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I am really attracted by the field of differential geometry which generalize analysis on euclidean spaces that I've been working with my whole life. However by learning the field I encountered different notion of derivatives, namely :

  1. The tangent vector ( directional derivative of a function )
  2. The ehresman connection ( derivative of a section )
  3. The covariant derivative
  4. The exterior derivative ( derivative on the graded antisymmetric multilinear algebra of differential forms )
  5. The exterior covariant derivative (Used to define curvature from a one form ?
  6. The Lie derivative ( derivative along curves of vector and tensors fields )

First of all I know I lack knowledge and intuition on all of them. I know the definition of them, I know their difference, to what object they act on, what info we need to use them etc. I am not looking here for definitions.

I would like to know why do we need so many definition and approach to derivatives. I would hope to have a general version of derivatives that will apply to all objects but that doesn't seem to be the case here.

What are the need for all of them?

In what case do we need one and not the others ?

Also I would really appreciate a references that would list all of thoses derivatives, minor intuition on them and also explain why and when do we have to use one or the other.

PS: I have already saw most of the other responses on this forum thanks.

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  • $\begingroup$ you are missing the functional or variational derivative $\endgroup$
    – janmarqz
    Commented Dec 31, 2016 at 18:07
  • $\begingroup$ Never heard of that but thanks ! $\endgroup$
    – user149705
    Commented Jan 1, 2017 at 0:23

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This is more of a extended comment than answer. I like to think every kind of derivative as boundary maps of some chain complex. Given a manifold $M$ you can construct various vector bundles over $M$. Now you can associate variuos chain complexes with these bundles (as a concrete example think of the de Rahm complex). These derivatives are some forms of derivative maps (e.g. exterior derivative for de Rahm complex) or composition of derivative maps with some special map (e.g Lie derivatives [via Cartan's formula]). You can also express curvature tensor as a composition of boundary maps (this actually shows that curvature is a generalization of second derivative). Chapter 13 of This book explain the above statements for curvature and connections. For complex manifolds, some of these are explained in the book "Differential Analysis on Complex Manifolds" by R. O. Wells. I don't know any reference where all of these are present.

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  • $\begingroup$ Thank you for your contribution. I was aware of the boundary point of view regarding the exterior derivative of differential forms. I wasn't aware that you could extend this idea to other derivatives. Unfortunately I am looking more for a practical answer, along the lines of why do we not settle down on a universal version of derivatives. And if this is not possible in what kind of application do we need each one more than the other. $\endgroup$
    – user149705
    Commented Jan 1, 2017 at 23:58

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