   Chapter 16.8, Problem 13E

Chapter
Section
Textbook Problem

Verify that Stokes’ Theorem is true for the given vector field F and surface S.13. F(x, y, z) = -y i + x j -2 k,S is the cone z2 = x2 + y2, 0 ≤ z ≤ 4, oriented downward

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+xj2k and

Consider the expression for surface of cone, 0z4 is,

z2=x2+y2 (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=ScurlFdS (3)

Here,

S is surface.

Consider surface S, z=g(x,y) is in upward orientation. Write the expression for surface integral of F over surface S.

ScurlFdS=D(PgxQgy+R)dA (4)

Here,

A is area.

The boundary value of surface cone is a circle. The maximum value of z is 4.

Substitute 4 for z in equation (1),

42=x2+y2x2+y2=16

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

x=4costy=4sintz=4,0t2π

Write the expression for vector function r(t) .

r(t)=xi+yj+zk

Substitute 4cost for x, 4sint for y, 4 for z,

r(t)=4costi4sintj+4k

Differentiate the expression with respect to t,

r(t)=ddt(4costi4sintj+4k)=ddt(4cost)iddt(4sint)j+ddt(4)k=4(sint)i(4cost)j+(0)k=4sinti4costj

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

F(r(t))=(4sint)i+(4cost)j2k=4sinti+4costj2k

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

F(r(t))r(t)=4sinti+4costj2k(4sinti4costj)=(4sint)(4sint)+(4cost)(4cost)+(2)(0)=16sin2t16cos2t+0=16(sin2t+cos2t)

F(r(t))r(t)=16(1){sin2t+cos2t=1}=16

Write the expression for CFdr

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