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[tex] \frac{1}{|\partial B(x,r)|}\int_{\partial B(x,r)}u(y)\,dS(y)=\frac{1}{|\partial B(0,1)|}\int_{\partial B(0,1)} u(x+rz)\,dS(z) [/tex]

Why does [tex]dS(y)\to dS(z)[/tex] and not [tex] dS(y)\to dS(x+rz)[/tex]?

If you want more information, it comes from http://www.stanford.edu/class/math220b/handouts/laplace.pdf on page 8, it's used to prove the mean value formula for the laplacian.

Is it because it doesn't matter what radius the surface we are integrating over is because we are taking the average?

Why does [tex]dS(y)\to dS(z)[/tex] and not [tex] dS(y)\to dS(x+rz)[/tex]?

If you want more information, it comes from http://www.stanford.edu/class/math220b/handouts/laplace.pdf on page 8, it's used to prove the mean value formula for the laplacian.

Is it because it doesn't matter what radius the surface we are integrating over is because we are taking the average?

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