Curl of a vector field

In the plane

Following @needham2021visual page 262, we define first the circulation of a vector field $V$ along a closed simple loop $L$ :

C_{L} (V) \equiv \oint_{L} V \cdot d r = \oint_{L} V_{L} d s = \oint_{L} [P d u + Q d v]

Then, it can be shown that

C_{L} (V) ≍ {Q (c) - Q (a)} δ v - {P (d) - P (b)} δ u ≍

≍ (\partial_{u} Q - \partial_{v} P) δ A

where $≍$ means "ultimately equal".

Now, the curl is defined as the "local circulation per unit area", so

c u r l (V) = \partial_{u} Q - \partial_{v} P

which is the traditional definition.

Nemotecnic rule

\nabla \times F = | \begin{array}{ccc} \hat{ı} & \hat{ȷ} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{x} & F_{y} & F_{z} \end{array} |

which expands as

\nabla \times F = (\frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z}) \hat{ı} + (\frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x}) \hat{ȷ} + (\frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y}) \hat{k} = [\begin{array}{l} \frac{\partial F_{z}}{\partial y} - \frac{\partial F_{y}}{\partial z} \\ \frac{\partial F_{x}}{\partial z} - \frac{\partial F_{z}}{\partial x} \\ \frac{\partial F_{y}}{\partial x} - \frac{\partial F_{x}}{\partial y} \end{array}]

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Author of the notes: Antonio J. Pan-Collantes

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