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Differential vs. Derivative of a multivariable function

  1. Oct 4, 2009 #1
    Consider a (possibly complex-valued) function [itex]F(z) = F(x,y)[/itex] of two variables. Can it make sense to talk about the differential [itex]dF[/itex] of this function without it having a derivative [itex]dF/dz[/itex]? Or must [itex]F[/itex] be differentiable before we can even start talking about [itex]dF[/itex]?
     
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  3. Oct 5, 2009 #2

    HallsofIvy

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    Yes, F must be "differentiable" in order to have a "differential"!
     
  4. Oct 5, 2009 #3

    quasar987

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    Mh, why is that?

    I thought that by definition, dF is the formal expression

    [tex]dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy[/tex]

    So existence of partial derivatives is sufficient to make sense of dF.
     
  5. Oct 5, 2009 #4
    This is precisely what I thought! We only need the partials to exist to make sense out of [itex]dF[/itex]. But as we all know, the existence of partials is insufficient to guarantee differentiability.
     
  6. Oct 5, 2009 #5

    HallsofIvy

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    Well, you can write
    [tex]df= \frac{\partial f}{\partial x}dx+ \frac{\partial f}{\partial y}[/tex]
    as long as the partial derivatives exist but to what point? None of the properties of a differential work unless f is differentiable.
     
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