   Chapter 16.8, Problem 3E

Chapter
Section
Textbook Problem

Use Stokes’ Theorem to evaluate ∫∫s curl F · dS.3. F(x, y, z) = zey i + x cos y j + xz sin y k, S is the hemisphere x2 + y2 + z2 = 16, y ≥ 0, oriented in the direction of the positive y-axis

To determine

To evaluate: The expression ScurlFdS by using Stokes’ theorem.

Explanation

Given data:

Consider the expression for the vector field F(x,y,z) ,

F(x,y,z)=zeyi+xcosyj+xzsinyk (1)

And S is the hemisphere that is,

x2+y2+z2=16,y0 (2)

Formula Used:

Consider the expression for the Stokes’ theorem,

ScurlFdS=CFdr (3)

Consider the boundary curve C is the circle x2+y2+z2=16,y=0 where the hemisphere intersects the xy-plane. This boundary curve C must be oriented in the counter-clockwise direction. Therefore, the vector equation of C is,

r(t)=4cos(t)i+4sin(t)k=4costi4sintk,0t2π

Differentiate the equation with respect to t,

r(t)=ddt(4costi4sintk)=4sinti4costk

Find the expression for F(r(t)) .

Substitute 4cost for x, 0 for y and 4sint for z in equation (1) to find F(r(t)) ,

F(r(t))=(4sint)e0i+(4cost)cos0j+(4cost)(4sint)sin0k=4sinti+4costj<

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