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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

Use the Divergence Theorem to calculate the surface integral ∫∫s F · dS; that is, calculate the flux of F across S.

12. F(x, y, z) = (xy + 2xz) i + (x2 + y2) j + (xy - z2) k, S is the surface of the solid bounded by the cylinder x2 + y2 = 4 and the planes z = y - 2 and z = 0

To determine

To calculate: The flux of vector field F(x,y,z)=(xy+2xz)i+(x2+y2)j+(xyz2)k across the surface of the solid S which is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 .

Explanation

Given data:

The vector field is F(x,y,z)=(xy+2xz)i+(x2+y2)j+(xyz2)k .

The surface of solid S is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 .

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.

x2+y2=4

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

x=rcosθ,y=rsinθ,0r2,0θ2π

As the surface S is bounded by the cylinder x2+y2=4 , and the planes z=y2 and z=0 , the limits of z, r, and θ are written as follows.

y2z00r20θ2π

Calculation of divF :

Substitute (xy+2xz) for P , (x2+y2) for Q , and (xyz2) for R in equation (2),

divF=x(xy+2xz)+y(x2+y2)+z(xyz2)=(y+2z)+(0+2y)+(02z)=y+2z+2y2z=3y

Calculation of flux of vector field:

Substitute 3y for divF in equation (1),

SFdS=E(3y)dV

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

SFdS=02π02rsinθ20(3rsinθ)dz(rdr)dθ=02π023r2sinθdrdθrsinθ20(1)dz=02π023r2sinθdrdθ[z]rsinθ20=02π023r2sinθdrdθ[0(rsinθ2)]

Simplify the expression as follows

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Chapter 16 Solutions

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