# Integration by Parts in Several Variables

My professor gave me the following formula for integration by parts in my multivariable calculus class. He said that we wouldn't find it in our book, and he didn't provide a proof. I have tried to work through it, but I am still left with one question: Why is it necessary that the curve is closed (the line integral)?

$$\int\int_{D}f(x,y)\frac{\partial g}{\partial x,y}dA=\oint_{\Sigma}f(x,y)g(x,y)\mathbf{n}\cdot d\mathbf{s}-\int\int_{D}g(x,y)\frac{\partial f}{\partial x,y}dA$$

For lack of a better notation, I used $$\frac{\partial f}{\partial x,y}$$ to represent the fact that the derivative could be with respect to either x or y.

Dr Transport
Gold Member
Gauss's theorem states

$$\int\int\int_{V} \vec{\bigtriangledown} \cdot \vec{F} d \tau = \oint_{S} \vec{F}\bullet\textbf{n}dS$$

substitute $$F = fg$$ anad do the algebra

Ok, by Gauss's Theorem do you mean the Divergence Theorem? I haven't heard of it referred to as that before and wanted to make sure they're the same $$\iint\limits_S\mathbf{F}\cdot d\mathbf{S}=\iiint\limits_V\operatorname{div}\mathbf{F}dV$$

Thanks for the help.

Last edited:
TD
Homework Helper
apmcavoy said:
Ok, by Gauss's Theorem do you mean the Divergence Theorem?
That is correct 