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Why does the multi variable chain rule need to be proven?

  1. Sep 22, 2013 #1
    It seems to me to follow from the definition of a partial derivative.

    If f(x,y) = fx + fy, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy? Then in the limiting case, the changes in f, x and y becomes differentials instead. All this seems to be given by the definitions themselves, is it not?

    So we have

    df = ∂f/∂x*dx + ∂f/∂y*dy

    All that's left is dividing by the parameter differential dt and we have our chain rule. I seem to recall there being entire lectures devoted to 'proving' this however. What am I missing?
     
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  3. Sep 22, 2013 #2

    arildno

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    What you've written is intuitive hand-waving of no rigour.
    Just because something strikes you as "obvious", doesn't mean that it is proven.
     
  4. Sep 23, 2013 #3
    I'm not hand waving, I'm referring to the definition of a partial derivative. Definitions doesn't need proving, right?

    By "If f(x,y) = fx + fy, then what else can Δf be other than ∂f/∂x*Δx + ∂f/∂y*Δy?", I'm saying that it follows from the definition that a change in f to be however much we change x by times the rate of change of f with respect to x, plus the change in y times the rate of change of f with respect to y. That's just vectorial addition.
     
  5. Sep 23, 2013 #4

    Office_Shredder

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    I don't understand this claim. Are you saying f can be written as the sum of a function depending only on x and a function depending only on y?
     
  6. Sep 23, 2013 #5
    No I was referring to a vector valued function with a î and a ĵ component, which I guess was a mistake. I don't think it's a necessary condition for my 'claim' though?
     
  7. Sep 23, 2013 #6
    Maybe I'm phrasing it incorrectly. Can we pose this question instead:

    If we define the partial of f wrt xi as the rate of change in f wrt xi, keeping everything but xi constant, in what other way can we define a change in f other than the sum of all ∂f/∂xi*Δxi? What part of this definition lacks rigour?
     
  8. Sep 23, 2013 #7

    Office_Shredder

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    The part where you say "In what other way?". The answer is that I don't know what other ways we could do this, but if you want to be rigorous you have to prove that. If someone said "in what other way could we model the real numbers?" you'd probably be stumped, but it turns out there's a whole branch of mathematics devoted to studying alternative models of the real numbers

    As one more concrete example, you seem to be assuming that if there is a small change in x, it does not have an appreciable effect on what a small change in y will do to my function (and vice versa), without even stating that it's a point of interest, much less proving it to be the case.
     
  9. Sep 23, 2013 #8

    UltrafastPED

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  10. Sep 23, 2013 #9
    Good point with the second paragraph, I guess that isn't self evident at all. If you don't mind me asking, is there such functions, exhibiting the behaviour you're talking about or are you just making a point?
     
  11. Sep 23, 2013 #10
    Differentials aren't "limiting cases" of changes in a variable. Differential geometry gives a more formal meaning to differentials and differential forms, but for now it should suffice for you to know that what you have said is mostly nonsensical.

    The multivariate chain rule needs to be proved because it is important to the understanding of multivariate calculus.
     
  12. Sep 23, 2013 #11
    Last edited: Sep 23, 2013
  13. Sep 23, 2013 #12

    Office_Shredder

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    Consider the function

    f(x,y) = xsin(1/y) for y not equal to zero
    f(x,0) = 0 for all x.

    Then let's look at f(0,0). If I jiggle just x a little bit, my function is identically zero, so the partial derivative in the x direction is zero. If I jiggle just y a little bit, my function is identically zero, so the partial derivative in the y direction is zero. However, if I jiggle x a little bit, then the function

    g(y) = xsin(1/y) for y not zero, and g(0) = 0

    is not even a continuous function, so a little change in y is going to cause a change of magnitude up to size x, regardless of how small my change in y is.
     
  14. Sep 24, 2013 #13
    Excellent attitude, I'll just creep back into my dumb person hole then. Apologies for the curiosity.
     
  15. Sep 24, 2013 #14
    (Most of you should also note that the question wasn't "why does anything that makes intuitive sense geometrically need proving ever", which I suppose is a tiresome question, but "why doesn't this follow from the definition".)
     
  16. Sep 24, 2013 #15
    Thank you. I take my attitude to be very fun. :tongue:

    The point of this is to say that what you have written is wrong, not that you are stupid. Don't be oversensitive to what we say in that regard. You're still learning (and so am I!) so it's okay to make mistakes. I make plenty of mistakes (you can see plenty of them here and here).

    Given, my dumb person hole has 10 2-liter bottles of Dr. Pepper (my Muddah loves me), so I think I'll go back there. :biggrin:
     
  17. Sep 24, 2013 #16

    WannabeNewton

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    The problem is that what you're stating is not a definition. It is, as many have pointed out, just hand-waving "physics" logic, especially the sinful act of dividing by differentials.
     
  18. Sep 24, 2013 #17

    lurflurf

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    I depends on specifics. If the single variable chain rule is carefully formulated (ie not as in most calculus books) the multiple variable chain rule is obvious as the same proof can be used with vector variables in place of real numbers. Certain definitions of partial derivatives can help as well, but I think using cute definitions obscures the idea.
     
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