   Chapter 16.9, Problem 13E

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.13. F = |r| r, where r = x i + y j + z k, S consists of the hemisphere z = 1   −   x 2 + y 2 and the disk x2 + y2 ≤ 1 in the xy-plane

To determine

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

Explanation

Given data:

The vector field is given as follows.

F=|r|r (1)

Here,

r=xi+yj+zk

The surface S consists of the hemisphere z=1x2y2 and the disk x2+y21 in 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 (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|(xi+yj+zk)=x2+y2+z2(xi+yj+zk)=xx2+y2+z2i+yx2+y2+z2j+zx2+y2+z2k

Calculation of divF :

Substitute xx2+y2+z2 for P , yx2+y2+z2 for Q , and zx2+y2+z2 for R in equation (3),

divF=[x(xx2+y2+z2)+y(yx2+y2+z2)+z(zx2+y2+z2)]=[xx(x2+y2+z2)+x2+y2+z2x(x)+yy(x2+y2+z2)+x2+y2+z2y(y)+zz(x2+y2+z2)+x2+y2+z2z(z)]=[x2x2+y2+z2(2x)+x2+y2+z2+y2x2+y2+z2(2y)+x2+y2+z2+z2x2+y2+z2(2z)+x2+y2+z2]=x2+y2+z2x2+y2+z2+3x2+y2+z2

Simplify the expression as follows.

divF=x2+y2+z2+3(x2+y2+z2)x2+y2+z2=4(x2+y2+z2)x2+y2+z2=4x2+y2+z2

Calculation of flux of vector field:

Substitute 4x2+y2+z2 for divF in equation (2)

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