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Derivative and Jacobian of a transformation

  1. Feb 13, 2012 #1

    ElijahRockers

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    1. The problem statement, all variables and given/known data

    For the transformation, draw the lattice lines, calculate the derivative, and calculate the Jacobian.

    [itex]x=rcos\theta[/itex]
    [itex]y=rsin\theta[/itex]

    3. The attempt at a solution

    I drew the lattice lines correctly. What I am confused about is the derivative. Since x and y are both functions of r and theta, what derivative are they talking about? Wouldn't I have to take the partial with respect to R or theta? This section is supposed to be on calculating determinants.

    I understand that the Jacobian is the determinant of a particular matrix, but where does this matrix come from?
     
  2. jcsd
  3. Feb 13, 2012 #2

    lanedance

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    the Jacobian matrix is the matrix of partial derivatives
    [tex]
    \begin{pmatrix}
    \frac{\partial x}{\partial r} & \frac{\partial x}{\partial \theta} \\
    \frac{\partial y}{\partial r} & \frac{\partial y}{\partial \theta} \\
    \end{pmatrix}
    [/tex]

    the Jacobian determinant is the determinant of that matrix and is probably what you're referring to with the shorthand "Jacobian"
     
  4. Feb 13, 2012 #3

    lanedance

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  5. Feb 13, 2012 #4

    ElijahRockers

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    Alright thanks. Yea I looked at the Jacobian wikipedia earlier, before I posted this, but it all just seemed greek to me. I am about to try to finish the assignment.

    Thanks again, I'll let you know how it goes.
     
    Last edited: Feb 13, 2012
  6. Feb 14, 2012 #5

    lanedance

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    note if you had a small displacement (dr,dtheta)^T, multiplying this by the jacobian would give you the corresponding (dx,dy), similar to the chain rule
     
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