Curl

The curl of a vector field \(\require{physics} \vb{F}\) in space can be defined using line integrals. The circulation of \(\vb{F}\) around curve \(C\) is \[\begin{equation} \require{physics} \int_C \vb{F} \dd{ \vb{s} } \thinspace , \end{equation}\] which is nothing more than a line integral of a vector function. To get an expression for the circulation per unit area inside \(C\), we divide the line integral by area \(A\) \[\begin{equation} \frac{1}{A} \int_C \vb{F} \dd{ \vb{s} } \thinspace . \end{equation}\] The curl of \(\vb{F}\) at a point along any unit vector \(\vb{ \hat{u} }\) is defined to be the value of a closed line integral in a plane orthogonal to \(\vb{ \hat{u} }\) in the limit of a contracted area \(A\) at that point. Formally, this is written down as \[\begin{equation} \require{physics} (\text{curl}~\vb{F}) \vdot \vb{ \hat{u} } = \lim_{A \to 0} \frac{1}{A} \int_C \vb{F} \dd{ \vb{s} } \thinspace . \end{equation}\] Since the line integral depends on the path of integration, \(C\) is defined via the right-hand rule. The above formula signifies the curl of a vector field as the infinitesimal area density of the circulation of that field.

By integrating over \(C\), we finally obtain a formula for the curl of a vector field: \[\begin{align} \text{curl}~\vb{F} &= \boldsymbol{\nabla} \times \mathbf{F} % \\ &= \qty( \pdv{}{x},\pdv{}{y},\pdv{}{z} ) % \times (F_x,F_y,F_z) % \\ &= \begin{vmatrix} \vb{i} & \vb{j} & \vb{k} \\ \pdv{}{x} & \pdv{}{y} & \pdv{}{z} \\ F_x & F_y & F_z \end{vmatrix} \thinspace , \end{align}\] where we have introduced a similar notation for \(\qty(\pdv{}{x},\pdv{}{y},\pdv{}{z})\) as in the divergence theory. The \(3 \times 3\) determinant is often used as a mnemonic device to fall back on.