I'd like to understand why i cannot seem to be able to define unit polar basis vectors. Let me explain:(adsbygoogle = window.adsbygoogle || []).push({});

We have our usual polar coordinates relation to Cartesian:

x = r cosθ ; y = r sinθ

if I define [itex]\hat{e_{r}}[/itex], [itex]\hat{e_{\vartheta}}[/itex] as the polar basis vectors, then they should be contravariant, meaning that they can be obtained from [itex]\hat{u_{x}}[/itex], [itex]\hat{u_{y}}[/itex] as:

[itex]\hat{e_{r}}[/itex] = [itex]\delta x/ \delta r\ \hat{u_{x}} + \delta y / \delta r \ \hat{u_{y}}[/itex] = cosθ [itex]\hat{u_{x}}[/itex] + sin θ [itex]\hat{u_{y}}[/itex]

and

[itex]\hat{e_{\vartheta}}[/itex] = [itex]\delta x/ \delta \vartheta \ \hat{u_{x}} + \delta y / \delta \vartheta \ \hat{u_{y}} [/itex] = -r sinθ [itex]\hat{u_{x}}[/itex] + r cosθ [itex]\hat{u_{y}}[/itex]

which implies that |[itex]\hat{e_{\vartheta}}[/itex]| = r, rather than being a unit vector as usually considered.

Is this right?

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# 2-dimension polar basis vectors

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