How to generalize the various vector calculus theorems to distributions?

Question

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate integrals, however, sometimes the function may has singularities in the domain of integration (for example $\frac{1}{r}$ has 0 as a singularity), and the div of such a function contains a Dirac delta function (e.g. $\mathop{\mathrm{div}}\left(\frac{𝑟}{|𝑟|^3}\right)=4\pi 𝛿(𝑥)$ ), and in this case we must regard the div of this function as a distribution (a linear functional on the vector space of all smooth real-valued functions with compact support). My question is the following: do the identities of vector calculus still hold? And if so, what is the meaning of curl, div in this case, do they mean distribution?

Answer 1

This may be borderline for this site, but here goes: What you are looking for may be the notion of weak derivative. A function $f$ defined on some $d$-dimensional set $\Omega$ has a weak partial derivative with respect to the $i$-th coordinate if there exists a function $g_i$ such that for all smooth functions $\phi$ with compact support in $\Omega$ it holds that $$ \int_\Omega f(x)\partial_i \phi(x) dx = -\int_\Omega g_i(x)\phi(x)dx. $$ This looks very much like the distributional derivative and in fact one can say that $f$ has weak partial derivatives if the respective distributional derivatives can be identified with locally integrable functions.

Similarly, you can define other differential operators: A locally integrable (vector-valued) function $\vec g$ is a weak gradient of $f$ if for all vector valued smooth functions $\vec \phi$ with compact support it holds that $$ \int_\Omega f(x)\operatorname{div}\vec\phi(x) dx = -\int_\Omega \vec g(x)\cdot\vec\phi(x)dx $$ and so on.

However, this is still not satisfying, since you need a certain notion of integral over the boundary of merely integrable functions. The theory you are looking for is the theory of Sobolev spaces and their traces. For the boundary integrals you may even want to consider integrals with respect to the Hausdorff measure. One reference is the book "Geometric integration theory" by Krantz and Parks.

Answer 2

This is an addition to Dirk's answer. It is of course impossible to give an all-embracing answer to your question—there are so many identities. But the general answer is a resounding YES. The proofs are usually very simple, using a density argument—distributions can be approximated by smooth functions in their natural topologies so one can take limits of the classical versions for smooth functions. By the way this means that one can avoid some of the complications involved in using minimal smoothness assumptions.

There are, of course, some caveats (already hinted at above).

1. If products are involved there can be difficulties. However, all is not lost—avoid the common fallacy that the fact that one cannot always multiply distributions means that one can NEVER multiply them. Usually one can! There are many situations where one can a rigorous and elementary definition—for starters the product of a smooth scalar function and distributional vector field (or vice versa) is well-defined and the usual rules (product formula, etc.) hold.

2. The question of integration (e.g., in the Stokes theorem). Distributions, say on the line, always have primitives but not necessarily definite integrals. Again this doesn’t mean that they NEVER have—again they “usually do in practice” and have the sort of properties you would expect from the classical case. These concepts were developed decades ago using elementary methods (no functional analysis or duality theory for locally convex spaces. Not that I have anything against the latter—they just don’t usually occur in the toolbox of many mathematicians who want to use distributions).

3. I could go on and on about this but I will give two further remarks: Evaluation at points: it is not true that every distribution has a value at every point but again this does NOT mean that one can NEVER evaluate a distribution at a point. Again this concept was investigated in detail (in the 50’s and 60’s of the last century) using elementary methods and most distributions of practical relevance can be evaluated at most points. Composition with functions. Again symbols like $T\circ f$, i.e., the substitution of a function in a distribution, are not always well-defined. However, there are general situations where it can be rigorously defined and then it has the familiar properties.

The theory of distributions was developed by Sobolev and Schwartz, the latter using the language and techniques of duality for locally convex spaces. A number of mathematicians were then motivated to develop the theory from a more direct and elementary standpoint (i.e., at the level of an advanced calculus, or in germanophone countries “Analysis”, course). However, most texts use the Schwartz approach and the latter seems to have slipped into oblivion.

All these approaches are on record and easily available online (some names: Mikusinski, Heinz König, Sebastião e Silva, Sikorski). I would be happy to supply more details.