The three axes of the skewed coordinate systems are $[1, 0, 0]$, [$1/\sqrt{2}$, $1/\sqrt{2}$, 0] and [$1/\sqrt{3}$, $1/\sqrt{3}$, $1/\sqrt{3}$]. What is the transformation matrix $t_{ij}$ and $g_{ij}$ that transform from Cartesian to the skewed system and from the skewed to the Cartesian respectively. I have read a book entitled mathematical Physics- Applied mathematics for Scientists and Engineering for a few days and still cannot figure it out. Please help! Thank you.
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Let consider the matrix
$$M =\begin{bmatrix} 1 & \frac1{\sqrt 2} & \frac1{\sqrt 3}\\ 0 & \frac1{\sqrt 2} &\frac1{\sqrt 3}\\ 0 & 0 & \frac1{\sqrt 2}\end{bmatrix}$$
then $M$ is the transformation matrix from the new coordinate system to the standard cartesian system and $M^{-1}$ is the transformation matrix from the standard cartesian system to the new coordinate system.
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$\begingroup$ Thank you for your prompt response. Does that mean for a tensor $\bar{\bar{\mu^\prime}}$ in the skewed system, then the tensor in the Cartesian $\bar{\bar{\mu}} = M^{-1} \bar{\bar{\mu^\prime}} M$ ? $\endgroup$– JunJun 1, 2018 at 14:29