[[Fundamental theorem of calculus]]
# Stokes's theorem

Let $\Sigma$ be an oriented piecewise-smooth surface bounded by a positively oriented, piecewise smooth, simple closed curve $\partial \Sigma$.
If $\vab F$ is a vector field differentiable in $\Sigma$ then #m/thm/calculus 
$$
\oint_{\partial \Sigma}\vab F \cdot d\vab r = \iint_{\Sigma}\vab{\nabla}\times \vab F \cdot d\vab a
$$
Note that the left hand side is equivalent to the [[circulation]] integral $\Gamma$.[^2016]

[^2016]: 2016\. [[Sources/@stewartCalculus2016|Calculus]], p. 174


## Practice problems
- 2016\. [[Sources/@stewartCalculus2016|Calculus]], pp. 1179–1180 (§16.9 exercises)

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