Question regarding tensors derive acceleration in polar form

[learningphysics]

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.

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

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

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

This is what A.3 says:
<<<<An object having components $$A^{ij...n}$$ in the $$x^i$$ system of coordinates and $$A^{i'j'...n'}$$ in the $$x^{i'}$$ system is said to behave as a contravariant tensor under the transformation $$\{x^i\}->\{x^{i'}\}$$ if $$A^{i'j'...n'}=A^{ij...n}{p_i}^{i'}{p_j}^{j'}...{p_n}^{n'}$$>>>>

I'm not sure how this is to be done. The $$a_{\theta}$$ 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!

 

[dextercioby]

$$A' ^{i}=\frac{\partial x' ^{i}}{\partial x^{j}} A^{j}$$ (1)

For

$$A^{j}=\left(\dot{x},\dot{y}\right)$$ (2)

So

$$A'^{1}=\frac{\partial\rho}{\partial x} \dot{x}+\frac{\partial\rho}{\partial y} \dot{y} =\cos\phi \ \left(\dot{\rho}\cos\phi-\rho\dot{\phi}\sin\phi\right)+\sin\phi \ \left(\dot{\rho}\sin\phi+\rho\dot{\phi}\cos\phi\right)=\dot{\rho}$$

This is for the first comp of the transformed velocity.

U do the other "3" (one for velocity & 2 for acceleration).

Daniel.

 

[learningphysics]

Cool! Thanks a bunch dexter!