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&#039;j&#039;...n&#039;} in the x^{i&#039;} system is said to behave as a contravariant tensor under the transformation \{x^i\}-&gt;\{x^{i&#039;}\} if A^{i&#039;j&#039;...n&#039;}=A^{ij...n}{p_i}^{i&#039;}{p_j}^{j&#039;}...{p_n}^{n&#039;}>>>>

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&#039; ^{i}=\frac{\partial x&#039; ^{i}}{\partial x^{j}} A^{j} (1)

For

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

So

A&#039;^{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!