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I am a little confused on how to apply a change of variables to a surface integral. If I have $ \int_\Sigma F\cdot N dS $, and a nice map to another surface, say $f$, do I apply the change of variables as $\int_{f^{-1}(\Sigma)}F\cdot N \ |J(f^{-1})|\ dS$, with $J(f)$ the Jacobian?

The particular problem is: $\Sigma = \{ (x,y,g(x,y))\in R^3 : x^2+y^2 \leq 1 \}$ where $g$ is $C^2 $ and the graph has the property that every ray from the origin in $R^3$ intersects $\Sigma$ at most once. We are given that $\Sigma$ is contained in some ball of radius R centered at the origin, and need to relate the integral $\int_\Sigma \nabla f \cdot N dS$, where $f=\frac{1}{\|x\|}$, to a formula involving R and the area of the image of $\Sigma$ under the projection to the boundary of the R-ball.

I notice that $\nabla f \cdot N$ becomes the constant $-\frac{1}{R^2}$, so can I say that the previous integral is equal to $-\frac{1}{R^2}Area(E)$, where $E$ is the area of the projection of $\Sigma$ to the surface of the ball? I know this is not quite right, but I have to be on the right track. Again, I am not confortable applying change of variables to a general surface integral, and any help would be appreciated. Thanks!

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1 Answer 1

The Jacobian is only needed because we need to work with the volume/area/length element. So if you know $$\int_{\sum}(F_{x}N_{x}+F_{y}N_{y}+F_{z}N_{z})ds$$ where $ds=|r_{x}\times r_{y}|dxdy$. Now when we change the area from $\sum$ to $f^{-1}(\sum)$, the function does not really change, but we need to change the domain of integration from $\sum$ to $f^{-1}\sum$ and the surface element from $|r_{x}\times r_{y}|dxdy$ to $|f^{-1}(r)_{u}\times f^{-1}(r)_{v}|dudv$. Here we know $x=f(u,v)_{x},y=f(u,v)_{y}$. So by chain rule you have $$\frac{d}{du}f^{-1}(r)=\frac{df^{-1}}{dx}\frac{dx}{du}+\frac{df^{-1}}{dy}\frac{dy}{du},\frac{d}{dv}f^{-1}(r)=\frac{df^{-1}}{dx}\frac{dx}{dv}+\frac{df^{-1}}{dy}\frac{dy}{dv}$$and multiplying this out should give you the desired formula.

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