Hi everyone, I am trying to self study some general relativity however I met some problem in the(adsbygoogle = window.adsbygoogle || []).push({});

contravarient and covarient basis.

In the lecture, or you can also find it on wiki page 'curvilinear coordinates',

the lecturer introduced the tangential vector e_{i}=∂r/∂xi and the gradient vector e^{i}=del x^{i}

and showed them to have a relation of e^{i}dot e_{j}= δ^{i}_{j}.

I totally follow and agree the algebraic steps, both on the lecture and on the wiki page.

However I cannot figure it out GEOMETRICALLY ! Let consider a 2D curvilinear coordinates,

Given we have two tangential basis e_{1}, e_{2}which are not parallel.

How can one geometrically (by drawing and inspection) contruct a third vector that is parallel to e_{1}but orthogonal to e_{2}?

However from algebraically we showed the del x^{1}is the wanted vector !!

Lastly, I tried to explicitly see what is going on in the polar coordinate (we are familiar with polar coordinate) however the tangential vector e_{r}and e_{θ}are orthogonal...

Appreciate you kindly help.

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# I Contracovarient and covarient basis form orthogonal basis

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