The definitions I learned in my calculus courses for curl and divergence were rather, at least to me, unintuitive and seemed to work only for $\mathbb{R}^3$.
I took a look on Wikipedia:
"The divergence of a vector field $F$ at a point $p$ is defined as the limit of the net flow of $F$ across the smooth boundary of a three-dimensional region $V$ divided by the volume of V as $V$ shrinks to $p$. Formally:
$$\operatorname{div}\,\mathbf{F}(p) =
\lim_{V \rightarrow \{p\}}
\iint_{S(V)} {\mathbf{F}\cdot\mathbf{n} \over |V| } \; dS$$
The curl of a vector field F, denoted by $\operatorname{curl} \, \mathbf{F}$, or $\nabla \times F$, at a point is defined in terms of its projection onto various lines through the point. If $\scriptstyle\mathbf{\hat{n}}$ is any unit vector, the projection of the curl of $F$ onto $\scriptstyle\mathbf{\hat{n}}$ is defined to be the limiting value of a closed line integral in a plane orthogonal to $\scriptstyle\mathbf{\hat{n}}$ as the path used in the integral becomes infinitesimally close to the point, divided by the area enclosed.
As such, the curl operator maps continuously differentiable functions $f : \mathbb{R}^3 \to \mathbb{R}^3$ to continuous functions $g : \mathbb{R}^3 \to \mathbb{R}^3$.
Implicitly, curl is defined by:
$$(\nabla \times \mathbf{F}) \cdot \mathbf{\hat{n}} \ \overset{\underset{\mathrm{def}}{}}{=} \lim_{A \to 0}\left( \frac{1}{|A|}\oint_{C} \mathbf{F} \cdot d\mathbf{r}\right)$$
"
To me, "intuitively", it seems that divergence at a point involves making a closed surface around the point, measuring the total flux, and dividing that by the volume of the closed surface. Even more simply put, it is the net measure of how much "flow" is going in or out.
For the curl at a point, we formulate a plane around the point, take an integral of the closed loop around the point and divide by the area enclosed by the loop. More simply, one could imagine a tiny sphere in $\mathbb{R}^3$. The curl tells us how that sphere is rotating by giving a vector. The vector tells us the direction and speed of rotation.
Does anyone perhaps have a better or more intuitive way of viewing curl and divergence? I was also wondering if there is a general definition that works for any given vector space over any field, even finite fields. But then integration and dot products and even limits might not make sense.
Best Answer
The generalization of scalar and vector fields is the differential form. The generalization of $\text{grad}$, $\text{div}$, $\text{curl}$ is the exterior differential. See the details in the section Exterior derivative in vector calculus.