Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.

F (x, y, z) = yzi + 9xzj + exyk, C is the circle x2 + y2 = 1, z = 3.

Question

Mathematics

Kline

25 March, 19:31

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.

F (x, y, z) = yzi + 9xzj + exyk, C is the circle x2 + y2 = 1, z = 3.

+1

Answers (1)

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Mirabel · Answer 1

S has a unit normal k, so we need the k component of curl F only

curl F (k direction) = (3z-z) = 2z at z=3, = 6

So INT_C F. dr = INT_S curlF. dS = INT:A curl F. (k) dA,

= INT6 dA = 6A

The circle (S) hasa radius R=4, so A = piR^2 = 16pi

Work = 6*16pi = 96pi (stokes gives Counterclockwise, so we use + k, clockwise, - k, or simply factor by - 1)

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.F (x, y, z) = yzi + 9xzj + exyk, C is the circle x2 + y2 = 1, z = 3.

Use Stokes' Theorem to evaluate C F · dr where C is oriented counterclockwise as viewed from above.

F (x, y, z) = yzi + 9xzj + exyk, C is the circle x2 + y2 = 1, z = 3.