I Validity of a differential expression with contravariant

1. Jan 2, 2017

eliot13

In the context of a program on the covariant derivative, I need to do the connection between the differences of each contravariant component into local basis ($$\text{d}v_{\theta}$$, $$\text{d}v_{\varphi}$$) and the terms involving Christoffel's symbol.

For a 2D spherical surface, by taking this metric tensor :

$$\begin{pmatrix} 1 & 0 \\ 0 & \sin(\theta)^2 \end{pmatrix}$$

I have found the following relations (parallel transport corresponds to $$\text{D}v^{\theta}=\text{D}v^{\phi}=0$$) :

$$\text{General expression} : \text{D}v^{i}= \text{d}v^{i} +v^{k}\Gamma_{jk}^{i}\text{d}y^{j}$$

Here are the Christoffel's symbols non vanishing :

$$\Gamma^{\theta}_{\varphi\varphi} = -\sin\theta\cos\theta$$

$$\Gamma^{\varphi}_{\theta\varphi} =\Gamma^{\varphi}_{\varphi\theta}=\cot\theta$$

So, I get the following expressions :

\notag \left.\begin{aligned} \Gamma^{\theta}_{\varphi\varphi} & = -\sin\theta\cos\theta\\ \Gamma^{\varphi}_{\theta\varphi} & =\Gamma^{\varphi}_{\varphi\theta}=\cot\theta\\ \\ \text{D}v^{\theta} & = \text{d}v^{\theta} + v^{\varphi}\Gamma^{\theta}_{\varphi\varphi}\,\text{d}\varphi\\ & = \text{d}v^{\theta} - v^{\varphi}\sin\theta\,\cos\theta\,\text{d}\varphi \quad\quad\quad\quad\quad (1)\\ \\ \text{D}v^{\varphi} & = \text{d}v^{\varphi} + v^{\theta}\Gamma^{\varphi}_{\varphi\theta}\,\text{d}\varphi + v^{\varphi}\Gamma^{\varphi}_{\theta\varphi}\,\text{d}\theta\\ & = \text{d}v^{\varphi} + \cot\theta\, (v^{\theta}\text{d}\varphi + v^{\varphi}\text{d}\theta) \quad\quad\quad\quad\quad (2) \end{aligned}\right.

Are the 2 relations for $$\text{D}v^\theta\quad\quad\quad (1)$$ and $$\text{D}v^\varphi\quad\quad\quad (2)$$ correct ?

A last question : If I take into account of $$r$$ (radius of the sphere), does it change the results on the 2 expressions above ? I mean if I start with the metric tensor :

$$\begin{pmatrix} r^2 & 0 \\ 0 & r^2\sin(\theta)^2 \end{pmatrix}$$

Thanks

Last edited: Jan 2, 2017