   Chapter 16.9, Problem 8E

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.8. F(x, y, z) = (x3 + y3) i + (y3 + z3) j + (z3 + x3) k, S is the sphere with center the origin and radius 2

To determine

To calculate: The flux of vector field F(x,y,z)=(x3+y3)i+(y3+z3)j+(z3+x3)k across the surface S .

Explanation

Given data:

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

The surface S is the sphere with center the origin and radius 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 expression for spherical coordinate system.

Ef(x,y,z)dV=ρ1ρ2ϕ1ϕ2θ1θ2ρ2sinϕf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)dρdϕdθ (3)

Here,

ρ is the radius of the sphere.

Calculation of divF :

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

divF=x(x3+y3)+y(y3+z3)+z(z3+x3)=(3x2+0)+(3y2+0)+(3z2+0)=3x2+3y2+3z2=3(x2+y2+z2)

Calculation of flux of vector field:

Substitute 3(x2+y2+z2) for divF in equation (1),

SFdS=3E(x2+y2+z2)dV (4)

Parameterize the sphere with radius ρ as follows.

x=ρsinϕcosθ,y=ρsinϕsinθ,z=ρcosϕ,0ρ2,0ϕπ,0θ2π

Substitute ρsinϕcosθ for x , ρsinϕsinθ for y , and ρcosϕ for z in equation (4),

SFdS=3E[(ρsinϕcosθ)2+(ρsinϕsinθ)2+(ρcosϕ)2]dV=3

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