Originally Posted by

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

$\displaystyle x=rcos \phi$, $\displaystyle y=rsin \phi + \eta cos \alpha$, $\displaystyle z=\eta sin \alpha$ and $\displaystyle \alpha$ 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:

$\displaystyle g_{ij}=

\begin{array}{ccc}

1&0& Sin \phi Cos \alpha

\\0&r^2&rCos \phi Cos \alpha

\\Sin \phi Cos \alpha&rCos \phi Cos \alpha&1

\end{array}

$

It seems to me that I need to either find $\displaystyle (r,\phi,\eta)$ 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.