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Math Help - Stoke's Theorem

  1. #1
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    Stoke's Theorem

    Let F = (y,-x,zx^3y^2) Evaluate \int \int_{S} (curlF) dS, where S is the surface x^2 + y^2 + z^2 = 1, z\geq 0

    I need help with the setup.
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  2. #2
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    Since you titled this "Stoke's theorem" can we assume you know that theorem? (By the way, it should be "Stokes' " theorem. His name was "Stokes", not "Stoke".)

    Stokes theorem says \int_C \vec{F}\cdot d\vec{r}= \int\int_S curl \vec{F}\cdot d\vec{S} where r is the closed curve bounding surface S.

    Since you are asked to find \int\int_S (curl F)dS, Stokes' theorem says that you can instead integrate \int_C \vec{F}d\vec{r} where C is the boundary of that hemi-sphere. When z= 0, x^2+ y^2= 1 so the boundary is the unit circle in the xy-plane. Of course, when z= 0, F= (y, -x, 0). I think I might be inclined to integrate around the unit circle by using the angle, \theta, as parameter: x= cos(\theta), y= sin(\theta). For the unit circle, dr= d\theta.
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  3. #3
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    Thank You that was great help
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