Why is the curl the limiting value of circulation density?

Question

The circulation of vector field around some region is the line integral of the field over closed path surrounding that region so why do we need the curl at all if we have a quantity that represents the amount of circulation and why the curl is the circulation density and not the circulation?

Answer 1

Given a vector field ${\bf F}$ in the plane ${\mathbb R}^2$ the circulation is a function $$\Phi_{\bf F}(\gamma):=\int_\gamma{\bf F}({\bf z})\cdot d{\bf z}$$ that produces a scalar value for each closed curve $\gamma\subset{\mathbb R}^2$, in particular for all boundary curves $\gamma:=\partial B$ of (reasonable) domains $B\subset{\mathbb R}^2$. Note that the set of all such curves is very huge, and a priori it would be difficult to make general statements about this function $\Phi$. But this function is important: It measures the nonconservativity of the field ${\bf F}$. A conservative field ${\bf F}$ (meaning ${\bf F}=\nabla f$) has $\Phi_{\bf F}(\gamma)=0$ for all closed curves $\gamma$.

It is a "wonder of mathematics" that the nonconservativity of ${\bf F}$ can be captured in a simple point-function $${\rm curl}\,{\bf F}:\quad{\mathbb R}^2\to{\mathbb R}, \qquad {\bf z}\mapsto{\rm curl}\,{\bf F}({\bf z})$$ in such a way that the circulation of ${\bf F}$ around the boundary $\partial B$ of a domain $B$ appears as integral of the density ${\rm curl}\,{\bf F}$ over $B$: $$\Phi_{\bf F}(\partial B):=\int_B{\rm curl}\,{\bf F}({\bf z})\>{\rm d}({\bf z})\ ,$$ whereby ${\rm d}({\bf z})$ denotes the area element at ${\bf z}=(x,y)$. This means that the nonconservativity of ${\bf F}$ is in a precise way localized to the points of the domain of ${\bf F}$.