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Stokes Theorem paraboloid intersecting with cylinder

  1. Jul 26, 2012 #1
    1. The problem statement, all variables and given/known data

    Use stokes theorem to elaluate to integral [itex] \int\int_{s} curlF.dS[/itex] where [itex] F(x,y,z)= x^2 z^2 i + y^2 z^2 j + xyz k [/itex] and s is the part of the paraboliod [itex]z=x^2+ y^2 [/itex] that lies inside the cylinder [itex] x^2 +y^2 =4 [/itex] and is orientated upwards

    2. Relevant equations



    3. The attempt at a solution

    so i use Stokes theorem [itex] \int\int_{s} curlF.dS = \oint_{c} F.dv[/itex]

    so i want to get parametric equations for C and so i get x=2cost y=2sint and z=4 i came up with these as the boundary curve c is a circle of raadius 2 and my z value of 4 because that is where the parabaloid and cylinder intersect...am i right in my thinking here?

    so then dx= -2sintdt ; dy=2costdt and dz=0

    so then i get [itex] \oint_{c} F.dv = \oint_{c} x^2 z^2 dx - y^2z^2 dy + xyzdz[/itex]

    which becomes [itex]\int^{2\pi}_{0} ((4cos^2 t )(16)(-2sint) - (4sin^2t)(16)(ccost)) dt[/itex]

    [itex]\int^{2\pi}_{0} ((-128cos^2 t )(sint) - (128sin^2 t)(cost)) dt[/itex]

    am i working on the right lines here?
     
    Last edited: Jul 26, 2012
  2. jcsd
  3. Jul 26, 2012 #2
    whoops hold up a sec...not finished putting up my attempt, im editing the post!
     
  4. Jul 26, 2012 #3
    the origional post is complete now. anyone got any ideas?
     
  5. Jul 26, 2012 #4
  6. Jul 26, 2012 #5

    sharks

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    I think the last part is wrong and it should be:
    ##\int^{2\pi}_{0} (-128cos^2 t sint + 128sin^2 tcost) .dt##
     
    Last edited: Jul 26, 2012
  7. Jul 26, 2012 #6

    LCKurtz

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    Check that - sign.

    Looks good except for that minus sign.
     
  8. Jul 26, 2012 #7

    sharks

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    The integration can be tricky but there is a simple trick to it. Let us know how it goes.

    However, i have a question myself, since i'm still learning. I suppose it is OK for me to ask here? What if the orientation of the surface S was downwards? Then, how would the calculations from post #1 change?
     
    Last edited: Jul 26, 2012
  9. Jul 26, 2012 #8
    Hi LCkurtz and Sharks
    Thanks a million for comments.

    To integrate i used substitution u=cost du=-sintdt and similar for sin.

    as in [itex]\int cos^2 t sint =\frac{1}{3} cos^3 t[/itex] i think this works

    so i get [itex]\frac{128}{3} (cos^3 t + sin^3 t)^{2\pi}_{0}[/itex]

    =-[itex]\frac{128}{3}[/itex] i hope :)
     
    Last edited: Jul 26, 2012
  10. Jul 26, 2012 #9
    i'm afraid i havent a clue, sorry!
     
  11. Jul 26, 2012 #10

    sharks

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    But isn't the whole integral this:
    ##\int^{2\pi}_{0} (-128cos^2 t sint + 128sin^2 tcost) .dt##
    Then, it would become:
    ##\int^{2\pi}_{0} (-128cos^2 t sint).dt + \int^{2\pi}_{0} (128sin^2 tcost).dt##
     
  12. Jul 26, 2012 #11

    LCKurtz

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    I was tempted in my original post to ask the OP if he had thought about the orientation or whether he just took a guess. You use the right hand rule. In this case the normal was pointed upwards, so the right hand rule would give you counterclockwise in the xy plane as viewed from above which gives ##t:\, 0\rightarrow 2\pi##. For downward orientation you go the other way around the circle which could be ##t:\, 2\pi\rightarrow 0##. The effect is to change the sign of the answer.
     
  13. Jul 26, 2012 #12

    sharks

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    Thank you for this quick and accurate answer, LCKurtz.
     
  14. Jul 26, 2012 #13
    [itex]\int^{2\pi}_{0} (-128cos^2 t sint)dt + \int^{2\pi}_{0} (128sin^2 t cost)dt[/itex]

    =128[[itex]\frac{1}{3} cos^3 t ]^{2\pi}_{0} + 128[\frac{1}{3} sin^3 t ]^{2\pi}_{0}[/itex]

    =[itex]\frac{128}{3}[-1] = \frac{-128}{3} [/itex]
     
  15. Jul 26, 2012 #14
    ahhh thanks a million i had wondered about that alright. that cleared it up. thanks a million.
     
  16. Jul 26, 2012 #15

    sharks

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    I don't think the evaluation part is correct.
     
  17. Jul 26, 2012 #16
    whoops should be [itex]\frac{128}{3}[1-1] = 0 [/itex]
     
  18. Jul 26, 2012 #17
    Thanks a million sharks! your brill! :smile:
     
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