   Chapter 16.9, Problem 3E

Chapter
Section
Textbook Problem

Verify that the Divergence Theorem is true for the vector field F on the region E.3. F(x, y, z) = ⟨z, y, x⟩, E is the solid ball x2 + y2 + z2 ≤ 16

To determine

To Verify: The divergence theorem for the vector field F(x,y,z)=z,y,x on the region E.

Explanation

Given:

The vector field is F(x,y,z)=z,y,x.

The region E is the solid ball x2+y2+z216.

Formula used:

The divergence theorem is given by,

SFdS=EdivFdV (1)

The expression to find divergence of vector field F(x,y,z)=Pi+Qj+Rk is,

divF=xP+yQ+zR (2)

The expression for SFdS ,where n is the unit outward normal of the surface S is as follows.

SFdS=SFndS (3)

Calculation:

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

divF=x(z)+y(y)+z(x)=0+1+0=1

Calculation of flux EdivFdV is as follows.

Substitute the value of divF in the expression EdivFdV,

EdivFdV=E(1)dV

EdivFdV=V(E) (4)

Write the expression for volume of the solid region E with the radius r.

V(E)=43πr3

As the radius of the region is 4, substitute 4 for r.

V(E)=43π(4)3=2563π

Substitute 2563π for V(E) in equation (4),

EdivFdV=2563π (5)

As the solid ball is a sphere with radius 4, consider the parametric equation equations as follows.

x=4sinϕcosθ,y=4sinϕsinθ,z=4cosϕ,0ϕπ,0θ2π

Write the position vector as ,

r(ϕ,θ)=4sinϕcosθ,4sinϕsinθ,4cosϕ (6)

Find the tangent vectors as follows.

To calculate the tangent vector rϕ differentiate on both sides of the expression in equation (6) with respect to ϕ.

ddϕ[r(ϕ,θ)]=ddϕ4sinϕcosθ,4sinϕsinθ,4cosϕ

Rewrite the expression as follows.

rϕ=4cosϕcosθ,4cosϕsinθ,4sinϕ

To calculate of tangent vector rθ differentiate on both sides of the expression in equation (6) with respect to θ.

ddθ[r(ϕ,θ)]=ddθ4sinϕcosθ,4sinϕsinθ,4cosϕ

Rewrite the expression as follows.

rθ=4sinϕsinθ,4sinϕcosθ,0

Write the expression for normal vector to the surface with the tangent vectors as follows.

n=rϕ×rθ=4cosϕcosθ,4cosϕsinθ,4sinϕ×4sinϕsinθ,4sinϕcosθ,0

Find the normal vector from the tangent vectors as follows

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