care of the boundary term on the integral. Any idea?

On the left in the first two lines it should be $dA$ or something like that as you are integrating over an area. — Ross Millikan, Apr 27 '19 at 14:32

score 0 · Accepted Answer · answered May 01 '19 at 15:07

I found out that Green's Theorem is okay to use, since the theorem itself can be obtained by integration by parts. So if we want to use integration by parts to obtain the equality, we might have to construct the integration on three types of domains as in the proof of Green's theorem.

How to take care of boundary term of integration by parts?

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