   Chapter 16.9, Problem 14E

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.14. F = |r|2r, where r = x i + y j+ z k, S is the sphere with radius R and center the origin

To determine

To calculate: The flux of vector field F=|r|2r across the surface S , where r=xi+yj+zk .

Explanation

Given data:

The vector field is given as follows.

F=|r|2r (1)

Here,

r=xi+yj+zk

The surface S is the sphere with radius R and centered the origin.

Formula used:

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

SFdS=EdivFdV (2)

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 (3)

Write the expression for spherical coordinate system.

Ef(x,y,z)dV=ρ1ρ2ϕ1ϕ2θ1θ2ρ2sinϕf(ρsinϕcosθ,ρsinϕsinθ,ρcosϕ)dρdϕdθ (4)

Here,

ρ is the radius of the sphere.

Calculation of vector field F :

Substitute xi+yj+zk for r in equation (1),

F=|xi+yj+zk|2(xi+yj+zk)=(x2+y2+z2)2(xi+yj+zk)=(x2+y2+z2)(xi+yj+zk)=x(x2+y2+z2)i+y(x2+y2+z2)j+z(x2+y2+z2)k

F=(x3+xy2+xz2)i+(x2y+y3+z2y)j+(x2z+y2z+z3)k

Calculation of divF :

Substitute (x3+xy2+xz2) for P , (x2y+y3+z2y) for Q , and (x2z+y2z+z3) for R in equation (3),

divF=[x(x3+xy2+xz2)+y(x2y+y3+z2y)+z(x2z+y2z+z3)]=(3x2+y2+z2)+(x2+3y2+z2)+(x2+y2+3z2)=5(x2+y2+z2)

Calculation of flux of vector field:

Substitute 5(x2+y2+z2) for divF in equation (2),

SFdS=E5(x2+y2+z2)dV (5)

Parameterize the sphere with radius R as follows

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