Monday Math 132

Continuing from last week, let us have a central vector field in n dimensions.

Using, as with this post the geometer’s definition of the n-sphere; let us then find the flux of our vector field over the n-sphere of radius R with center at the origin. Denoting this n-sphere as Σ, with outward normal and differential element , the flux is then
.
But since Σ is an n-sphere centered on the origin, the outward normal is the unit radial vector , and so . Thus, the flux is
. But r=R for all points on Σ, so the flux is then , where M(Σ) is the n-1 dimensional measure of Σ.
We recall from this post that the n-ball of radius R has n dimensional measure , where ; and that the n-sphere of radius R has n-1 dimensional measure . Thus, we see the flux is

Next, let . Our vector field is then undefined at the origin if k is negative. Our flux through the n-sphere Σ is then . Now, note that if k=1-n, then this is constant with respect to R: the flux becomes simply .

For any point r>0, the divergence of our vector field is found, using our result from last week, to be
.
Note, however, if k=1-n, then the term differentiated is a constant, and the divergence is then zero for all r>0. Thus, in this case, for any closed n-1 dimensional “surface” not enclosing the origin, so that the divergence theorem is valid, the flux is thus zero. Similarly, we combine this with the above result about the n-sphere Σ to see that if a closed n-1 dimensional “surface” encloses the origin, the flux through it is nCn. Thus, if we want to extend the validity of the divergence theorem as applied to F for all regions Ω of n-dimensional space, we need
,
with for x0; thus, we must have the divergence of F take the form of an n-dimensional Dirac delta distribution:
, and so
the function obeys
.
Now, we recall that there is a scalar function such that ; so long as gn‘(r)=f(r). Then, we see that
becomes
.
Thanks to the translation symmetry of the Laplacian, we see then that
satisfies

and so G(x,y)=Gn(xy) is a Green’s function for the n-dimensional Laplacian. This function Gn(x) is also sometimes called the Newtonian kernel. Let us now examine it for specific numbers of dimensions.
For n=1, nC_n=2, , so ; and since the distance from the origin in one-dimension is , we have .
For n=2, nC_n=2π, , so , and so .
For n≥3, has n-1>1, so it easily integrates to find , and so . Specifically, for n=3, nC_n=4π, so
(So is a Green’s function for the three-dimensional Laplacian and the only one which goes to zero as , due to the uniqueness theorem).
For n=4, nC_n=2π2, so .

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One Response to “Monday Math 132”

  1. Monday Math 135 « Twisted One 151's Weblog Says:

    […] irrotational term, and so our divergence condition is , Poisson’s equation. Next, we noted here, that in three-dimensions, the Green’s function for the Laplacian with our boundary […]

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