Exercise 1925Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

Stokes with a hemisphere

#Stokes#hemisphere#closed curve

Statement

Compute $\oint_C\vec F\cdot d\vec r$ for $\vec F=(z,0,0)$ where $C$ is the boundary of the hemisphere $x^2+y^2+z^2=1$ , $z\ge0$ .

Hint

$\nabla\times\vec F=(0,0,-1)$ if $P=z$ , $Q=0$ , $R=0$ : $\nabla\times\vec F=(0,-P_z,Q_x)=(0,-1,0)$ — recompute. $\nabla\times(P,Q,R)=(R_y-Q_z,\,P_z-R_x,\,Q_x-P_y)$ ; here $P=z,Q=0,R=0$ so curl $=(0,1,0)$ .

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Solution

Curl. $\nabla\times\vec F=(0,1,0)$ .

Use Stokes on flat disk at $z=0$ with $\vec n=(0,0,1)$ : curl· $\vec n=0$ .

Conclusion. $\oint_C\vec F\cdot d\vec r=0$ .

Stokes with a hemisphere

Statement

Solution

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