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EXAMPLE 2 Use Stokes' Theorem to compute the integral fſcurl F ds, where F(x, y, z) = zxi + yzj + xyk and S is the sphere x2 + y2 + z2 = 9 that lies inside the cylinder x2 + y? = 1 and above the xy-plane. (See the figure.) SOLUTION To find the boundary curve C we solve the equations x2 + y2 + z2 = 9 and x2 + y2 = 1. Subtracting, we get z? = 8 and so z = V8. A vector equation of C is r(t) = cos(t)i + sin(t)j + V8 k; 0sts 2n Video Example () r'(t) = -sin(t)i + cos(t)j Also, we have F(r(t)) = V8 i+ V8sin(t)j + cos(t)sin(t)k Therefore, by Stokes' Theorem, SS curl F. dS = SF. dr = F(r(t)) r'(t)dt V8 cos(t) + V8sin(t)cos(t))dt = V8 -V8 cos (tsimt) dt = 0

EXAMPLE 2 Use Stokes' Theorem to compute the integral fſcurl F ds, where F(x, y, z) = zxi + yzj + xyk and S is the sphere x2 + y2 + z2 = 9 that lies inside the cylinder x2 + y? = 1 and above the xy-plane. (See the figure.) SOLUTION To find the boundary curve C we solve the equations x2 + y2 + z2 = 9 and x2 + y2 = 1. Subtracting, we get z? = 8 and so z = V8. A vector equation of C is r(t) = cos(t)i + sin(t)j + V8 k; 0sts 2n Video Example () r'(t) = -sin(t)i + cos(t)j Also, we have F(r(t)) = V8 i+ V8sin(t)j + cos(t)sin(t)k Therefore, by Stokes' Theorem, SS curl F. dS = SF. dr = F(r(t)) r'(t)dt V8 cos(t) + V8sin(t)cos(t))dt = V8 -V8 cos (tsimt) dt = 0

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 12T
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16.8b6

EXAMPLE 2 Use Stokes' Theorem to compute the integral Sſcurl F ds, where F(x, y, z) = zxi + yzj + xyk and S is the sphere x2 + y2 + z2 = 9 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane. (See the figure.) SOLUTION To find the boundary curve C we solve the equations x2 + y2 + z? = 9 and x2 + y2 = 1. Subtracting, we get z? = 8 and so z = V8. A vector equation of C is r(t) = cos(t)i + sin(t)j + V8 k; 0sts 2x Video Example ) r'(t) = -sin(t)i + cos(t)j Also, we have F(r(t)) = V8 i+ v8sin(t)j + cos(t)sin(t)k Therefore, by Stokes' Theorem, ff curl F· dS = [F. dr = F(r(t)) · r'(t)dt - LV8 cos( ) + v8sin(t)cos(t))dt -V8 cos (tsimt) = V8 dt = 0 pad It
Transcribed Image Text:EXAMPLE 2 Use Stokes' Theorem to compute the integral Sſcurl F ds, where F(x, y, z) = zxi + yzj + xyk and S is the sphere x2 + y2 + z2 = 9 that lies inside the cylinder x2 + y2 = 1 and above the xy-plane. (See the figure.) SOLUTION To find the boundary curve C we solve the equations x2 + y2 + z? = 9 and x2 + y2 = 1. Subtracting, we get z? = 8 and so z = V8. A vector equation of C is r(t) = cos(t)i + sin(t)j + V8 k; 0sts 2x Video Example ) r'(t) = -sin(t)i + cos(t)j Also, we have F(r(t)) = V8 i+ v8sin(t)j + cos(t)sin(t)k Therefore, by Stokes' Theorem, ff curl F· dS = [F. dr = F(r(t)) · r'(t)dt - LV8 cos( ) + v8sin(t)cos(t))dt -V8 cos (tsimt) = V8 dt = 0 pad It
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