Exercise 1926Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Stokes' theorem

#Stokes theorem#curl#surface

Statement

Compute $\oint_C\vec F\cdot d\vec r$ for $\vec F=(-y,x,0)$ where $C$ is the unit circle in the $z=0$ plane, counterclockwise.

Hint

$\oint=\iint_S(\nabla\times\vec F)\cdot\vec n\,dS$ ; $\nabla\times\vec F=(0,0,2)$ .

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Curl. $\nabla\times\vec F=(0,0,2)$ .

Normal. $\vec n=(0,0,1)$ .

Stokes. $\oint=2\cdot\text{Area}(S)=2\pi$ .

Conclusion. $\oint_C=2\pi$ .