   Chapter 16, Problem 35RE

Chapter
Section
Textbook Problem

Verify that the Divergence Theorem is true for the vector field F(x, y, z) = x i + y j + z k, where E is the unit ball x2 + y2 + z2 ⩽ 1.

To determine

To verify: The Divergence Theorem is true for the vector field F(x,y,z)=xi+yj+zk where E is the unit ball x2+y2+z21 .

Explanation

Given data:

The vector field is F(x,y,z)=xi+yj+zk , where E is the unit ball x2+y2+z21 .

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 volume of sphere.

V=43πr3 (3)

Consider the value of radius as follows.

r=1

Substitute 1 for r in equation (3),

V=43π(1)3=43π

Find the value of divF .

Substitute x for P , y for Q , and z for R in equation (2),

divF=x(x)+y(y)+z(z)=1+1+1=3

Substitute 3 for divF in equation (1),

SFdS=E3dV

Substitute x2+y2+z21 for E ,

SFdS=x2+y2+z213dV=3V

Substitute 43π for V ,

SFdS=3(43π)

SFdS=4π (4)

Consider a parametric representation as follows.

r(ϕ,θ)=sinϕcosθi+sinϕsinθj+cosϕk (5)

Consider the vector field as follows.

F(x,y,z)=xi+yj+zk (6)

Modify equation (5) as follows.

F(r(ϕ,θ))=sinϕcosθi+sinϕsinθj+cosϕk

Compare equations (5) and (6).

x=sinϕcosθy=sinϕsinθz=cosϕ

Find the value of rϕ×rθ .

rϕ×rθ=|ijkxϕyϕzϕxθyθzθ|

Substitute sinϕcosθ for x , sinϕsinθ for y and cosϕ for z ,

rϕ×rθ=|ijk(sinϕcosθ)ϕ(sinϕsinθ)ϕ(cosϕ)ϕ(sinϕcosθ)θ(sinϕsinθ)θ(cosϕ)θ|=|ijkcosθcosϕsinθcosϕsinϕsinθsinϕsinϕcosθ0|=[(0+sin2ϕcosθ)i(0sinθsin2ϕ)j+(cos2θsinϕcosϕ+sin2θsinϕcosϕ)k]=sin2ϕcosθi+sinθsin2ϕj+sinϕcosϕk {cos2θ+sin2θ=1}

Find the value of F(r(ϕ,θ))(rϕ×rθ)

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