1. ## Metric Components

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.

2. 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.
Once you have written the metric tensor as a matrix, the "conjugate" is just the inverse matrix: $\displaystyle g^{ij}g_{jk}= g^i_k$ is the identity matrix.

3. But finding the inverse of this matrix seems unreasonably difficuilt given the value of the question.