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Thread: Partial Derivatives

  1. #1 Partial Derivatives 
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    Jan 2007
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    Hello community, firstly I wish you a Merry Christmas since it's the 2nd day of Christmas celebration.

    Then I've got a question:

    When I want to differentiate a function with more than 1 variable, then, for the total differential, I need to do the following:

    Let f(x1, x2, ....., xn) be a function of n variables. Then

    Df/Dx1 + Df/Dx2 + ... + Df/Dxn = df/d(x1,...,xn)

    (while D stands for the partial derivative symbol).

    Why is it NOT

    sqrt [(Df/Dx1) + ... + (Df/Dxn)] ???

    Since, when I consider it in a geometrical sense, for example in 2 dimensions,
    then I've got the change in x-direction and at the same time in y-direction.
    Now these changes would be, as far as I know, added vectorically, i.e.
    sqrt ((dx) + (dy)) , given a Euclidean distance between two points.

    Could somebody explain to me where my mstake is, please? And further, one could also choose still other metrics or norms, which would yield yet other formulas.

    Thanks a lot


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  3. #2 Re: Partial Derivatives 
    . DrRocket's Avatar
    Join Date
    Aug 2008
    Posts
    5,486
    Quote Originally Posted by mastermind
    Hello community, firstly I wish you a Merry Christmas since it's the 2nd day of Christmas celebration.

    Then I've got a question:

    When I want to differentiate a function with more than 1 variable, then, for the total differential, I need to do the following:

    Let f(x1, x2, ....., xn) be a function of n variables. Then

    Df/Dx1 + Df/Dx2 + ... + Df/Dxn = df/d(x1,...,xn)

    (while D stands for the partial derivative symbol).

    Why is it NOT

    sqrt [(Df/Dx1) + ... + (Df/Dxn)] ???

    Since, when I consider it in a geometrical sense, for example in 2 dimensions,
    then I've got the change in x-direction and at the same time in y-direction.
    Now these changes would be, as far as I know, added vectorically, i.e.
    sqrt ((dx) + (dy)) , given a Euclidean distance between two points.

    Could somebody explain to me where my mstake is, please? And further, one could also choose still other metrics or norms, which would yield yet other formulas.

    Thanks a lot
    It is neither.

    See discussion of the derivative in several variables here. http://www.thescienceforum.com/A-question-28302t.php

    In the case you have posed the derivative at a point is an nx1 matrix whic in your notation would be [Df/Dx1, Df/Dx2, ... , Df/Dxn ]


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