Covariant Derivative: A^μₛᵦ Definition & Use

[matt grime]

And back again into the loop: what exactly is the definition of covariant derivative you're starting from then? Because the problem seems to be you can't translate from your definition to this one.

 

[pervect]

For the "pictoral" approach I mentioned in more detail, if you can get a hold of MTW's book "Gravitation", it's discussed around pg 244 in chapter 10.

As far as the detailed answer to your question

A general vector x can be represented as $x^i e_i$, where the $e_i$ are the basis vectors.

Thus in a cartesian coordinate system we would have as basis vectors e_x, e_y, e_z, in a cylindrical coordinate system we would have e_r, e_theta, e_z, etc etc.

The definition of the Christoffel symbols

[tex]\Gamma^{\mu}{}_{\sigma \alpha}[/itex] is that they describe how the basis vectors transform in terms of the basis vectors.<br /> <br /> i.e we take<br /> <br /> $$\nabla_{\sigma} e_{\alpha}$$ <br /> <br /> and express it in terms of the basis vectors as<br /> <br /> $$\nabla_{\sigma} e_{\alpha} = \Gamma^{\mu}{}_{\alpha \sigma} e_{\mu}$$<br /> <br /> The rest is the chain rule. $$\nabla_{\sigma} A^{\mu}e_{\mu} = (\nabla_{\sigma}A^{\mu}) e_{\mu} +(\nabla_{\sigma}e_{\mu})A^{\mu}$$<br /> <br /> The first term gives the partial derivative, the second term gives the Christoffel symbols.[/tex]