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I want to change a integral over the ball $B(x,r)\in\mathbb{R}^n$ (with center in $x\in\mathbb{R}^n$ and radius r>0) to the unit ball $B(0,1)$.

The integral is: $\displaystyle\int_{\partial B(x,r)}u\;dS(y)$

Let $y:=x+rz$- The jacobian of this transformation is: $r^n$.

In addition,

$y\in\partial B(x,r)\Leftrightarrow |x-y|=r\Leftrightarrow |x-(x+(x+rz))|=r\Leftrightarrow |z|=1\Leftrightarrow z\in\partial B(0,1)$.

With this:

$\boxed{\displaystyle\int_{\partial B(x,r)}u\;dS(y)=r^n\int_{\partial B(0,1)}u(x+rz)\;dS(z)}$

Is this ok?

Thanks!

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P.D. (edit 1)

I calculated the jacobian as follow:

$y=x+rz\Leftrightarrow\left\{\begin{array} yy_1=x_1+rz_1\\ y_2=x_2+rz_2\\ \vdots \\ y_n=x_n+rz_n\end{array}\right.$

Then, calculating the partial differentials I obtain the jacobian:

$J=\left|\begin{array}{cccc}r&&&\\ &r&&\\ \vdots&&&\\ &&&r\end{array}\right|=r^n$

What is wrong here?

yemino
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  • I think that the Jacobian should be $r^{n-1}$, since the sphere is $n-1$ dimensional. – detnvvp Oct 17 '13 at 01:41
  • thanks detnvvp. I edited explaining how to calculate the jacobian. What is wrong with that? – yemino Oct 17 '13 at 01:53
  • This only holds if your transformation goes from an open set of $\mathbb R^n$ to an open set of $\mathbb R^n$, which is not the case here. – detnvvp Oct 17 '13 at 01:55
  • please, may you explain to me how to did it, or give me a link or book related? I want to learn, – yemino Oct 17 '13 at 01:57
  • I think you can find this in any book about integration on manifolds, which is the case here. – detnvvp Oct 17 '13 at 01:58
  • I really don't know where to find this. I'm reading right now the Tromba, and I just see transformations from R^n in R^n. – yemino Oct 17 '13 at 02:29

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