Monday Math 129

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 \phi(\mathbf{x})=e^{-\imath\mathbf{k}\cdot\mathbf{x}}, this means
and since \mathbf{\nabla}e^{-\imath\mathbf{k}\cdot\mathbf{x}}=-\imath\mathbf{k}e^{-\imath\mathbf{k}\cdot\mathbf{x}}, 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 \mathbf{\hat{n}} 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=ψv, we see

Now, letting \psi(\mathbf{x})=e^{-\imath\mathbf{k}\cdot\mathbf{x}}, 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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4 Responses to “Monday Math 129”

  1. Monday Math 130 « Twisted One 151's Weblog Says:

    […] 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 […]

  2. m.diehl Says:

    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?

    • twistedone151 Says:

      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.

      • Martin Diehl Says:

        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

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