We have also the same notions of derivation, curl, etc... for functions that are less regular. When you write Maxwell's equations, you are writing a system of partial differential equations.

To investigate them, you have to specify the type of solution you look for (in the language of PDEs: classic, mild, weak...) and the functional space you set your theory in. A natural space for the electric and magnetic fields is $L^2(\mathbb{R}^3)$, because this is the energy space (where the energy $\int_{\mathbb{R}^3}(E(x)^2+B(x)^2)dx$ is defined). Also more regular subspaces, such as the Sobolev spaces with *positive* index, or bigger spaces as the Sobolev spaces with *negative* index are often considered.

These spaces rely on the concept of almost everywhere, i.e. they can behave badly, but only in a set of points with zero measure. Also, the Sobolev spaces generalize, roughly speaking, the concept of derivative. I suggest you take a look at some introductory course in PDEs and functional spaces. A standard reference may be the book by Evans, or also the monumental work by HÃ¶rmander.

*Comment to the edit*: it is **not true** that

the maxwell equations in differential form, will always give a nicely behaved continuous and differentiable vector field solution

Consider, e.g. the static equation
\begin{equation*}
\nabla\cdot E=\rho \; .
\end{equation*}
To investigate this equation, you have to give it a precise meaning. What are $E$ and $\rho$? Let's assume, as you said, that $\rho$ is some discontinuous function. Then it is quite strange to look for solutions of $E$ that are smooth and well behaved! We have mathematical objects that can behave even worse than discontinuous functions, and are called distributions. In particular, we are interested in the distributions dual to functions of rapid decrease, that are called $\mathscr{S}'(\mathbb{R}^3)$. Without entering into details, all functions in $L^p(\mathbb{R}^3)$, $1\leq p \leq \infty$ are distributions in $\mathscr{S}'$, as well as Dirac's delta function and its derivatives. And mathematically, **it is perfectly legitimate to look at the divergence equation above in the sense of distributions**: i.e. to search a distribution $E\in(\mathscr{S}'(\mathbb{R}^3))^3$ such that its distributional divergence $\nabla\cdot E \in \mathscr{S}'(\mathbb{R}^3)$ is equal to $\rho\in\mathscr{S}'(\mathbb{R}^3)$. Suppose that equation admits a solution, then this solution would not, in general, be a regular function, but a **distribution**. It may be, for example, a discontinuous function in $L^1$, or a sum of derivatives of the delta function.

Anyways, as I already wrote, it is necessary that you understand better the concept of **Cauchy and boundary value problems for PDEs in functional spaces**, and also the concept of **classical, mild and weak solutions** to understand fully the machinery behind Maxwell's equations, and the mathematical meaning of a solution for such a problem.

You were pretty close already. There is a handy table on Wikipedia for a variety of coordinate systems. But for the polar system:

$$ \vec{\nabla} \cdot \vec{U} = \frac{\partial U_r}{\partial r} + \frac{1}{r} \frac{\partial U_\theta}{\partial \theta} $$

and you can look up the curl in the same table.

These can be derived from the Cartesian definitions by considering the total differentials:

$$ dr = \frac{\partial r}{\partial x} dx + \frac{\partial r}{\partial y} dy$$

and

$$ d\theta = \frac{\partial \theta}{\partial x} dx + \frac{\partial \theta}{\partial y} dy$$

Or to put it another way, you end up with:

$$ \frac{\partial \phi}{\partial r} = \frac{\partial \phi}{\partial x}\frac{\partial x}{\partial r} + \frac{\partial \phi}{\partial y}\frac{\partial y}{\partial r}$$

which is just the chain-rule for derivatives. $\phi$ is any variable. A similar expression can be written for the $\theta$ coordinate. You can use this to go between any coordinate systems you would like to go between and relate it all back to the Cartesian system by successive chaining.

As for a physical insight, that's a little bit harder. But the way I like to look at it for the polar (aka cylindrical) system is that the unit length of $\theta$ changes as $r$ changes. If you draw a big circle vs a small circle, the arc length of the wedge defined by an angle $\theta$ is certainly different. And so it is natural that the derivative in the $\theta$ direction will have some radial dependence.

## Best Answer

Curl is a measure of the rate at which a(n infinitesimally small) region of fluid rotates about its own centre. You might measure it by inserting a (very) small paddlewheel in the fluid - the speed at which it rotates is the curl. For example, on a fairground Ferris wheel, the big wheel rotates (non-zero curl) the gondolas gyrate (zero curl). Swirl some beer in a glass; the beer rotates (non-zero curl), the glass gyrates (zero curl).

BTW, 'divergence' is flux over a closed curve/surface. If the curve/surface doesn't enclose a source or sink, and the fluid is incompressible, the divergence is zero. Essentially 'what goes in, comes out'