I have been working on this for a long time but with no results Could someone please solve this for me.

Thanks

KC

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- May 6th 2009, 08:36 AMigodspeedProove
I have been working on this for a long time but with no results Could someone please solve this for me.

Thanks

KC - May 6th 2009, 01:41 PMNonCommAlg
you need to change to the generalized spherical coordinates, i.e. put $\displaystyle x_1=r \cos \phi_1, \ x_j=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{j-1} \cos \phi_j, \ \ 2 \leq j \leq n-2,$ and

$\displaystyle x_{n-1}=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \cos \theta, \ x_n=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{n-2} \sin \theta,$ where $\displaystyle 0 \leq \phi_j \leq \pi, \ 0 \leq \theta \leq 2 \pi,$ and $\displaystyle 0 \leq r \leq R.$

now find the Jacobian of this transformation and finally using gamma function your multiple integral would be pretty easy to evaluate. - May 6th 2009, 05:44 PMigodspeed
see we did not learn the method that you are referring to, is there another way to solve it?

- May 7th 2009, 07:36 AMNonCommAlg
there's also an inductive solution. see here.

- May 7th 2009, 01:03 PMMoo
Wow !

I finished this pdf in around 2 hours... That was long (Crying) (hey I'm still a beginner at creating pdf lol). Tell me if there is any mistake.

I hope this will help ~

Prerequisites : Integral and transformation in $\displaystyle \mathbb{R}^n$, integration by parts, reduction formula for integrals, and being brave (Rofl) - May 7th 2009, 10:59 PMmr fantastic