Re: Off Topic - Jacobian and Change of Variables
- To: mathgroup at smc.vnet.net
- Subject: [mg35881] Re: Off Topic - Jacobian and Change of Variables
- From: Selwyn Hollis <slhollis at earthlink.net>
- Date: Mon, 5 Aug 2002 06:01:52 -0400 (EDT)
- References: <aiiu4b$qrt$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Kevin,
It comes down to a fact about determinants, whose proof can be found in
any intermediate-level linear algebra text:
volume(L(R)) = |det L|* volume(R)
Use that with R = dV and L = the linearization of the integrand, i.e.
the Jacobian. By the way, a wonderful reference is ``Linear Algebra Done
Right" by Sheldon Axler (Springer). See also ``Advanced Calculus of
Several Variables" by C.H. Edwards (Dover).
----
Selwyn Hollis
slhollis at mac.com
Kevin J. McCann wrote:
> This problem has bothered me for years. In elementary calculus we change
> integration variables. For 1d this is straightforward. For 2d we make use of
> the cross product to determine transformed areas. For 3d we use the
> dot-cross product to represent a "cube" of volume. So far so good. Now when
> we have more than 3 we use the Jacobian determinant which is on "obvious"
> extension to the 3d case. I have never seen a proof of the multidimensional
> case. Can someone point me to a reference?
>
> Thanks,
>
> Kevin
>
>
>