Question

Verify that the Divergence Theorem is true for the vector field F on the region E. Give the flux. F(x, y, z) = x2i + xyj + zk, E is the solid bounded by the paraboloid z = 4− x2 − y2 and the xy-plane.

Answer 1

Answer:

We are given the vector field F(x,y,z)=x2i+xyj+zkF(x,y,z)=x2i+xyj+zk and the solid bounded by the paraboloid z=9−x2−y2z=9−x2−y2 and the xy-plane. The divergence of FF is divF=2x+x+1=3x+1.divF=2x+x+1=3x+1. The integral on the right hand side of the divergence theorem becomes

∭QdivFdV=∬Q3x+1dV.∭QdivFdV=∬Q3x+1dV.

We will use cylindrical coordinates to evaluate this integral. The region QQ becomes 0≤θ≤2π,0≤r≤3,0≤z≤9−r20≤θ≤2π,0≤r≤3,0≤z≤9−r2 and the integral becomes

∬Q3x+1dV=∫2π0∫30∫9−r20(3rcosθ+1)rdzdrdθ=∫2π0∫30∫9−r20(3r2cosθ+r)dzdrdθ=∫2π0∫30(3r2cosθ+r)z∣∣∣9−r20drdθ=∫2π0∫30(3r2cosθ+