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Homework Help: Test Stokes' Theorem for the function

  1. Sep 28, 2006 #1
    Test Stokes' Theorem for the function
    [itex] \vec{v} = (xy) \hat{x} + (2yz) \hat{y} + (3yz) \hat{z} /[/itex]

    for the triangular shaded region

    [tex] \int_{S} (\grad \times v) \dot da = \oint_{P} v \bullet dl [/tex]

    for the left hand side

    [tex] \int_{0}^{2} \int_{0}^{2} (-2y \hat{x} - 3z \hat{y} - x \hat{z}) \hat{x} dy dz = \int_{0}^{2} \int_{0}^{2} (-2y) dy dz = \left[ \frac{-y^2} \right]_{0}^{2} \int_{0}^{2} dz = (-4)(2) = -8 [/tex]

    so far so good??

    ok for hte right hand side

    i) x=0, z=0, [itex] v \bullet dl = 2y dy[/itex] integration limits 0 to 2

    ii) x=0, y=2, [itex] v \bullet dl = 0 dz [/itex] integration from 0 to 2

    i am not sute about the last one though....

    am i right with i and ii though??

    thank you for any advice!
     

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    Last edited by a moderator: Sep 29, 2006
  2. jcsd
  3. Sep 29, 2006 #2

    HallsofIvy

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    Science Advisor

    No, your [itex]\nabla \vec{v}[/itex] is wrong. The [itex]\hat{y}[/itex] component is 0.

    That should be [itex]v \bullet dl= 2yz dy= 0[/itex]

    for the last one, you can take y= t, z= 2- t integrating from t= 0 to 2.

    By the way, to get [itex]\nabla[/itex] use "\nabla", not "\del".
     
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