Covectors, Metric Tensors and Unit Vectors

Asked Mar 26 '24 at 22:01

Active Mar 27 '24 at 10:45

Viewed 53 times

In orthogonal cartesian coordinates $c\cdot c=|c|^2$ is the square of length of the vector $c$.

In oblique coordinates the square of the length of $c = c * g * c$ where g is the metric tensor and $g*c$ is a covector.

If $e_x$ and $e_y$ are the basis vectors of vector $c$ then how does one calculate the new basis vectors $e_x'$ and $e_y'$ of $g*c$?

Am I correct in assuming that $e_x'$ will be perpendicular to $e_y'$?

A relevant video can be seen here.

edited Mar 27 '24 at 10:45

Daniele Tampieri

asked Mar 26 '24 at 22:01

R. Emery

Can you please write this question more clearly? You will get a much better answer if you explain and set things up more coherently. – KyleBroder Mar 26 '24 at 22:25
I think the answer is here: https://en.wikipedia.org/wiki/Skew_coordinates – R. Emery Mar 26 '24 at 22:59
1

Your dual basis, ${e^i}$, will be completely determined by your vector basis, ${e_i}$, and the identity;
$$e^i(e_j)=\delta^i_j$$
– Volk Mar 27 '24 at 04:01
@Volk Thank you. That was the answer I was looking for. – R. Emery Mar 27 '24 at 18:51
https://en.wikipedia.org/wiki/Dual_basis – R. Emery Mar 29 '24 at 22:51

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