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Coding > WebGL Simulation - Parallel Transport and Covariant Derivative on Spherical Surface



  • Tested on Firefox 60.0, Chrome 66.0 and Opera 53.0
  • Possible overlapping bug or no first view loaded for sphere on Safari 11.0.3
\begin{equation} \begin{aligned} \definecolor{White}{RGB}{255,255,255}\color{White} \text{Absolute Differential} &\definecolor{White}{RGB}{255,255,255}\color{White}\,:\,(\text{D}\vec{V})^{i}= \text{d}v^{i}+v^{k}\Gamma_{jk}^{i}\text{d}y^{j} \end{aligned} \end{equation} \begin{equation} \left.\begin{aligned} \definecolor{White}{RGB}{255,255,255}\color{White} \Gamma^{\theta}_{\varphi\varphi} &\definecolor{White}{RGB}{255,255,255}\color{White}= -\sin\theta\cos\theta\\ \definecolor{White}{RGB}{255,255,255}\color{White} \Gamma^{\varphi}_{\theta\varphi} &\definecolor{White}{RGB}{255,255,255}\color{White}= \Gamma^{\varphi}_{\varphi\theta}=\cot\theta\\ \definecolor{White}{RGB}{255,255,255}\color{White} (\text{D}\vec{V})^{\theta} &\definecolor{White}{RGB}{255,255,255}\color{White}= \text{d}v^{\theta} + v^{\varphi}\Gamma^{\theta}_{\varphi\varphi}\,\text{d}\varphi\\ \definecolor{White}{RGB}{255,255,255}\color{White} &\definecolor{White}{RGB}{255,255,255}\color{White}= \text{d}v^{\theta} - v^{\varphi}\sin\theta\,\cos\theta\,\text{d}\varphi\\ \definecolor{White}{RGB}{255,255,255}\color{White} (\text{D}\vec{V})^{\varphi} &\definecolor{White}{RGB}{255,255,255}\color{White}= \text{d}v^{\varphi} + v^{\theta}\Gamma^{\varphi}_{\varphi\theta}\,\text{d}\varphi + v^{\varphi}\Gamma^{\varphi}_{\theta\varphi}\,\text{d}\theta\\ \definecolor{White}{RGB}{255,255,255}\color{White} &\definecolor{White}{RGB}{255,255,255}\color{White}= \text{d}v^{\varphi} + \cot\theta\,(v^{\theta}\text{d}\varphi + v^{\varphi}\text{d}\theta) \end{aligned}\right. \end{equation}

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