Hi, I'm working through Schutz's intro to GR on my own, and I'm trying to do problems as I go to make sure it sinks in. I've encountered a bump in chapter 5, though. I don't think this is a tough problem at all, I think it's just throwing me off because x and y are coordinates as well as variables, if you know what I mean.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

Hopefully my latex tags work!

Let [tex] f= x^2 +y^2 + 2xy [/tex]

[tex] V \rightarrow (x^2 +3y, y^2+3x) [/tex]

[tex] W \rightarrow (1,1) [/tex]

So, among other things, I have to express f as a function of r and theta, and find the components of V and W in the r, theta basis. I thought I knew what I was doing, but the new components of my V vector are pretty ugly, which always makes me suspicious.

2. Relevant equations

[tex] x^2 + y^2 = r^2 [/tex]

[tex] x = rcos\theta[/tex]

[tex] y=rsin\theta[/tex]

[tex] V^{\alpha '} = \Lambda^{\alpha '}_{\beta} V^{\alpha} [/tex]

3. The attempt at a solution

Well, I'm pretty sure f is just [tex] f(r, \theta) = r^2 + 2r^2\cos\theta [/tex],

I'd rather not write out the matrix equation, but using the equation above for the transformation of vector components, and writing out V in r and theta as [tex] V \rightarrow (r^2\cos^2\theta +3r\sin\theta, r^2\sin^2\theta +3r\cos\theta) [/tex], and I've calculated the transformation matrix between cartesian and polar to have components from left to right, top to bottom, of [tex] \cos\theta, \sin\theta, -\sin\theta /r, \cos\theta/r [/tex], I get pretty ugly vector components for the new coordinate system (complete with things that don't simplify much, like cos^3 + sin^3, etc).

Does it look like I'm doing it right? Completely wrong matrix?

Thanks.

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# Vector and Basis Transformations

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