Exercise 2.06

Consider $\mathbb{R}^{3}$ as a manifold with the fiat Euclidean metric, and coordinates $\{x, y, z\}$. Introduce spherical polar coordinates $\{r, \theta, \phi\}$ related to $\{x, y, z\}$ by $$ \begin{align} x&=r\,\sin \theta\,\cos \phi \\ y&=r\,\sin \theta \sin \phi \\ z&=r\,\cos \theta \end{align} $$ so that the metric takes the form $$ ds^{2}=dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin ^{2}\theta \,d\phi^{2} $$ (a) A particle moves along a parameterized curve given by $$ x(\lambda)=\cos \lambda,\quad y(\lambda)=\sin\lambda,\quad z(\lambda)=\lambda. $$ Express the path of the curve in the $\{r, \theta, \phi\}$ system.
(b) Calculate the components of the tangent vector to the curve in both the Cartesian and spherical polar coordinate systems.

(a) 

$$ \begin{cases} r(\lambda)=\sqrt{ 1+\lambda^{2} } \\ \theta(\lambda)=\arctan(1/\lambda) \\ \phi(\lambda)=\lambda \end{cases} $$ 

(b) 

In cartesian coordinates 

$$ V^{\mu}=\frac{d}{d\lambda}\begin{pmatrix} \cos\lambda \\ \sin\lambda \\ \lambda \end{pmatrix} =\begin{pmatrix} -\sin\lambda \\ \cos\lambda \\ 1 \end{pmatrix} $$ 

 In spherical polar coordinates 

$$ V^{\mu}=\frac{d}{d\lambda}\begin{pmatrix} \sqrt{ 1+\lambda^{2} } \\ \arctan(1/\lambda) \\ \lambda \end{pmatrix} =\begin{pmatrix} \frac{\lambda}{\sqrt{ 1+\lambda^{2} }} \\ \frac{-1}{1+\lambda^{2}} \\ 1 \end{pmatrix} $$