Differential vs. Derivative of a multivariable function

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Discussion Overview

The discussion revolves around the relationship between the differential of a multivariable function and its differentiability. Participants explore whether the existence of partial derivatives is sufficient to define the differential of a function without requiring the function to be differentiable.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant questions if it is possible to discuss the differential dF of a function F(z) without the function having a derivative dF/dz, suggesting a dependency on differentiability.
  • Another participant asserts that differentiability is necessary for the existence of a differential.
  • A participant challenges this view by stating that the differential can be expressed using partial derivatives, implying that the existence of partial derivatives alone should suffice for defining dF.
  • Further, it is noted that while one can write the expression for df using partial derivatives, the properties of a differential may not hold unless the function is differentiable.

Areas of Agreement / Disagreement

Participants express differing views on whether the existence of partial derivatives is adequate for defining the differential of a function, indicating a lack of consensus on the relationship between differentiability and the differential.

Contextual Notes

The discussion highlights the distinction between the existence of partial derivatives and the requirement for differentiability, with participants acknowledging that the existence of partial derivatives does not guarantee differentiability.

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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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Yes, F must be "differentiable" in order to have a "differential"!
 
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.
 
quasar987 said:
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.

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.
 
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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