   Chapter 16.8, Problem 15E

Chapter
Section
Textbook Problem

Verify that Stokes’ Theorem is true for the given vector field F and surface S.15. F(x, y, z) = y i + z j + x k,S is the hemisphere x2 + y2 + z2 = 1, y ≥ 0, oriented in the direction of the positive y-axis

To determine

To verify: Whether Stokes’ theorem is true for given vector filed F and surface S.

Explanation

Given data:

The field is F(x,y,z)=yi+zj+xk and

Consider the expression for the hemisphere surface S, y0 .

x2+y2+z2=1 (1)

Formula Used:

Write the expression for curl of F(x,y,z)=Pi+Qj+Rk .

curlF=|ijkxyzPQR|

curlF=(RyQz)i(RxPz)j+(QxPy)k (2)

Write the expression for the Stokes’ theorem.

CFdr=ScurlF(rϕ×rθ)dS (3)

Here,

S is surface,

ϕ,θ are spherical coordinates.

The hemisphere surface is oriented in the positive direction of y-axis. Hence consider y=0 .

Substitute 0 for y in equation (1).

x2+02+z2=1x2+z2=1

Consider the parametric equations since the orientation of surface is upward as,

x=costy=0z=sint,0t2π

Write the expression for vector function r(t) .

r(t)=xi+yj+zk (4)

Substitute cost for x, 0 for y, sint for z,

r(t)=costi+(0)j+(sint)k=costisintk

Differentiate the expression with respect to t.

r(t)=ddt(costisintk)=ddt(cost)iddt(sint)k=(sint)i(cost)k=sinticostk

Find the value of F(r(t)) .

F(r(t))=(0)i+(sint)j+(cost)k=sintj+costk

Find the value of F(r(t))r(t) .

F(r(t))r(t)=(sintj+costk)(sinticostk)=(0)(sint)+(sint)(0)+(cost)(cost)=cos2t

Write the expression for CFdr .

CFdr=02πF(r(t))r(t)dt

Substitute 8sin2t+4sintcost for F(r(t))r(t) ,

CFdr=02π(cos2t)dt=02π(12(cos2t+1))dt{cos2t=2cos2t1}=1202π(cos2t+1)dt=12[sin2t2+t]02π

Apply limit values and simplify the equation.

CFdr=12[(sin2(2π)2+2π)(sin2(0)2+0)]=12(0+2π0)=π

Find the value of curlF by the use of equation (2).

curlF=(xyzz)i(xxyz)j+(zxyy)k=(01)i(10)j+(01)k{t(k)=0,t(t)=1}=ijk

Consider the parametric equations of hemisphere surface S, D={(ϕ,θ)|0ϕπ,0θ2π} .

x=sinϕcosθy=sinϕsinθz=cosϕ

Re-modify the equation (4) for spherical coordinate system.

r(ϕ,θ)=xi+yj+zk

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

r(ϕ,θ)=sinϕcosθi+sinϕsinθj+cosϕk

Write the expression to find rϕ×rθ

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Solve the equations in Exercises 126. 4xx313x4x31x31=0

Finite Mathematics and Applied Calculus (MindTap Course List)

In Exercises 4148, find the indicated limit given that limxaf(x)=3 and limxag(x)=4 41. limxa[f(x)g(x)]

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

If y = f (g(x)) then y′ = f′(g′(x)).

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

The tangential component of acceleration for at t = 0 is: 0

Study Guide for Stewart's Multivariable Calculus, 8th 