Use the Divergence Theorem to evaluate the following integral S F · N dS and find the outward flux of F through the surface of the solid bounded by the graphs of the equations. Use a computer algebra system to verify your results. F(x, y, z) = 2(x + y + z) S: z = 0, z = 4 − x2 − y2

Result;

Step-by-step explanation:

Where:

F(x, y, z) = 2(x·i +y·j +z·k) and

S: z = 0, z = 4 -x² - y²

For the solid region between the paraboloid

z = 4 - x² - y²

div F        

For S: z = 0, z = 4 -x² - y²

We have the equation of a parabola

To verify the result for F(x, y, z) = 2(x·i +y·j +z·k)

We have for the surface S₁ the outward normal is N₁ = -k and the outward normal for surface S₂ is N₂ given by

Solving we have;

Plugging the values for N₁ and N₂, we have

Where:

F(x, y, z) = 2(xi +yj +zk) we have

= 32π.