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nLab > Latest Changes: Differentiable maps

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    • CommentRowNumber1.
    • CommentAuthorTobyBartels
    • CommentTimeNov 4th 2011

    I’ve rather greatly expanded differentiable map, defining variations.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeNov 4th 2011
    • (edited Nov 4th 2011)

    There is also smooth function. Somehow these entries should eventually be merged, or at least interlinked.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeNov 4th 2011

    Thanks for pointing out. On the other hand, I think we should keep them separated. Smooth is differentiable infinitely many times. It is good to have the elementary differentiability questions separated from the infinitely differentiable case. For example into differentiable one should eventually write examples of various pathologies, laws like Schwarz law for interchange of partial derivatives, various kinds of differentiabilities in infinite-dimensional case, questions of completion of spaces of C kC^k-functions and so on. Smooth case is more the ideal best situations where one could concentrate on geometrical applications and the extension to more general types of spaces; over there various kinds of differentiability should be avoided.

    • CommentRowNumber4.
    • CommentAuthorTobyBartels
    • CommentTimeNov 7th 2011
    • (edited Nov 7th 2011)

    The entries do link to one another. I agree with Zoran about keeping them separate. Also beware the cache bug: use smooth map.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeApr 20th 2014

    I added some definitions, and further examples, to differentiable map.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeMay 10th 2014

    I added a comment about symmetry of second derivatives to differentiable map, plus the standard counterexample in the absence of continuity. I may have more to add soon.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeMay 11th 2014

    I’ve added to differentiable map a couple of definitions of “strong twice-differentiability” that don’t require continuity of the second derivative. Any suggestions of better names for these definitions would be welcome.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeMay 13th 2014

    One last edit: I’ve added a proof that every twice-differentiable function has a symmetric second derivative (and removed the ad hoc names for stronger twice-differentiability). A friendly person at MSE pointed me to a proof of this in Dieudonné’s treatise on analysis. Why do we not learn this theorem in multivariable calculus, instead of the weaker and more random-looking statement that the mixed partial derivatives are symmetric if they are continuous?

    • CommentRowNumber9.
    • CommentAuthorTobyBartels
    • CommentTimeMay 21st 2014

    Well, I mention this theorem in my class, at least!

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMay 21st 2014

    Does your textbook?

    • CommentRowNumber11.
    • CommentAuthorTobyBartels
    • CommentTimeMay 21st 2014

    No.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeJun 7th 2017

    I have touched differentiable map:

    1. added background remarks on Euclidean space to beginning of Definition-section,

    2. added numbered environments to the Definition-section and pointers to them from the Examples-section

    3. added the definition of differetiable functions between differentiable manifolds

1 to 12 of 12

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