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 *dΣ*, 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 *nC _{n}*. 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

**x**≠

**0**; 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

*g*‘(

_{n}*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**)=

*G*(

_{n}**x**–

**y**) is a Green’s function for the

*n*-dimensional Laplacian. This function

*G*(

_{n}**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*=2π

^{2}, so .

Tags: Central Field, Gamma Function, Green's Function, Math, Monday Math, n-ball, n-dimensional, n-sphere, Newtonian Kernel

September 13, 2010 at 12:17 am |

[…] 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 […]