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?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# 2-dimension polar basis vectors

Loading...

Similar Threads - dimension polar basis | Date |
---|---|

I Levi-Civita properties in 4 dimensions | Apr 23, 2017 |

I Ricci curvatures determine Riemann curvatures in 3-dimension | Nov 29, 2016 |

I Example of computing geodesics with 2D polar coordinates | Aug 6, 2016 |

I Can there be a bounded space w/o a boundary w/o embedding? | Jul 19, 2016 |

Maxwell's Equations in N Dimensions | Feb 22, 2016 |

**Physics Forums - The Fusion of Science and Community**