Exercise 2223Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Stokes on a cone

#Stokes#cone#surface

Statement

Compute the circulation of $\vec F=(y,z,x)$ on the boundary of the cone $z=\sqrt{x^2+y^2}$ , $z\le1$ .

Hint

$\nabla\times\vec F=(-1,-1,-1)$ ; use the flat disk at $z=1$ to simplify.

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Curl. $\nabla\times\vec F=(-1,-1,-1)$ .

Use disk. $S$ : disk $x^2+y^2\le1$ at $z=1$ , $\vec n=(0,0,1)$ , with boundary circle at $z=1$ .

Stokes. $\oint=\iint_D(-1)\,dS=-\pi$ .

Conclusion. Circulation $=-\pi$ .