- The boundary of a surface \(S\) is denoted \(\partial S\).
We say that \(S\) is closed if \(\partial S\) is empty.
- Suppose that \(S\) is oriented (a continuously varying unit normal
is specified at each point of \(S\)). The boundary
orientation of \(\partial S\) is defined as follows: If you walk
along the boundary in the positive direction with your head pointing
in the normal direction, then the surface is on your left.
- \begin{align*}
{\bf curl}({\bf F}) &= \left|
\begin{array}{ccc}
{\bf i}& {\bf j} & {\bf k} \\
\dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z}\\
F_1 & F_2 & F_3
\end{array}
\right| \\
&= \left(\dfrac{\partial {F_3}}{\partial{y}} - \dfrac{\partial {F_2}}{\partial{z}}\right){\bf i} -
\left(\dfrac{\partial {F_3}}{\partial{x}} - \dfrac{\partial {F_1}}{\partial{z}} \right){\bf j} +\left( \dfrac{\partial{F_2}}{\partial{x}} -
\dfrac{\partial{F_1}}{\partial{y}}\right){\bf k}
\end{align*}
Symbolically, \({\bf curl}({\bf F})=\nabla\times{\bf F}\) where \(\nabla\) is the del operator
\[
\nabla= \langle \dfrac{\partial}{\partial{x}} , \dfrac{\partial}{\partial{y}} , \dfrac{\partial}{\partial{z}} \rangle
\]
- Stokes’ Theorem relates the circulation around the boundary to the
surface integral of the curl:
\[
\oint_{\partial S} {\bf F}\cdot d{\bf s} = \iint_{S\,} {\bf curl}({\bf F})\cdot
d{\bf S}
\]
- If \({\bf F} = \nabla V \), then \({\bf curl}({\bf F})=\mathbf{0}\).
- Surface Independence:
If \({\bf F} = {\bf curl}({\bf A})\), then the flux of \({\bf F}\) through a surface
\(S\) depends only on the oriented boundary \(\partial S\) and
not on the surface itself:
\begin{equation*}
\iint_{S\,} {\bf F} \cdot d{\bf S} = \oint_{\partial
S} {\bf A} \cdot d{\bf s}
\end{equation*}
In particular, if \(S\) is closed (that is, \(\partial S\) is
empty) and \({\bf F} = {\bf curl}({\bf A})\), then
\(\displaystyle{
\iint_{S\,} {\bf F}\cdot d{\bf S} = 0}\).
- The curl is interpreted as a vector that encodes circulation per unit area: If \(P\) is any point and \({\bf e}_{\bf n}\) is a unit normal vector, then
\[
\int_{C} {\bf F}\cdot d{\bf s}\approx
({\bf curl}({\bf F})(P)\cdot{\bf e}_{\bf n})\,\textrm{Area}(D)
\]
where \(C\) is a small, simple closed curve around \(P\) in the plane through \(P\) with normal vector \({\bf e}_{\bf n}\), and \(D\)
is the enclosed region.