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Math Help - Tough calculus...

  1. #1
    Junior Member
    Joined
    Nov 2008
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    Tough calculus...

    I have the following from Vector Calculus:

    \frac{\delta\phi}{\delta x}=\frac{yz(y+z)}{(x+y+z)^2}



    \frac{\delta\phi}{\delta y}=\frac{xz(x+z)}{(x+y+z)^2}


    \frac{\delta\phi}{\delta x}=\frac{xy(x+y)}{(x+y+z)^2}


    I have to determine whether this system has solutions or not. If there is a solution I have to find it. I also have to determine the domain on which solution is defined. Also some words about uniqueness must be said...


    Can anyone give me some help?
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  2. #2
    MHF Contributor
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    France
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    Hi

    Integrating

    \frac{\delta\phi}{\delta x}=\frac{yz(y+z)}{(x+y+z)^2}

    gives

    \phi(x,y,z)=-\frac{yz(y+z)}{x+y+z} + f(y,z)

    \phi(x,y,z)=-yz + \frac{xyz}{x+y+z} + f(y,z)

    Derivation with respect to y gives

    \frac{\delta\phi}{\delta y} = -z + \frac{xz(x+z)}{(x+y+z)^2} + \frac{\delta f}{\delta y}

    But \frac{\delta\phi}{\delta y} = \frac{xz(x+z)}{(x+y+z)^2}

    Therefore
    \frac{\delta f}{\delta y} = z and f(y,z) = yz + g(z)

    The same approach for z leads to

    \phi(x,y,z)=\frac{xyz}{x+y+z} + constant
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  3. #3
    Member
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    Jun 2007
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    I think this is where you might use the formulas for gradient, curl, divergence and laplacian equations:









    You may need to wrte out teh Jacobian matrix first to help you.

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