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Math Help - curl

  1. #1
    MHF Contributor arbolis's Avatar
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    curl

    If \vec F=\vec \nabla \varphi, show that \vec \nabla \times \vec F=\vec 0. \vec F is a vector field and \varphi is a scalar field.
    I know it's an obvious question but I can't remember how to formally prove it.
    I know it's true if  \frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2  \varphi }{\partial z \partial x} and \frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2  \varphi }{\partial y \partial x} in Cartesian coordinates, I've showed it for \varphi (x,y,z)=f(x)g(y)h(z) but not for a general form of \varphi. I think I'm over-complicating this.
    Could you give me a tip?
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by arbolis View Post
    If \vec F=\vec \nabla \varphi, show that \vec \nabla \times \vec F=\vec 0. \vec F is a vector field and \varphi is a scalar field.
    I know it's an obvious question but I can't remember how to formally prove it.
    I know it's true if  \frac{\partial ^2 \varphi}{\partial y \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial y} ,\frac{\partial ^2 \varphi}{\partial x \partial z} =\frac{\partial ^2 \varphi }{\partial z \partial x} and \frac{\partial ^2 \varphi}{\partial x \partial y} =\frac{\partial ^2 \varphi }{\partial y \partial x} in Cartesian coordinates, I've showed it for \varphi (x,y,z)=f(x)g(y)h(z) but not for a general form of \varphi. I think I'm over-complicating this.
    Could you give me a tip?
    It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...
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  3. #3
    MHF Contributor arbolis's Avatar
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    Quote Originally Posted by Failure View Post
    It follows from Stokes Theorem, for example. But maybe that's not the kind of tip that you have been expecting...
    I'm not expecting a particular kind of help so I appreciate yours.
    I do not know how to relates Stokes' theorem with this problem.
    I know that \int _{\gamma} \vec F \cdot  d\vec r = \int _{S} ( \vec \nabla \times \vec F ) \cdot d\vec S.
    In the example given, I do not know where \vec F and \varphi are defined.
    By intuition I could take \gamma as a circle with radius R tending to infinite and S being the upper half sphere of radius R but I'm not sure it makes sense nor how do I use \varphi. Am I, at least, on the right direction? If so I'll think more about it.

    Edit: Now I'm thinking about using Clairaut's theorem. But I'd have to show that \varphi is continuous and it is not given...
    Last edited by arbolis; April 16th 2010 at 09:29 PM.
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