Use the Divergence Theorem to find the flux of F across the surface sigma with outward orientation.
F(x,y,z) = 2xzi + yzj + z^2k, where sigma is the surface of the solid bounded above by z = sqrt(a^2-x^2-y^2) and below by the xy-plane.
Use the Divergence Theorem to find the flux of F across the surface sigma with outward orientation.
F(x,y,z) = 2xzi + yzj + z^2k, where sigma is the surface of the solid bounded above by z = sqrt(a^2-x^2-y^2) and below by the xy-plane.
So you want to evaluate bounded above by a sphere of radius and bounded below by the plane .
Use spherical coordinates (where is the radius, is the azimuth angle, and is the angle around the circle.
The bounds:
So you want to evaluate
Evaluating this integral should be straightforward. I have included the final answer below.
Spoiler: