Question

Use the Divergence Theorem to calculate the surface integral S F · dS; that is, calculate the flux of F across S. F(x, y, z) = (2x3 + y3)i + (y3 + z3)j + 3y2zk, S is the surface of the solid bounded by the paraboloid z = 1 − x2 − y2 and the xy-plane.

Answer 1

$\vec F(x,y,z)=(2x^3+y^3)\,\vec\imath+(y^3+z^3)\,\vec\jmath+3y^2z\,\vec k$

has divergence

$\mathrm{div}\vec F(x,y,z)=6x^2+3y^2+3y^2=6(x^2+y^2)$

Then by the divergence theorem, the flux of $\vec F$ across $S$ is equal to the integral of $\mathrm{div}\vec F$ over the interior of $S$. In cylindrical coordinates, this integral is

$\displaystyle6\int_0^{2\pi}\int_0^1\int_0^{1-r^2}r^3\,\mathrm dz\,\mathrm dr\,\mathrm d\theta=\boxed{\pi}$