#### learningphysics

Homework Helper

- 4,099

- 5

<<<<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 a

*contravariant*tensor 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!