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Math Help - Applying Stokes theorem

  1. #1
    MHF Contributor arbolis's Avatar
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    Applying Stokes theorem

    Let S_G be the surface of the cap (see figure). Let C be the differentiable curve of the cap in the xy plane. Let S_A be the portion of the plane xy that C contains. And let A be the value of the area of S_A. Consider S_G oriented outward. Determine with arrows the sense of the curve C and the value of \iint _{S_G} curl F dS, where F=((\ln (z+1))^2xy, (\ln (z+1))^2 \frac{x^2}{2}+x(z+1), \cos z ).
    -------------------------------------------------------
    My attempt :
    The arrow is easy to draw : counter clockwise.
    I've calculated the curl of F, but now I realize it is likely useless. Anyway here it comes : curl F =\left ( \frac{x^2 \ln (z+1)}{2z+2}   \right ) \vec i + \left( \frac{xy \ln (z+1)}{z+1} \right) \vec j +(z+1) \vec k.
    I'm sure I have to calculate \int _C FdS (which is worth \iint _{S_G} curl F dS by Stokes' theorem) but how to proceed? I have almost no information about C, I mean I don't know its parametrization. The only way I think I might solve the problem is to realize something obvious that imply that \int _C FdS=0, but I don't see anything.
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  2. #2
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    Quote Originally Posted by arbolis View Post
    Let S_G be the surface of the cap (see figure). Let C be the differentiable curve of the cap in the xy plane. Let S_A be the portion of the plane xy that C contains. And let A be the value of the area of S_A. Consider S_G oriented outward. Determine with arrows the sense of the curve C and the value of \iint _{S_G} curl F dS, where F=((\ln (z+1))^2xy, (\ln (z+1))^2 \frac{x^2}{2}+x(z+1), \cos z ).
    this is a nice problem! in order to find the integral, you'll need to use Stoke's theorem first and then Green's theorem. note that in xy plane we have z = 0 and thus \ln(z + 1) = dz = 0.

    so by Stoke's theorem: \iint_{S_G} curl (F) \cdot \bold{n} \ dS = \int_C F \cdot d \bold{r}=\int_C xdy. on the other hand, by Green's theorem: \int_C xdy = \int_C 0 \cdot dx + xdy = \iint_{S_A} dxdy =A. thus \iint_{S_G} curl (F) \cdot \bold{n} \ dS =A.
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  3. #3
    MHF Contributor arbolis's Avatar
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    Thanks for the answer NCA, I wouldn't have find it by myself within the next 6 remaining days my exam. The problem is indeed nice because it is different from most.
    Just a little clarification of my first post : I mean C has the clockwise sense and not anticlockwise as I said. I sketched it clockwise but I don't know why I said anticlockwise.
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