My book presents this differenciability definition and says that it's due to Frechet and Stols:

Given $f:U\to \mathbb{R}$, with $u\subset \mathbb{R}^n$, let $a\in U$. We say that our functions is differentiable at point $a$ when there are constants $A_1,\cdots, A_n$ such that, for every vector $v = (\alpha_1,\cdots,\alpha_n)\in\mathbb{R}^n$, with $a+v\in U$, we have:

$$f(a+v) = f(a) + A_1\alpha_1 + \cdots + A_n\cdot \alpha_n + r(v)$$

when $\lim_{v\to 0}\frac{r(v)}{|v|} = 0$

However, in this question, the derivative is different, it's a limit. Which one is the right one? If they're different, then what's the explanation for this one in my book? I liked the explanation given in that question.


1 Answer 1


The short answer is that they are the same. $A_1\alpha_1 + \ldots + A_n\alpha_n$ plays the role of the derivative. If you move it to the left hand side and divide by |v| you get: \begin{equation} \frac{ f(a+v) - f(a) - (A_1\alpha_1 + \ldots + A_n\alpha_n) }{|v|} = \frac{r(v)}{|v|} \to 0 \text{ as } |v|\to 0 \end{equation} You can also show that if their definition is satisfied then you can find a function $r$ that you describe.

More generally, the derivative of a function should be a linear map. You might be familiar with this: given a function $f:\mathbb{R}^2\to \mathbb{R}$ you have multiple directions in which you can differentiate. For example, if you differentiate in the $x$-direction then the derivative is what we usually denote $\frac{\partial f}{\partial x}$. The linear map then takes a vector (which should represent the direction we want to differentiate in) and spit out the derivative in that direction. To connect this with what you have, the derivative at a point $a\in U$ should be a linear map $\mathbb{R}^n \to \mathbb{R}$ and so (once you fix a basis) is given just given by $n$ real numbers.

It turns out that the definition you gave is very useful, as it makes more clear the idea that the derivative should be the "best linear approximation to the function $f$ at the point $a$". That is, $f$ at points near $a$ is given by its value at $a$, plus a linear bit, plus a little bit of error given by $r(v)$. The benefit of the definition given in the other question is of course that it looks a lot like the normal definition of the derivative.

There's lots of interesting stuff to say about derivatives in higher dimensions, but I'll leave it there. If you want any more details or clarification, let me know.

  • $\begingroup$ Sorry, where's the limit definition in your answer? $\endgroup$ Apr 3, 2017 at 21:04
  • $\begingroup$ It is in the first expression given. The vector $v$ takes the role of $h$ and $A_1\alpha_1 + \ldots + A_n\alpha_n$ takes the role of (h times) the derivative. $\endgroup$
    – phunfd
    Apr 4, 2017 at 11:18

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