Integration by Parts in Several Variables

[amcavoy]

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.

Thanks for your help.

 

[Dr Transport]

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

 

[amcavoy]

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.

 

[TD]

Ok, by Gauss's Theorem do you mean the Divergence Theorem?

That is correct