I'm having trouble with this question. It's from Rindler's Introduction to Special Relativity which I'm going through myself. I'm just starting to learn about tensors.(adsbygoogle = window.adsbygoogle || []).push({});

<<<<i) A vector [tex]A^i[/tex] has components [tex]\dot{x}, \dot{y}[/tex] in rectangular Cartesian coordinates; what are its components in polar coordinates?>>>>

This part I believe I know. The components are [tex]\dot{r}, r\dot{\theta}[/tex]. The first component is the [tex]a_r[/tex] component and the second is the [tex]a_{\theta}[/tex] component.

<<<<ii) A vector [tex]B^i[/tex] has components [tex]\ddot{x}, \ddot{y}[/tex] in rectangular Cartesian coordinates; prove, directly from A.3 that its components in polar coordinates are [tex]\ddot{r}-r{(\ddot{\theta})}^2, \ddot{\theta}+2\dot{r}\dot{\theta}/r[/tex]>>>>

This is what A.3 says:

<<<<An object having components [tex]A^{ij....n}[/tex] in the [tex]x^i[/tex] system of coordinates and [tex]A^{i'j'...n'}[/tex] in the [tex]x^{i'}[/tex] system is said to behave as acontravarianttensor under the transformation [tex]\{x^i\}->\{x^{i'}\}[/tex] if [tex]A^{i'j'....n'}=A^{ij....n}{p_i}^{i'}{p_j}^{j'}....{p_n}^{n'}[/tex]>>>>

I'm not sure how this is to be done. The [tex]a_{\theta}[/tex] coordinate in part ii) seems to be divided by r. I don't know if this is a mistake in the book or there is some reason for it.

How do I use the definition of contravariant tensors to derive the formula for acceleration in polar coordinates? I really have no clue. I can derive the formula just using derivatives, but I don't see how to use tensors to derive it.

Thanks a bunch for your help!

**Physics Forums | Science Articles, Homework Help, Discussion**

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!

# Question regarding tensors derive acceleration in polar form

**Physics Forums | Science Articles, Homework Help, Discussion**