For what covariant derivative?

In summary, the conversation discusses the use of the differential form of position vector r, and the process of finding d\hat{r}. The theory of covariant derivative is necessary to understand Christoffel symbols, which appear in the process of differentiating on curved surfaces. The final result is d\vec{r}=dr\hat{r}+d\theta \hat{\theta}, but the questioner notes an error and the correct result should be d\vec{r}=dr\hat{r}+rd\theta \hat{\theta} with a factor of r in the second term. This use of covariant and contravariant terms is necessary for differentiation in curved spaces.
  • #1
Jhenrique
685
4
I will take the differential form of position vector r:

##\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?
 
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  • #2
You need Christoffel symbols to take derivatives on curved surfaces. Therefore they appear whenever you're varying any quantity in a curved space. It should look like this for differentiating covariant terms, [tex]\nabla_\mu x^\nu = \partial_\mu x^\nu + \Gamma_{\mu \alpha}^\nu x^\alpha [/tex] and [tex]\nabla_\mu x_\nu = \partial_\mu x_\nu - \Gamma_{\mu \nu}^\alpha x_\alpha [/tex] for contravariant terms.
 

What is a covariant derivative?

A covariant derivative is a mathematical concept used in differential geometry to describe how a vector field changes as it moves along a curved surface. It takes into account the curvature and other geometric properties of the surface, and is used to define the concept of parallel transport.

Why is a covariant derivative needed?

A covariant derivative is needed because regular derivatives do not work well on curved surfaces. On a flat surface, the derivative of a vector field is simply the change in the vector's components as it moves along the surface. However, on a curved surface, this approach does not work because the vector's components also change due to the curvature of the surface. The covariant derivative takes into account this curvature and provides a more accurate way to describe how the vector field changes.

How is a covariant derivative calculated?

The calculation of a covariant derivative involves using a connection, which is a mathematical construct that describes how vectors change as they are transported along a surface. The connection is used to define the covariant derivative operator, which is then applied to the vector field to obtain the covariant derivative.

What is the difference between a covariant derivative and a regular derivative?

The main difference between a covariant derivative and a regular derivative is that the covariant derivative takes into account the curvature of the surface, while the regular derivative does not. This means that the covariant derivative is better suited for describing how vectors change on curved surfaces, while the regular derivative is better suited for flat surfaces.

What are some applications of covariant derivatives?

Covariant derivatives have many applications in mathematics and physics. They are used in differential geometry to study curved surfaces and in general relativity to describe the curvature of spacetime. They are also used in other areas of physics, such as quantum mechanics and electromagnetism, to describe how particles and fields behave on curved surfaces.

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