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Math Help - Metric Components

  1. #1
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    Metric Components

    Calculate the metric and conjugate metric components for oblique cylindrical coordinates (r,\phi,\eta) where:

    x=rcos \phi, y=rsin \phi + \eta cos \alpha, z=\eta sin \alpha and \alpha is a parameter.

    I can calculate the metric components but want to know how I would be expected to calculate the conjugate metric components.

    For the metric components I get:

    g_{ij}=<br />
\begin{array}{ccc}<br />
1&0& Sin \phi Cos \alpha<br />
\\0&r^2&rCos \phi Cos \alpha<br />
\\Sin \phi Cos \alpha&rCos \phi Cos \alpha&1<br />
\end{array}<br />

    It seems to me that I need to either find (r,\phi,\eta) in terms of x,y,z and then calculate the conjugate basis or alternatively I need to find the inverse of the matrix above. Either task seems to be too hard for the number of marks awarded to the question.
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  2. #2
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    Quote Originally Posted by Kiwi_Dave View Post
    Calculate the metric and conjugate metric components for oblique cylindrical coordinates (r,\phi,\eta) where:

    x=rcos \phi, y=rsin \phi + \eta cos \alpha, z=\eta sin \alpha and \alpha is a parameter.

    I can calculate the metric components but want to know how I would be expected to calculate the conjugate metric components.

    For the metric components I get:

    g_{ij}=<br />
\begin{array}{ccc}<br />
1&0& Sin \phi Cos \alpha<br />
\\0&r^2&rCos \phi Cos \alpha<br />
\\Sin \phi Cos \alpha&rCos \phi Cos \alpha&1<br />
\end{array}<br />

    It seems to me that I need to either find (r,\phi,\eta) in terms of x,y,z and then calculate the conjugate basis or alternatively I need to find the inverse of the matrix above. Either task seems to be too hard for the number of marks awarded to the question.
    Once you have written the metric tensor as a matrix, the "conjugate" is just the inverse matrix: g^{ij}g_{jk}= g^i_k is the identity matrix.
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  3. #3
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    But finding the inverse of this matrix seems unreasonably difficuilt given the value of the question.
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