Curl Operator and Stokes Theorem

From HvWiki

The curl operator is often written as

$\textrm{curl}\textbf{A}\,$

$\nabla\times\textbf{A}.\,$

Sometimes, in older (usually European) texts, it is written as

$\textrm{rot}\textbf{A}. \,$

It is defined as the "circulation per unit area" of a field:

$\nabla\times\textbf{A} = \lim_{S\rightarrow 0}\frac{\oint_{\partial S} \textbf{A}\cdot d\textbf{l}}{S}$

In the usual coordinate systems, we have:

Rectangular:

$\nabla\times\textbf{A} = \textbf{a}_x\left (\frac{\partial A_z}{\partial y} - \frac{\partial A_y}{\partial z}\right ) + \textbf{a}_y \left (\frac{\partial A_x}{\partial z} - \frac{\partial A_z}{\partial x}\right ) + \textbf{a}_z \left (\frac{\partial A_y}{\partial x} - \frac{\partial A_x}{\partial y}\right ).$

This can be conveniently written as a determinant

$\nabla\times\textbf{A} = \left \vert \begin{matrix} \textbf{a}_x & \textbf{a}_y & \textbf{a}_z \\ \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ \\ A_x & A_y & A_z \end{matrix} \right \vert$

Circular cylindrical:

$\nabla\times\textbf{A} = \frac{1}{r} \left \vert \begin{matrix} \textbf{a}_r & r\textbf{a}_\phi & \textbf{a}_z \\ \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \phi} & \frac{\partial}{\partial z} \\ \\ A_r & rA_\phi & A_z \end{matrix} \right \vert$

Spherical coordinates:

$\nabla\times\textbf{A} = \frac{1}{r^2\sin\theta} \left \vert \begin{matrix} \textbf{a}_r & r\textbf{a}_\theta & r\sin\theta\textbf{a}_\phi \\ \\ \frac{\partial}{\partial r} & \frac{\partial}{\partial \theta} & \frac{\partial}{\partial \phi} \\ \\ A_r & rA_\theta & r\sin\theta A_\phi \end{matrix} \right \vert$

Orthogonal curvilinear coordinates:

$\nabla\times\textbf{A} = \frac{1}{h_1h_2h_3} \left \vert \begin{matrix} h_1\textbf{a}_1 & h_2\textbf{a}_2 & h_3\textbf{a}_3 \\ \\ \frac{\partial}{\partial q_1} & \frac{\partial}{\partial q_2} & \frac{\partial}{\partial q_3} \\ \\ h_1A_1 & h_2A_2 & h_3A_3 \end{matrix} \right \vert$

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Stokes Theorem

If we take any open orientable surface, the integral of the curl of a "smooth" vector function over that surface is equivalent to the integral of the function over a closed contour along the border of the surface.

$\int\!\!\int_S \nabla\times\textbf{A}\cdot d\textbf{S} = \oint_{\partial S} \textbf{A}\cdot d\textbf{l}$