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I Validity of a differential expression with contravariant

  1. Jan 2, 2017 #1
    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 :

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


    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
  2. jcsd
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