Integration by Parts in Several Variables

  • Thread starter amcavoy
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  • #1
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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)?

[tex]\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[/tex]

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

Thanks for your help.
 

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  • #2
Dr Transport
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Gauss's theorem states

[tex] \int\int\int_{V} \vec{\bigtriangledown} \cdot \vec{F} d \tau = \oint_{S} \vec{F}\bullet\textbf{n}dS [/tex]

substitute [tex] F = fg [/tex] anad do the algebra
 
  • #3
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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 :smile:

[tex]\iint\limits_S\mathbf{F}\cdot d\mathbf{S}=\iiint\limits_V\operatorname{div}\mathbf{F}dV[/tex]

Thanks for the help.
 
Last edited:
  • #4
TD
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apmcavoy said:
Ok, by Gauss's Theorem do you mean the Divergence Theorem?
That is correct :smile:
 

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