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Trying to understand covariant derivative, conceptually and visually. Help please?

  1. Dec 30, 2009 #1

    I registered here today because I'm quite curious about the covariant derivative, and although I've consulted several texts on the subject (and wikipedia, and other locations), I've found it somewhat difficult to piece together a visual understanding of the covariant derivative. The material I've found, while somewhat helpful, leaves much to be desired in terms of simplifying the subject.

    I've come here to ask you if I at least sort of have it right, and below are pictures which illustrate the best that I've been able to understand from the material I've read so far.

    I am eighteen years old and have no grounding in differential geometry, advanced analysis, or any extremely rigorous mathematics. I'm sure there are plenty of other subjects I should tackle in advance of tensor calculus, but I've set my sights on it and I figure that I'll learn anything I need to along the way. I understand the meaning of summation notation, and at this point, I think I've gotten to the point where I understand what tensors are and what meaning they have (hard-fought victories, I assure you). I am studying on my own, and I am not personally familiar with anyone qualified to teach the material.

    Now, I am not trying to deal with the matter of covariant derivative computation just yet. I've seen the formula, and I know what Christoffel symbols look like; I could learn it by rote if necessary. But I've never been satisfied learning by rote - without a visual and conceptual understanding of a subject, I can't feel comfortable carrying out computations.

    So, for this simple example I've imagined, can you tell me if I've gotten the gist of the covariant derivative, and if not (which I imagine is more probable than my having understood it), could you please show me a good visual representation of the covariant derivative?

    (The division at the end isn't mean to express real computation, just the gist of the change with respect to the variable.)
  2. jcsd
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