   Chapter 16.9, Problem 7E

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.7. F(x, y, z) = 3xy2 i + xez j+ z3 k, S is the surface of the solid bounded by the cylinder y2 + z2 = 1 and the planes x = -1 and x = 2

To determine

To calculate: The flux of vector field F(x,y,z)=3xy2i+xezj+z3k across the surface of the solid S, which is bounded by the cylinder y2+z2=1 , and the planes x=1 and x=2 .

Explanation

Given data:

The vector field is F(x,y,z)=3xy2i+xezj+z3k .

The surface of solid S is bounded by the cylinder y2+z2=1 , and the planes x=1 and x=2 .

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 cylinder as follows.

y2+z2=1

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

y=rcosθ,z=rsinθ,0r1,0θ2π

As the surface S is bounded by the cylinder y2+z2=1 , and the planes x=1 and x=2 , the limits of x, r, and θ are written as follows.

1x20r10θ2π

Calculation of divF :

Substitute 3xy2 for P , xez for Q , and z3 for R in equation (2),

divF=x(3xy2)+y(xez)+z(z3)=3y2x(x)+xezy(1)+3z2=3y2(1)+xez(0)+3z2=3(y2+z2)

Calculation of flux of vector field:

Substitute 3(y2+z2) for divF in equation (1),

SFdS=E3(y2+z2)dV

Apply the limits of bounded region of surface and substitute rcosθ for y , rsinθ for z , and rewrite the expression as follows

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