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Homework Help: Using the Jacobian to Prove Laplace's 2D Eq.

  1. Feb 4, 2010 #1
    1. The problem statement, all variables and given/known data

    I apologize in advance for my inability to present formal equations here. I'll do my best to be clear with the representation using simple text.

    "Use the Jacobian Matrix to Prove Laplace's 2D Eq.: (partial^2 u)/(partial x^2) + (partial^2 u)/(partial y^2) = 0"

    2. Relevant equations

    Laplace Terms:

    (partial u)/(partial x) = (partial v)/(partial y)

    (partial u)/(partial y) = -(partial v)/(partial x)

    3. The attempt at a solution

    I attempted to place in the various Laplace Terms mentioned above into a 2x2 matrix, and find the determinant. However, this did not appear to work, as it resulted in:

    det | (partial u)/(partial x), (partial u)/(partial y)|
    | -(partial v)/(partial x), (partial v)/(partial y)|

    = (partial^2 u)/(partial x^2) - (partial^2 u)/(partial y^2)
     
  2. jcsd
  3. Feb 4, 2010 #2

    Dick

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    Science Advisor
    Homework Helper

    Are you sure you understand what a partial derivative is? The first set of equations you gave are the Cauchy-Riemann equations for an analytic complex function. The determinant doesn't help. That just gives you products of first derivatives. Not the second derivatives you need for the Laplace equation. Differentiate Cauchy-Riemann.
     
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