Exercise 1903Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Gauss divergence theorem for a cylinder

#divergence#cylinder#flux

Statement

Compute the outward flux of $\vec F=(x,y,0)$ through the cylinder $x^2+y^2\le R^2$ , $0\le z\le h$ .

Hint

$\operatorname{div}\vec F=2$ ; multiply by the volume.

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Divergence. $\operatorname{div}\vec F=1+1+0=2$ .

Volume. $\pi R^2h$ .

Flux. $2\pi R^2h$ .

Conclusion. Flux $=2\pi R^2h$ .