Definition of differentiability for multivariable functions

Question

My question is motivated by the two recent questions

• Use definition to prove that the function $f(x,y)=xye^{xy}$ is differentiable at all points in $\mathbb{R}^2$

• Let $g: \mathbb R \rightarrow \mathbb R$ a differentiable function in $\mathbb R$. $f(x,y) = \frac{g(y)}{1+g^2(x)}$ is differentiable in its domain?

Both question deal with functions $\phi : \mathbb R^2 \to \mathbb R$ which are defined to be differentiable at $(x_0,y_0)$ if

1. The partial derivatives $\dfrac{\partial \phi }{\partial x}(x_0,y_0)$ and $\dfrac{\partial \phi }{\partial y}(x_0,y_0)$ exist

2. There exists a function $r : \mathbb R^2 \to \mathbb R$ such that $$\phi((x_{0}, y_{0}) + (h_{1}, h_{2})) = f(x_{0}, y_{0}) + \dfrac{\partial \phi }{\partial x}(x_0,y_0)h_{1} +\dfrac{\partial \phi }{\partial y}(x_0,y_0)h_{2} + r(h_{1}, h_{2})$$ and
   $$\lim_{(h_{1}, h_{2}) \rightarrow (0,0)} \frac{r(h_{1}, h_{2})}{\left \lVert (h_{1}, h_{2}) \right \rVert} = 0 .$$

Of course this definition can be generalized to functions $\phi : U \to \mathbb R$, where $U \subset \mathbb R^n$ is open, but let us restrict to the above situation.

I have never seen this definition in the literature and I am interested in references. In the comments to Use definition to prove that the function $f(x,y)=xye^{xy}$ is differentiable at all points in $\mathbb{R}^2$ it is claimed that the above definition of differentiability is quite standard and a reference is given to "Multivariate Calculus: concepts and contexts" by James Stewart, second edition, page 782, definition 7. I do not have access to this book, but no matter: In my opinion the definition is not standard. Of course the above definition is equivalent to the standard definition saying that $\phi$ is differentiable at $(x_0,y_0)$ if there exist a linear map $d\phi(x_0,y_0) : \mathbb R^2 \to \mathbb R$ such that $$\lim_{(h_1,h_2) \to (0,0)} \frac{\phi((x_{0}, y_{0}) + (h_{1}, h_{2}))- \phi(x_0,y_0) - d\phi(x_0,y_0) (h_1,h_2)}{\left \lVert (h_{1}, h_{2}) \right \rVert} = 0 .$$ Simply observe that $d\phi(x_0,y_0)$, if it exists, is represented by the Jacobian matrix with entries being the partial derivatives of $\phi$.

As a final soft question: What could be the benefit of basing the definition on the partial derivatives instead of using a "coordinate free" approach via best affine approximation of $\phi$ at $(x_0,y_0)$?

Answer 1

This should be a comment, but I do not have enough reputation. I think that the reason some authors prefer the definition involving partial derivatives is that it provides a direct way of finding the differential.

This isn't really a reason to not to give the "best linear approximation" definition, but I've seen shortcuts like this a lot in my real analysis courses:

1. A sequence of functions $ (f_n)_n $ is said to converge uniformly to $ f $ is $ \lim_{n\to \infty}\lVert f_n - f\rVert = 0 $. (A better definition would be: $ f_n\to f $ if... $ f_n\to f $ in the normed space induced by the uniform norm);
2. A function is $ \mathcal C^1 $ is all the partial derivatives exists and are continuous (better: a function is $ \mathcal C^1 $ if it is differentiable everywhere and the function mapping a point to the differential calculated at that point is continuous);
3. Etc.