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Help with Stokes' Theorem

  1. Apr 30, 2014 #1

    BiGyElLoWhAt

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    OK, so I know stokes theorem states that I can turn ##\int \int_{S}F\cdot dS## into ##\int \int (-P\frac{\partial g(x,y)}{\partial x} -Q\frac{\partial g(x,y)}{\partial y} + R)dxdy##

    but I have a problem that I'm working on that's screaming change of variables (to spherical), but I'm not sure if I can apply stokes' this way...

    my region is ##x^2 +y^2 + z^2 = 64##, and I really want to make my region a spherical region (if possible, as I need to understand this for my final) but I'm not sure how to define g.

    What's throwing me off is which partials to take. My F is ##<x,-y,z>## and upon making the proper substitutions, ##- \rho <cos(\theta )sin(\phi ), -sin(\theta )sin(\phi ), cos(\ phi)>##
    and I suppose g would be ##\rho <cos(\theta )sin(\phi ), sin(\theta )sin(\phi ), cos(\ phi)>## , since that's my region? It's the region in the first quadrant oriented inwards, that's where the negative sign came from, I guess it should probably be in g, but either way it's going make it into the integral.

    But now, how do I define my g and my partials? I understand P is my ##\hat{i}## and Q j and R z, but I'm uncertain as to which partials I'm multiplying by which functions.

    I'm not sure if I can even do this, I'm just trying to understand this better. any help would be greatly appreciated =]
     
  2. jcsd
  3. Apr 30, 2014 #2

    BiGyElLoWhAt

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    Ok, I didn't want to, but I watched the "how to" video, which basically works through the problem, and they ended up solving "g" for z, but I'm still wondering if there is a way to do this in spherical? Or do I need ##F = P\hat{\theta} + Q \hat{\phi} + R \hat{\rho}## in order to make this work like I'm wanting it to?
     
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