Following @needham2021visual page 262, we define first the circulation of a vector field $V$ along a closed simple loop $L$:
$$ \mathcal{C}_{\mathrm{L}}(\mathbf{V}) \equiv \oint_{\mathrm{L}} \mathbf{V} \cdot \mathrm{d} \mathbf{r}=\oint_{\mathrm{L}} \mathrm{V}_{\mathrm{L}} \mathrm{d} s=\oint_{\mathrm{L}}[\mathrm{P} \mathrm{d} u+\mathrm{Q} \mathrm{d} v] $$Then, it can be shown that
$$ \mathcal{C}_{\mathrm{L}}(\mathbf{V})\asymp \{\mathrm{Q}(\mathrm{c})-\mathrm{Q}(\mathrm{a})\} \delta v-\{\mathrm{P}(\mathrm{d})-\mathrm{P}(\mathrm{b})\} \delta \mathrm{u}\asymp $$ $$ \asymp (\partial_u Q- \partial_v P) \delta A $$where $\asymp$ means "ultimately equal".
Now, the curl is defined as the "local circulation per unit area", so
$$ curl(V)=\partial_u Q- \partial_v P $$which is the traditional definition.
Nemotecnic rule
$$ \nabla \times \mathbf{F}=\left|\begin{array}{ccc} \hat{\boldsymbol{\imath}} & \hat{\jmath} & \hat{\boldsymbol{k}} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_x & F_y & F_z \end{array}\right| $$which expands as
$$ \nabla \times \mathbf{F}=\left(\frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z}\right) \hat{\imath}+\left(\frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x}\right) \hat{\jmath}+\left(\frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y}\right) \hat{\boldsymbol{k}}=\left[\begin{array}{l} \frac{\partial F_z}{\partial y}-\frac{\partial F_y}{\partial z} \\ \frac{\partial F_x}{\partial z}-\frac{\partial F_z}{\partial x} \\ \frac{\partial F_y}{\partial x}-\frac{\partial F_x}{\partial y} \end{array}\right] $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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