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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
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

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Chapter 16 Solutions

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