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What's the difference between dx and ∂x?

  1. Jun 20, 2013 #1
    I understand from the equation:
    [tex] df = {\frac{\partial f}{\partial x}} dx + {\frac{\partial f}{\partial y}} dy [/tex]
    that: [itex] dx \neq \partial x [/itex].

    I understand why [itex] df \neq \partial f [/itex]
    ([itex]df[/itex] is the change in f when the change in all the variables is infinitesimal, and [itex] \partial f [/itex] is the change in f when the change in one vriable is infinitesimal and the others are constant, right?).

    but doesn't both [itex] dx [/itex] and [itex] \partial x [/itex] mean an infinitesimal change in x? if so, why aren't they equal? if not what do they mean?
     
  2. jcsd
  3. Jun 20, 2013 #2

    mfb

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    x could depend on y, for example. Or both x and y could depend on some common other variables. In the same way as ##df \neq \partial f##, ##dx \neq \partial x##
     
  4. Jun 20, 2013 #3
    how could x depend on y? aren't they both independent variables?
     
  5. Jun 20, 2013 #4

    mfb

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    Who said that?
     
  6. Jun 20, 2013 #5
    does it really matter?

    btw i meant that for f(x,y) if that wasn't clear.
     
  7. Jun 21, 2013 #6

    Fredrik

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    If you want to make sense of this for a function ##f:\mathbb R^2\to\mathbb R## (without using definitions from differential geometry), then you should define ##df:\mathbb R^4\to\mathbb R## by
    $$df(x,y,z,w)=D_1f(x,y)z+D_2f(x,y)w$$ for all ##x,y,z,w\in\mathbb R##. This ensures that for all ##x,y,dx,dy\in\mathbb R##, we have
    $$df(x,y,dx,dy)=D_1f(x,y)dx+D_2f(x,y)dy.$$ What you wrote can be considered a sloppy notation for this result.

    This is a way to make sense of df, dx and dy. I don't think I have ever seen anyone try to make sense of ##\partial x##. It's just a small part of the notation ##\partial f/\partial x##, which means ##D_1f##. I find the notation ##\partial f/\partial x## misleading, since it hides the fact that ##D_1f## (the partial derivative of f with respect to the first variable slot) is the same function no matter what symbols we usually use to represent the numbers we plug into ##f##.

    Note that with this definition, there's no need for dx and dy to be "infinitesimal". What does that even mean? I know that there's a definition of that term, but I haven't studied it, and I'm pretty sure that almost no one of the authors of physics books who use that term have either. If you see the term "infinitesimal" in a physics book, you should assume that it has nothing to do with infinitesimals. You should interpret it as a code that let's you know that the next thing that follows is a first-order approximation. For example, to say that for infinitesimal x, we have ##e^x=1+x##, is to say (in a weird and confusing way) that there's a function ##R:\mathbb R\to\mathbb R## such that ##e^x=1+x+R(x)## for all ##x\in\mathbb R##, and ##R(x)/x\to x## as ##x\to 0##.

    df(x,y,z,w) is a first-order approximation of the difference f(x+z,y+w)-f(x,y). So if we want to be weird and confusing, we can say that for infinitesimal z and w, we have f(x+z,y+w)-f(x,y) = df(x,y,z,w). If we want to cause some additional confusion, we can use the notations dx and dy for the real numbers z and w, and then drop the (x,y,z,w) from df(x,y,z,w), and the (x,y) from ##D_1f(x,y)## and ##D_2f(x,y)##. Then we would be saying that for infinitesimal dx and dy, we have
    $$f(x+dx,y+dy)-f(x,y)=df=D_1f dy+ D_2f dx =\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy.$$
     
    Last edited: Jun 21, 2013
  8. Jul 24, 2016 #7
    Δ = difference

    d = Δ but small difference, infinitesimal

    δ = d but along a curve

    Mathematical symbols are always graphics.


    I’m not sure if that will be true, but it would be beautiful.
     
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