EvaluateF.dS,where F = and S is the helicoid with vector equationr(u, v) = < u cos v, u sin v, v > 0

Given that, F¯=y,-x,z5 and ru,v=u cos v, u sin v, v where 0≤u≤3 and 0≤v≤π. Find ru×rv as follows.…

Answered: Evaluate F.dS, where F = and S is the…

Trigonometry (MindTap Course List)

8th Edition

ISBN:9781305652224

Author:Charles P. McKeague, Mark D. Turner

Publisher:Charles P. McKeague, Mark D. Turner

Chapter7: Triangles

Section7.2: The Law Of Cosines

Problem 65PS

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Question

Evaluate
F.dS,where F = <y, – x, z > and S is the helicoid with vector equation
r(u, v) = < u cos v, u sin v, v > 0 <u < 3, 0 < v< TWwith upward orientation.

Expert Solution

Step 1

Given that, $\bar{F} = <y, - x, z^{5}>$ and $r (u, v) = <u \cos v, u \sin v, v>$ where $0 \leq u \leq 3$ and $0 \leq v \leq π$ .

Find $r_{u} \times r_{v}$ as follows.

$\begin{array}{rcl} r (u, v) & = & <u \cos v, u \sin v, v> \\ r_{u} & = & <\cos v, \sin v, 0> \\ r_{v} & = & <- u \sin v, u \cos v, 1> \\ r_{u} \times r_{v} & = & |\begin{matrix} i & j & k \\ \cos v & \sin v & 0 \\ - u \sin v & u \cos v & 1 \end{matrix}| \\ = & \sin v i - \cos v j + (u \cos^{2} v + u \sin^{2} v) k \\ = & \sin v i - \cos v j + u k \end{array}$

Therefore, the normal upward vector to the surface is , $n = r_{u} \times r_{v} = <\sin v, - \cos v, u>$ .

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