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I'm taking a first course in General Relativity but I've been struggling with coordinate system transformation. For example, if I have a Vector defined in Cartesian (x,y) coordinates as $V_x=x^2+3y$ and $V_y=y^2+3x$, and I want to transform it into a polar coordinate system. First I should achieve the parameterization in the polar coordinates i.e. substituting $x=rcos(\theta)$ and $y=rsin(\theta)$ and then apply the transformation tensor?

By this process I would obtain the next: $$V^\mu(x,y)=\begin{bmatrix}x^2+3y\\y^2+3x\end{bmatrix}$$ $$V^\mu(r,\theta)=\begin{bmatrix}r^2cos^2(\theta)+3rsin(\theta)\\r^2sin^2(\theta)+3rcos(\theta)\end{bmatrix}$$ And after that apply the transformation $\Lambda^{\bar{\mu}}_{\mu}$

I feel like this is such an elementary question and yet it troubles me somehow.

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closed as off-topic by ACuriousMind, Gert, Martin, Kyle Kanos, user36790 Mar 10 '16 at 12:04

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  • $\begingroup$ I think you have to multiply it by the Jacobian Matrix as well. $\endgroup$ – Arif Burhan Mar 7 '16 at 19:56
  • $\begingroup$ That's what i meant by the transformation $\Lambda_{\mu}^{\bar{\mu}}$. So, that would be it? $\endgroup$ – RaulJimenez Mar 7 '16 at 20:45