Chapter 15.6, Problem 30ES

Let $x,y,\text{and}z$ be measured in meters, and suppose that $\text{F(}x\text{,}\text{y,}\text{z)}=-y\text{j}+z\text{j}+\text{3}x\text{k}$ is the velocity (in m/s) of a fluid particle at the point $(x,y,z)$ in a steady-state in-compressible fluid flow.

(a) Find the net volume of fluid that passes in the upward direction through the hemisphere $z=\sqrt{9-x^2-y^2}$ is 1s.

(b) Assuming that the fluid has a mass density of 1060 ${\text{kg/m}}^{\text{3}}$ , find the net mass of fluid that passes in the upward direction through the surface in part (a) in 1s.