## Monday Math 128

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=φψ, 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.