# I Validity of a differential expression

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1. Jan 1, 2017

### eliot13

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

For a 2D spherical surface, I have found the following relations (parallel transport corresponds to $$\text{D}v_{\theta}=\text{D}v_{\phi}=0$$) :

Here are the Christoffel's symbols non vanishing :

$$\text{General expression} : \text{D}v_{i}= \text{d}v_{i}-v_{k}\Gamma_{ij}^{k}\text{d}y^{j}$$

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

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

So, I get the following expressions :

$$\text{D}v_{\theta} = \text{d}v_{\theta} - v_{\varphi}\Gamma^{\varphi}_{\varphi\theta}\,\text{d}\varphi = \text{d}v_{\theta} - v_{\varphi}\cot\theta\,\text{d}\varphi\quad\quad\quad (1)$$
$$\text{D}v_{\varphi} = \text{d}v_{\varphi} - v_{\theta}\Gamma^{\theta}_{\varphi\varphi}\,\text{d}\varphi - v_{\varphi}\Gamma^{\varphi}_{\theta\varphi}\,\text{d}\theta$$
$$= \text{d}v_{\varphi} + v_{\theta}\sin\theta \cos\theta\,\text{d}\varphi - v_{\varphi}\cot\theta\,\text{d}\theta\quad\quad\quad (2)$$

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 1, 2017