One of the basic properties of the Laplacian is that given a compact Riemannian manifold-with-boundary (to which all this business applies equally), then for vanishing on the boundary, the inner product is fairly large relative to . As an immediate corollary, if satisfies the Laplace equation and vanishes on the boundary, then is identically zero.

It turns out that the proof of this will require the divergence theorem. This is a familiar fact from multivariable calculus, but it generalizes to -dimensions nicely as a corollary of Stokes theorem and some of the other machinery thus developed.

So, let’s choose an oriented Riemannian manifold of dimension with boundary . There is a volume form because of the choice of orientation globally defined. On , there is an induced Riemannian metric and an induced orientation, with a corresponding volume form on . If is a compactly supported vector field, the **divergence theorem** states that

where is the outward-pointing normal on . The key idea is to use Cartan’s magic formula; since , Stokes’ theorem implies

Now can be seen to be by definition of the volume forms. Indeed, if is an oriented basis of for , then is an oriented basis for . This proves the divergence theorem.

A simple corollary of the divergence theorem occurs when we replace by , and use the identity proved earlier; we find

However, since we are interested in the Laplacian, we naturally enough try . In which case, since :

where is the **outward normal derivative**. Why is this interesting? Let’s consider the case when vanish on . Let’s also consider the vector spaces defined in the following way. is the vector space of smooth functions supported outside , and is the vector space of smooth vector fields supported outside . These are inner product spaces because of the volume element. For instance, given , define . We have operators:

The identity above says that

In particular . What will ultimately turn out is that the right-hand-side of this will be approximately the Sobolev norm of squared, while the left side will be bounded by the Sobolev norm of times the Sobolev norm of . So in particular, will become a bounded-*below* operator between suitable Sobolev spaces. More on this later.

As another consequence, we find **Green’s identity**

One can apply this to prove the mean-value-property for harmonic functions on open regions of . Given harmonic on a region containing the closure of , choose , and apply Green’s identity to an annuluar region around . I will cover this in detail at some point in the future.

The main final point I want to make here is that the use of an orientation on is actually irrelevant. As we saw, is defined without regards to an orientation, as is . The point is that we can, on any Riemannian manifold, define a **volume element**

this is a map from with appropriate homogeneity properties. The point is that, in local coordinates, this volume element transforms kind of like an -form, but with an absolute value sign being added to the determinant. In particular, it makes sense to integrate with respect to a volume element.

So to prove the divergence theorem and its corollaries here for an unoriented manifold, note that the claims are local (since one can use a partition of unity), and choose an orientation locally. Then piece everything together.

January 11, 2010 at 6:53 pm

Hi

Just wondering, once you save your TeX file and the latex2wp python file in the same directory, how do you convert it into an html file?

Thanks

January 11, 2010 at 7:00 pm

python latex2wp.py sampletexfile.tex

should do the trick.

January 11, 2010 at 7:03 pm

Where would you type that?

Thanks

January 11, 2010 at 7:13 pm

I just type it in the command window.

January 11, 2010 at 7:20 pm

I get the error that ‘python’ is not recognized an an internal or external command

January 11, 2010 at 7:42 pm

Have you installed python on your computer?

January 11, 2010 at 7:48 pm

yes

I am typing this on cmd command prompt.

January 11, 2010 at 7:50 pm

you are not supposed to type it in the python command line box right?

January 11, 2010 at 7:52 pm

You might need to prefix “python” with the full directory where python.exe is stored.

I’ve never used the python command line box though.