What is the difference between a partial differental and an ordinary differential?

  1. I have been wanting to ask this for a while.

    In Calc I, I was introduced to differentials. It seemed like they act like quantities(please corrected me if I'm wrong). For example dx/dx=1. You can obtain this by differentiating x or by eliminating the dx in the numerator and denominator(I do not know why this worked).

    What convinced me that differentials where quantities was the chain rule. dy/dx=(dy/du)(du/dx). The proof is a bit tough, but you will obtain the same result by eliminating the du.(I may be making a TREMENDOUS mathematical blunder here, but it seemes to work)

    In Calc III, I was introduced to [tex]\partial[/tex]x and[tex]\partial[/tex]y. Obviously I found out that [tex]\partial[/tex]x[tex]\neq[/tex]dx or else the chain rule for multiple variables would not simplify to dz/du.

    So, why are these two infinitesimals so different?
     
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,667
    Staff Emeritus
    Science Advisor

    Re: What is the difference between a partial differental and an ordinary differentia

    No, you were not "introduced to [itex]\partial x[/itex] and [itex]\partial y[/itex] in Calc III. You were introduced to the partial derivatives [itex]\partial f/\partial x[/itex] and [itex]\partial f/\partial y[/itex]. There is no such thing as a "[itex]\partial x[/itex]".

    One important reason is that the partial derivatives themselves just don't tell you enough about the function. If the derivative of a function of one variable exists at a point, then it is differentiable (and so continuous) at that point. A function of several variables can have all its partial derivatives at a point and still not be differentiable nor even continuous at that point.

    Take f(x,y)= 0 if xy= 0, 1 otherwise. It is easy to show that [itex]\partial f/\partial x= \partial f/\partial y= 0[/itex] at (0,0) but f is not even continuous there.
     
  4. Re: What is the difference between a partial differental and an ordinary differentia

    Thanks, that cleared things up for me, HallsofIvy.
     
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