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Dx and delta(x) (in partial derivative)

  1. Dec 6, 2012 #1
    I have a question to ask, is dx = δx, can they cancel each other like [itex]\frac{dx}{δx}[/itex]=1
    and is it mean that:

    [itex]\frac{δf}{δx}[/itex][itex]\frac{dx}{dt}[/itex]=[itex]\frac{df}{dt}[/itex]?
    (f = f (x,y,z))
     
  2. jcsd
  3. Dec 6, 2012 #2
    I'm not sure what you mean with [itex]\delta x[/itex] in the first place.
     
  4. Dec 6, 2012 #3

    Mark44

    Staff: Mentor

    Assuming that x, y, and z are all differentiable functions of t, it does make sense to talk about df/dt, but it is not equal to $$ \frac{\partial f}{\partial x} \frac{dx}{dt}$$

    For f as you have defined it,
    $$ \frac{df}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt} + \frac{\partial f}{\partial z} \frac{dz}{dt}$$
     
  5. Dec 6, 2012 #4
    δx mean the denominator if taking partial derivative of f wrt x. I wonder both dx and δx equal, since they both mean infinitesimal amount of x.
     
  6. Dec 6, 2012 #5
    Ah, ok! Usually, they denote this by ∂ instead of a delta.

    But anyway, if you want to be very rigorous, then things like "dx" or "∂x" don't exist. The only thing that exists are the notations

    [tex]\frac{df}{dx}~~\text{and}~~\frac{\partial f}{\partial x}.[/tex]

    But these are not fractions since things like df and dx are undefined. Furthermore, notations like [itex]\frac{dx}{\partial x}[/itex] don't make sense.

    That said, there is a way to give dx a rigorous meaning. There are several ways, actually. One of these ways is through nonstandard calculus. Another way is by differential forms. But I won't confuse you with these things. Just remember that if you are in a standard calculus class, then things like dx and df don't really have any meaning. They are very handy and useful notations however.
     
  7. Dec 7, 2012 #6
    Thanks micromass :)
     
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