## Local differential operators

In this post we define a differential operator on Euclidean space and give some familiar examples.

For , the differential operator takes a smooth function on to a smooth function on . We call the **order** of this differential operator.

A linear map is a **(linear) differential operator of order ** if it is of the form , where is an matrix over . Given , the **symbol** is an matrix over such that , where .

We say that is **elliptic** if and for all , is an isomorphism of to itself.

**Examples**

Grad: ,

We have

Div: ,

We have

.

Curl: ,

We have

.

Exterior derivative ,

We have

Now, with respect to the ordered bases and for and , respectively, we may write this as

,

so by the previous example we see that .

Laplacian : ,

We have

.

All of the examples above are order 1 except for the Laplacian, which is order 2.

**Which of the above are elliptic?**

For dimensional reasons, the only candidates are , and . The first two are not elliptic since, for example, and has determinant . is elliptic, on the other hand, since .

Hi Ash,

Grad: ,

Your examples should be rewritten, in order to fit your notation from the first paragraph. For example, in your grad example, you could write instead:

Thus grad

.

jfdavisNovember 13, 2012 at 1:17 am