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) = (cos(z) + xy2) i + xe−z j + (sin(y) + x2z) k, S is the surface of the solid bounded by the paraboloid z = x2 + y2 and the plane z = 4.

Answer 1

The divergence of the vector field $\vec F$ is

$\nabla\cdot\vec F=y^2+0+x^2=x^2+y^2$

By the divergence theorem,

$\displaystyle\iint_S\vec F\cdot\mathrm d\vec S=\iiint_V(x^2+y^2)\,\mathrm dV$

where $V$ denotes the region with boundary $S$. Convert to cylindrical coordinates:

$x=u\cos v$

$y=u\sin v$

$z=z$

The integral is then

$\displaystyle\int_0^{2\pi}\int_0^2\int_{u^2}^4u^3\,\mathrm du\,\mathrm dv=\frac{32\pi}3$