# Definition of differentiability for multivariable functions

My question is motivated by the two recent questions

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 $$(x_0,y_0)$$?

• At the final paragraph, you mention a “final soft question”. Which other question(s) do you have? Commented Aug 12, 2022 at 16:14
• @JoséCarlosSantos My question is essentially a reference request. See the tags. Commented Aug 12, 2022 at 19:01
• @PaulFrost Your question? Are you and Kritiker der Elche the same user? Commented Aug 12, 2022 at 19:04
• @JoséCarlosSantos No, but I asked the same question in a comment to the first quoted question. I should have written "The question is ... " Commented Aug 12, 2022 at 19:06
• I can't recall references, but I have seen this several times when glossing over various textbooks. As you said, this is equivalent to the 'usual' definition of (Frechet) differentiability, so from a logical perspective, there's nothing superior about one definition over another in this case. However, I think such a phrasing of the definition is not good because it overemphasizes the partial derivative concept, rather than the concept of linear approximations, which is the key idea of differential calculus (and can be easily rephrased even for arbitrary Banach spaces). Commented Aug 13, 2022 at 22:10

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);