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Differences in presentation of ordinary partial derivatives of tensors

  1. Jan 17, 2014 #1
    Ok folks, I've taken a stab at the Latex thing (for the first time, so please bear with me).

    I've mentioned before that I'm teaching myself relativity and tensors, and I've come across a question.

    I have a few different books that I'm referencing, and I've seen them present the ordinary partial derivative in 3 different ways. It took me an hour to put together what I have below, so I have omitted the 3rd presentation, but I think I'm getting the message across with the examples below.

    1)

    [itex]V_{k'}=\frac{\partial u^{i}}{\partial u^{k'}}V_{k}[/itex]

    [itex]\frac{\partial}{\partial u^{l'}}(\frac{\partial u^{i}}{\partial u^{k'}}V_{i})=\frac{\partial^{2}u^{i}}{\partial u^{l'}\partial u^{k'}}V_{i}+\frac{\partial V_{i}\partial u^{i}}{\partial u^{l'}\partial u^{k'}}[/itex]

    2)

    [itex]A_{s'}=\frac{\partial x^{r}}{\partial x^{s'}}A_{r}[/itex]

    [itex]\frac{\partial A_{s'}}{\partial x^{k'}}=\frac{\partial^{2}x^{r}}{\partial x^{k'}\partial x^{s'}}A_{r}+\frac{\partial A_{r}\partial x^{p}\partial x^{r}}{\partial x^{p}\partial x^{k'}\partial x^{s'}}[/itex]

    In the second example, there seems to be an additional partial derivative being conducted in the second term on the RHS of the equation. This is the first step (in the text) of explaining the need for the covariant derivative, and I'm trying to understand why some texts have this additional partial derivative, and indeed the third text has 3 sets of partial derivatives in the first term of the RHS of the equations. Can anyone help me understand why this term is here, and why the presentation is inconsistent?
     
  2. jcsd
  3. Jan 18, 2014 #2
    see attachment
     

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  4. Jan 18, 2014 #3
    Right, I know they're the same from the chain rule.

    My question really I guess is why is it necessary to add the additional term? Is it because[itex]A_{r}(x^{p}(x^{k'}))[/itex]?

    Since the text seems to present this in different ways, I'd really like to understand the mechanics of why this extra term is there (and then cancelled) so that I can make sure I understand the derivations and tensor operations further along.
     
  5. Jan 18, 2014 #4
    You are guessing correctly.
     
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