Question

Using Gauss’ theorem calculate the flux of the vector field A = x³i + y³I + z³k
through the surface of a hemisphere of radius A which has the centre of its base at the
origin.

Answer 1

The Gauss' theorem is $\iint_S(A\sdot n)ds=\iiint_V(\nabla \sdot A)dv$ as $\nabla\sdot A=3(x^2+y^2+z^2)$ then

$\iiint_V(\nabla\sdot A)dv=3\sdot4\int_{0}^{\pi/2}d\phi\int_{0}^{\pi/2}\sin\theta d\theta\int_{0}^{A}r^2dr=$

$=12\pi/2(-\cos\theta)|_0^{\pi/2}A^3/3=2\pi A^3$