Exercise 1904Calculus › Multivariable calculus › Vector calculusL1Added 3 Nov 2026

The divergence theorem

#divergence theorem#flux#surface

Statement

Compute the flux of $\vec F=(x,y,z)$ through the sphere $S$ of radius $R$ centered at the origin, oriented outward.

Hint

$\iint_S\vec F\cdot\vec n\,dS=\iiint_V\operatorname{div}\vec F\,dV$ .

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Divergence. $\operatorname{div}\vec F=1+1+1=3$ .

Theorem. Flux $=3\cdot\text{Vol}(V)=3\cdot\frac43\pi R^3=4\pi R^3$ .

Conclusion. Flux $=4\pi R^3$ .