I will take the differential form of position vector r:(adsbygoogle = window.adsbygoogle || []).push({});

##\vec{r}=r\hat{r}##

##d\vec{r}=dr\hat{r}+rd\hat{r}##

So, now I need find ##d\hat{r}##

##d\hat{r}=\frac{d\hat{r}}{dr}dr+\frac{d\hat{r}}{d\theta}d\theta##

##\frac{d\hat{r}}{dr}=\Gamma ^{r}_{rr}\hat{r}+\Gamma ^{\theta}_{rr}\hat{\theta}=0\hat{r}+0\hat{\theta}=\vec{0}##

##\frac{d\hat{r}}{d\theta}=\Gamma ^{r}_{r\theta}\hat{r}+\Gamma ^{\theta}_{r\theta}\hat{\theta}=0\hat{r}+\frac{1}{r}\hat{\theta}=\frac{1}{r}\hat{\theta}##

So...

##d\hat{r}=\vec{0}dr+\frac{1}{r}\hat{\theta}d\theta=\frac{1}{r}d\theta \hat{\theta}##

Resulting in:

##d\vec{r}=dr\hat{r}+r\frac{1}{r}d\theta \hat{\theta}=dr\hat{r}+d\theta \hat{\theta}##

So, I have 2 question:

1) What the theory of covariant derivative has to do with this? Why I need understand covariante derivative? Where it appears? What expression it simplifies?

To understand the christofell's symbols is necessary because it appears in the process. But I don't see the covariant derivative in process...

2) Why my result is ##d\vec{r}=dr\hat{r}+d\theta \hat{\theta}##? It's wrong! Because ##d\vec{r}=dr\hat{r}+rd\theta \hat{\theta}## (with a factor r in 2nd term). However, I did all computation correctly. Where is the wrong?

**Physics Forums - The Fusion of Science and Community**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# For what covariant derivative?

Loading...

Similar Threads - covariant derivative | Date |
---|---|

I What is the covariant derivative of the position vector? | Feb 18, 2018 |

I Covariant derivative of Ricci scalar causing me grief! | Nov 26, 2017 |

A Covariant derivative only for tensor | Sep 23, 2017 |

I Product rule for exterior covariant derivative | May 15, 2017 |

I Conservation of dot product with parallel transport | Jan 15, 2017 |

**Physics Forums - The Fusion of Science and Community**