We have previously discussed the Fourier transform (here and here, especially). In this post, we noted that (using the symmetric angular convention) the space transform for an *n* dimensional space is

and the inverse is

.

We can also do the same for a vector field:

and

.

We note from vector calculus then that for a vector field, the components of the transform are the transforms of the components:

.

We also used integration by parts here to show that for a one-dimensional function *f*(*x*), with , the derivative has Fourier transform:

.

Similarly, we can use vector integration by parts for our multi-dimensional transforms. Working in three dimensions from here:

First, one form of the divergence theorem states:

,

where *S* is the boundary of the volume *V*, with outward normal.

Letting *f*=*φ**ψ*, and using the gradient product rule ,

.

Letting , we see

.

and since , we have

.

Now, if as , *φ*(**x**) goes to zero faster than , then, as we expand the volume *V* to cover all space, the surface integral will go to zero, and we obtain

,

which means

,

in analogy to our one-dimensional rule.

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Tags: Divergence Theorem, Fourier Transform, Math, Monday Math, Wavenumber

This entry was posted on July 26, 2010 at 12:14 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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August 2, 2010 at 12:16 am |

[…] Math 129 By twistedone151 Continuing from last week, we now consider the Fourier transform of the divergence of a vector field in three dimensions. […]