Chapter 15.6, Problem 29ES

Let $x,y,\text{and}z$ be measured in meters, and suppose that $\text{F(}x,y,z)=2x\text{i}-3y\text{j}+z\text{k}$ is the velocity vector (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 portion of the plain $x+y+z=1$ in the first octant in 1 s.

(b) Assuming that the fluid has a mass density of 806 ${\text{kg/m}}^{\text{3}},$ find the net mass of fluid that passes in the upward direction through the surface in part (a) in 1s.