Exercise 1908Calculus › Multivariable calculus › Vector calculusM1Added 3 Nov 2026

Divergence theorem with variable divergence

#divergence#sphere#variable density

Statement

Compute the flux of $\vec F=(x^3,y^3,z^3)$ through the sphere of radius $R$ .

Hint

$\operatorname{div}\vec F=3(x^2+y^2+z^2)$ ; integrate in spherical.

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Divergence. $\operatorname{div}=3(x^2+y^2+z^2)=3r^2$ .

Spherical. $\iiint 3r^2\cdot r^2\sin\varphi\,dr\,d\varphi\,d\theta=3\cdot4\pi\cdot\frac{R^5}{5}$ .

Result. Flux $=\frac{12\pi R^5}{5}$ .

Conclusion. Flux $=\frac{12\pi R^5}{5}$ .