Re: fourier transform in higher dimensions
- From: Stephen Montgomery-Smith <stephen@xxxxxxxxxxxxxxxxx>
- Date: Sun, 02 Apr 2006 17:52:04 GMT
iredshift@xxxxxxxxx wrote:
Hi,
I am studying transform methods for solving pdes and I am having
trouble seeing how transform properties for one variable generalize to
higher dimensions. For instance, i can show that:
F( u'(x) ) = (i*w)*F( u(x) )
F( u''(x) ) = (i*w)^2*F( u(x) )
or something similar, depending on how you define the fourier
transform. I can also see that in multiple dimensions:
F( laplacian(u(x1,...,xn)) ) = (I*w)^2*F( u(x1,..,xn) )
since the laplacian just gives a scalar for u: R^n->R and the transform
is linear
But what about vector functions? For instance,
is F( Grad(u(x1,..,xn)) ) = (i*w)*F( u(x1,..,xn) ) ?
If so, how can I see that this indeed should be the case.
Thank you for your help!
The Fourier transform of u(x1,...,xn) is a function of w1,...,wn. Thus
F(Grad u) = I (w1,...,wn) F(u)
and
F(Lap u) = ((Iw1)^2+...(Iwn)^2) F(u).
.
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