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Math Help - Jacobian determinants and area integrals.

  1. #1
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    Jacobian determinants and area integrals.

    I have the following past exam question.

    Write down the Cartesian coordinates x and y in terms of the plane polar coordinates r and \theta

    Which is easy.

    Next, evaluate the jacobian determiant


    By considering the matrix equation relating the vectors
    \begin{pmatrix}{dx}\\{dy}\end{pmatrix}
    and
    \begin{pmatrix}{dr}\\{d\theta}\end{pmatrix}
    use this result to obtain an expression for the area element dxdy in plane polar coordinates.

    b)Work out the numerical value of the same area integral

    \int(r^2+1)dA

    over the interior of the circle of radius 2 centred at the origin.
    Last edited by Confused77; August 25th 2008 at 04:41 AM.
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  2. #2
    Super Member wingless's Avatar
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    Transformation from cartesian coordinates to polar:
    x = r \cos \theta
    y = r \sin \theta

    Can you do it now? If you can't, where are you stuck?
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  3. #3
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    That's the part I did get, I seem unable to evaluate that determinant correctly. and then do the part just after.
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  4. #4
    Super Member wingless's Avatar
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    OK.

    I'll only work for two dimensions. Let us have a transform from (x,y) to (a,b) :
    x = f(a,b) and y=g(a,b).

    The Jacobian matrix is defined as

    J = \frac{\partial (x,y)}{\partial (a,b)} = \begin{bmatrix} \dfrac{\partial x}{\partial a} & & \dfrac{\partial x}{\partial b} \\   \\ \dfrac{\partial y}{\partial a} & & \dfrac{\partial y}{\partial b}\end{bmatrix}

    |J| = \left | \begin{array}{ccc} \dfrac{\partial x}{\partial a} & & \dfrac{\partial x}{\partial b} \\ \\ \dfrac{\partial y}{\partial a} & & \dfrac{\partial y}{\partial b}\end{array} \right |


    In our example, this is,

    |J| = \left | \begin{array}{ccc} \dfrac{\partial r \cos \theta }{\partial r} & & \dfrac{\partial r \cos \theta}{\partial \theta} \\ \\ \dfrac{\partial r \sin \theta}{\partial r} & & \dfrac{\partial r \sin \theta}{\partial \theta}\end{array} \right |

    And remember that the determinant of a 2x2 matrix is \left | \begin{array}{cc} a & b \\ c & d \end{array} \right | = ad - bc
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  5. #5
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    Thank you very much, have done that bit and part b now.

    I'm still stuck on this part however.

    By considering the matrix equation relating the vectors
    \begin{pmatrix}{dx}\\{dy}\end{pmatrix}
    and
    \begin{pmatrix}{dr}\\{d\theta}\end{pmatrix}
    use this result to obtain an expression for the area element dxdy in plane polar coordinates.
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  6. #6
    Super Member wingless's Avatar
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    Let R be our region that we integrate f on.

    \int\int\limits_R f(x,y)~dy~dx = \int\int\limits_R f(x,y)~dx~dy = \int\int\limits_R f(r\cos\theta,r\sin\theta) |J|~dr~d\theta
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