I've heard of something called a covariant derivative. what motivates it and what is it?
That's not quite right. When one calculates the directional derivative of a vector you need two things. The vector field and a vector which determines the direction you're interested.Ok this question is stupid, but can't we just use the chain rule to calculate the directional derivative of a tensor field in an arbitrary direction(byt that I mean can the directional derivative be written as a linear combination of the covariant derivative along corrdinate axis)? I heard that you can't but don't know why you wouldn't be able to.
Calculating the Christoffel symbols can be laborious at times but once you've done it a dozen or so times it will become second nature to you.If so how do we calculate it in a rbitrary direction. please don't tear me apart. There's something weird about the covariant derivative. the christoffel symbols make computation seem impossbile.
http://eom.springer.de/c/c026870.htmhow does one derive the general formula for the covariant derivative of a tensor field? To be more precise I took out sean carolls book at the library but did not understand equation 3.17 on page 97. Could someone derive it or prove it, or at the very least give me a better hint?