# Proove

• May 6th 2009, 08:36 AM
igodspeed
Proove
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 PM
NonCommAlg
Quote:

Originally Posted by igodspeed

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

you need to change to the generalized spherical coordinates, i.e. put $x_1=r \cos \phi_1, \ x_j=r \sin \phi_1 \sin \phi_2 \cdots \sin \phi_{j-1} \cos \phi_j, \ \ 2 \leq j \leq n-2,$ and

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

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 PM
igodspeed
see we did not learn the method that you are referring to, is there another way to solve it?
• May 7th 2009, 07:36 AM
NonCommAlg
Quote:

Originally Posted by igodspeed

see we did not learn the method that you are referring to, is there another way to solve it?

there's also an inductive solution. see here.
• May 7th 2009, 01:03 PM
Moo
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 $\mathbb{R}^n$, integration by parts, reduction formula for integrals, and being brave (Rofl)
• May 7th 2009, 10:59 PM
mr fantastic
Quote:

Originally Posted by igodspeed
I have been working on this for a long time but with no results Could someone please solve this for me.
Thanks
KC

Thread closed due to OP's habit of deleting posts after getting replies.