Differential vs. Derivative of a multivariable function

[AxiomOfChoice]

Consider a (possibly complex-valued) function $F(z) = F(x,y)$ of two variables. Can it make sense to talk about the differential $dF$ of this function without it having a derivative $dF/dz$? Or must $F$ be differentiable before we can even start talking about $dF$?

 

[HallsofIvy]

Yes, F must be "differentiable" in order to have a "differential"!

 

[quasar987]

Mh, why is that?

I thought that by definition, dF is the formal expression

$$dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy$$

So existence of partial derivatives is sufficient to make sense of dF.

 

[AxiomOfChoice]

Mh, why is that?

I thought that by definition, dF is the formal expression

$$dF=\frac{\partial F}{\partial x}dx+\frac{\partial F}{\partial y}dy$$

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 $dF$. But as we all know, the existence of partials is insufficient to guarantee differentiability.

 

[HallsofIvy]

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