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I Validity of a differential expression

  1. Jan 1, 2017 #1
    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 :

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

    r^2 & 0 \\
    0 & r^2\sin(\theta)^2

    Last edited: Jan 1, 2017
  2. jcsd
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