   Chapter 16.9, Problem 11E

Chapter
Section
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.11. F(x, y, z) = (2x3 + y3) i + (y3 + z3)j + 3y2z k, S is the surface of the solid bounded by the paraboloid z = 1 - x2 - y2 and the xy-plane

To determine

To calculate: The flux of the vector field F(x,y,z)=(2x3+y3)i+(y3+z3)j+3y2zk across the surface of the solid S, which is bounded by the paraboloid z=1x2y2 , and the xy-plane.

Explanation

Given data:

The vector field is F(x,y,z)=(2x3+y3)i+(y3+z3)j+3y2zk .

The surface of solid S is bounded by the paraboloid z=1x2y2 , and the xy-plane.

Formula used:

Write the expression to find flux of the vector field F(x,y,z) across the surface S .

SFdS=EdivFdV (1)

Here,

E is the solid region.

Write the expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk .

divF=xP+yQ+zR (2)

Write the equation of paraboloid as follows.

z=1x2y2 (3)

From the equation of paraboloid, the parameters are considered as follows.

x=rcosθ,y=rsinθ,0r1,0θ2π

Substitute rcosθ for x and rsinθ for y in equation (3),

z=1(rcosθ)2(rsinθ)2=1r2(cos2θ+sin2θ)=1r2(1) {cos2θ+sin2θ=1}=1r2

As the surface S is bounded by the paraboloid z=1x2y2 , and the xy-plane, the limits of z, r, and θ are written as follows.

0z(1r2)0r10θ2π

Calculation of divF :

Substitute (2x3+y3) for P, (y3+z3) for Q, and 3y2z for R in equation (2),

divF=x(2x3+y3)+y(y3+z3)+z(3y2z)=(6x2+0)+(3y2+0)+3y2=6x2+6y2=6(x2+y2)

Substitute rcosθ for x and rsinθ for y ,

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