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Math Help - Stoke's Theorem - Jacobian Problem

  1. #1
    Member Altair's Avatar
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    Stoke's Theorem - Jacobian Problem

    Question asks to evaluate the surface integral for the given F and S with stokes theorem.

    F= [z^2, x^2, y^2], s: z^2 = x^2 + y^2 for y >= 0, 0<=z<=2

    My question is that in transformation from x,y to u,v shouldn't there be jacobian multiplied ? I dont see it anywhere in the solution.
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by Altair View Post
    Question asks to evaluate the surface integral for the given F and S with stokes theorem.

    F= [z^2, x^2, y^2], s: z^2 = x^2 + y^2 for y >= 0, 0<=z<=2

    My question is that in transformation from x,y to u,v shouldn't there be jacobian multiplied ? I dont see it anywhere in the solution.
    There is no trouble or mystery here.

    The surface is being represented by the parametrised position vector \vec{r}(u, v) = x(u, v) \vec{i} + y(u, v) \vec{j} + z(u, v) \vec{k}. Using this representation, \vec{dS} = \frac{\partial \vec{r}}{\partial u} du \times \frac{\partial \vec{r}}{\partial v} dv.

    This is a formula that should be somewhere in your notes or textbook.
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