Continuing from last week, we now consider the Fourier transform of the divergence of a vector field in three dimensions. First, vector integration by parts tells us that for vector field **v** and scalar field *φ*

.

Letting , this means

,

and since , as before, we then have

.

Now, if **v** goes to zero sufficiently fast as *r* goes to infinity (|**v**| goes to zero faster than 1/*r*), then the surface integral will vanish as the volume is expanded to all space, and so

,

and so

.

Now, to find the Fourier transform of the curl of a vector field, we use a form of the divergence theorem which states that

,

where is the outward normal to the surface *S* bounding the volume *V*.

Now, the product rule for the curl of the product of a scalar field and a vector field is

,

so letting **P**=*ψ***v**, we see

,

so

.

Now, letting , and using

,

we get

,

and thus,

.

Again, we can expand the volume to all space, and if **v** goes to zero fast enough as *r* goes to infinity, then the surface integral vanishes, so

,

and thus

.

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Tags: Fourier Transform, Math, Monday Math

This entry was posted on August 2, 2010 at 12:16 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 9, 2010 at 1:07 am |

[…] of F(x), its divergence, and its curl. With the Fourier transform of F, then our results from last week tell us that . and . Now, the ability to decompose a vector into components parallel and […]

August 10, 2011 at 11:39 pm |

I have a question concerning the calculation of divergence in Fourier space.

I have a 3D field of 3×3 tensors in real space that I transform to Fourier space using FFT. Now I calculate the divergence by multiplying the tensor field in Fourier space with the corresponding frequency vector. The result should be the divergence in fourier space that I back transform.

My Problem is now, that I have imaginary parts in this divergence field. If I do a direct divergence calculation, using finite differences, the divergence is real only. That makes much more sense.

Any Ideas?

October 27, 2011 at 12:07 pm |

Unfortunately, I’m not familiar with how the Fourier transform extends to (rank 2 or higher) tensors. However, I would note that FFT is an algorithm for computing a discrete Fourier transform, not the continuous Fourier transform, so I’m not sure it works for computing divergence.

October 27, 2011 at 1:15 pm

Well, I found the solution.

When differentiating (e.g. calculating the divergence) you shouldn’t use the highest frequencies as they are not correctly represented using the DFT/FFT. Just set the to zero and use the formulas derive for the continuous FT