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Math Help - Vector Calculus

  1. #1
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    Vector Calculus

    Show that if \sigma(x,y) satisfies Laplace's equation,

     \frac{\partial \sigma}{\partial x^2}+\frac {\partial \sigma}{\partial y^2}=0  on a simply connected region R, then \forallclosed curves C of R, we have
    \int_c (\sigma_y dx - \sigma_x dy) =0
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    Quote Originally Posted by Andreamet View Post
    Show that if f(x,y) satisfies Laplace's equation,

     \frac{\partial \sigma}{\partial x^2}+\frac {\partial \sigma}{\partial y^2}=0 on a simply connected region R, then \forallclosed curves C of R, we have
    \int_c (\sigma_y dx - \sigma_x dy) =0
    Use greens theorem

    \iint_D -\sigma_{xx}-\sigma_{yy}dA=-\iint_D 0 dA=0
    Last edited by TheEmptySet; March 28th 2009 at 02:35 PM. Reason: I cant spell theorem I guess :(
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    Quote Originally Posted by TheEmptySet View Post
    Use greens theorem

    \iint_D -\sigma_{xx}-\sigma_{yy}dA=-\iint_D 0 dA=0

    Hi, thanks a lot. However, from the Question, I know that we need to apply Green's theorem. However, I am confused about what you did. I am very confused
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    Quote Originally Posted by Andreamet View Post
    Hi, thanks a lot. However, from the Question, I know that we need to apply Green's theorem. However, I am confused about what you did. I am very confused

    The formal statement of Green's theorem is (Stuart Calculus)

    Let C be a positivley oriented, piecwise-smooth, simple closed curve int he plance and let D be the region bounded by C. If P and Q have contionous partial derivatives on an open region that contains D, then

    \oint_C Pdx+Qdy=\iint_D \left( \frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dA


    You started with

    \int_c (\sigma_y dx - \sigma_x dy)

    note that P=\sigma_y \mbox{ and } Q=-\sigma_x

    Now using Greens theorem

    \int_c (\sigma_y dx - \sigma_x dy) =\iint_D -\sigma_{xx}-\sigma_{yy} dA=-\iint \sigma_{xx}+\sigma_{yy}dA=-\iint 0 dA=0
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