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When first learning the subject and when doing simple calculations, it's convenient to describe derivatives in terms of variables, i.e. $$ \frac{d}{dx} f(x) = f'(x)$$ and we say the derivative takes a "function" (expression) and maps it to another "function" (expression). But more rigorously, we can say that given an $n$-manifold $M$, a chart $x:U\subseteq M\rightarrow\mathbb{R}^n$, and a function $f:M\rightarrow \mathbb{R}$, $$ \frac{\partial}{\partial x_i} f \equiv \partial_i(f\circ x^{-1})\circ x$$ This is just one way of reformalizing derivatives, but what we have done is taken out the reliance on a naive notion of "variables". I was wondering if we have made any similar re-formulations for the limit. From the above equivalence, we need the notion of a limit to fully describe the partial derivative on $\mathbb{R}^n$. I.e. for a function $g:\mathbb{R}^n\rightarrow\mathbb{R}$, $$ \partial_ig(a_1,...,a_i,...,a_n)\equiv \lim_{h\to 0} \frac{g(a_1,...,a_i-h,...,a_n)-g(a_1,...,a_i,...,a_n)}{h} $$ You could use an "epsilon-delta" definition of limits if $n=1$ and you want to totally-order $\mathbb{R}$ or you could define limits using neighborhoods or open sets, but the notation is what I'm stuck on. Is there a way to formalize the limit as a map $$ \lim_{i,\ b}:\ C^0(\mathbb{R}^n)\ \to\ C^0(\mathbb{R}^n) $$ $$ \left(\lim_{i,\ b}g\right)(a_1,...,a_i,...,a_n)\equiv\ ``\,\lim_{a_i\to b}\left(g(a_1,...,a_i,...,a_n)\right)" $$ similar to the many ways we have reformatted the derivative? For clarity, my question is this:

Question: Is there a definition of the limit on Euclidean $\mathbb{R}^n$ space that doesn't require variables? If not, is there simply a notion of a limit that doesn't use variables in its notation?

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  • $\begingroup$ How familiar are you with the topological definitions of limits, particularly the usage of nets and such? That seems at least along the lines of what you're looking for... $\endgroup$ Apr 19, 2020 at 19:29
  • $\begingroup$ @StevenStadnicki I'm familiar with the definition of limit of a function $f:X\to Y$ for topological spaces $X$ and $Y$... For every open set $U$ of $X$ containing $p$, there's an open set $V$ of $Y$ such that the image of $U$ under $f$ is a subset of $V$. I'm pretty sure that's the definition I remember. I see that generalization to nets uses a variable-free notation, but how then would you notate the limit of the difference quotient? The traditional notation allows for "embedded" variables, but I can't find a way that represents this in a variable-free notation. $\endgroup$
    – Cam White
    Apr 19, 2020 at 20:08
  • $\begingroup$ "that doesn't require variables": what do you mean ? Even in the notation of derivatives there are variables. $\endgroup$
    – user65203
    Apr 20, 2020 at 9:23
  • $\begingroup$ @YvesDaoust I mean exactly what I say and the notation for derivatives in fact does not use variables; e.g. $\frac{\partial}{\partial x_i}$ only includes the symbols $\partial$, $x$, and $i$; $\partial$ is simply a symbol that helps us conceptually but it's certainly not a variable, $x$ is a chart which is a specific element of the atlas of the manifold so that's not a variable either, and $i$ is a specific integer between $1$ and $n$ so that's not a variable either. $\endgroup$
    – Cam White
    Apr 20, 2020 at 18:10
  • $\begingroup$ "$x$ is a chart which is a specific element of the atlas of the manifold so that's not a variable ": wow, I am afraid I have nothing to do here. $\endgroup$
    – user65203
    Apr 20, 2020 at 18:22

1 Answer 1

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A friend of mine, Jakub Marian, developed a coordinate free notation in calculus in his bachelor thesis: Alternative mathematical notation and its applications in calculus He was pretty good so there should be very good ideas.

I am looking at it now and he only treats calculus of 1 variable! You can do the case of $n$-variables yourself :-)

A crucial role is played by the identity function $\iota(x) = x$ which he uses to write compositions efficiently; e.g., the simple expression $(\iota^3+\iota^2 + \iota)\circ(\iota^3 + 2\iota)\circ(\iota^2 + 8\iota + 4\iota^2)$ in his notation corresponds to a crazy polynomial in the variable $x$. He views $\lim_a$ as an operator on functions (he uses the "undefined" symbol $\Omega$ if the limit does not exist). He has the "difference quotient with step $h$" operator $\Delta_h$ and defines "indefinite sums with step $h$" as $\sum_h = (\Delta_h)^{-1}$ (defined up to the set of $h$-periodic functions). Based on this, he develops a theory of the derivative $\partial$ and the integration $\int$ (in the sense of an inverse to $\partial$) and rewrites the classical theorems in this notation. He illustrates at many places how his notation simplifies some computations.

However, I can not imagine any real application of it in the sense of gathering more information about spaces of functions, etc. On the other hand, I was once thinking of using similar abstractions when I was supposed to compute some crazy integrals in my research and had the suspicion that there is a hidden structure which I just didn't see when looking at the crazy coordinate expressions. Such notation is also better if you communicate with a machine; e.g., from my experience, Mathematica and Haskell think about expressions in this way. It is probably very much related to $\lambda$-calculus but I am not an expert on that.

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  • $\begingroup$ This is interesting, but I did not see the definition of the notation $\Delta_\bullet$ (used in Definition 2.19). The definition $\partial = \lim_0 \Delta_h$ made sense, but it still has the variable $h$, while think the idea of this question is to eliminate $h$. $\endgroup$
    – David K
    Apr 19, 2020 at 22:06
  • $\begingroup$ He defines it in Definitions 2.10 and 2.11 (limit of a family of functions and a family of operators). It would be probably more elegant at this point to have the generalization for n variables. $\endgroup$
    – Pavel
    Apr 19, 2020 at 22:46
  • $\begingroup$ So the answer is actually that $\Delta_\bullet$ is not defined as an independent notation, the way $\Delta_h$ is; instead it is defined only in conjunction with the $\lim$ symbol. It seems a little strained, though the earlier definitions such as $\lim_a$ seemed intuitive enough. $\endgroup$
    – David K
    Apr 19, 2020 at 23:42
  • $\begingroup$ I would understand $\Delta_\bullet$ independenly as a family of operators $\Delta_{h}$ for $h\in \mathbb{R}$, i.e., a particular set. He defines the limit of a family of operators, e.g., $\lim_0 \Delta_\bullet$, as the operator associating to a function $f$ the function which is the limit of the family of functions $\Delta_h f$. The limit of a family of functions $g_h$ is then defined pointwise like $(\lim_0 g_h)(x) := \lim_0 (g_h(x))$. $\endgroup$
    – Pavel
    Apr 20, 2020 at 8:29
  • $\begingroup$ Correction of the last sentence: The limit of a family of functions $g_\bullet$ is then defined pointwise as the function $(\lim_0 g_{\bullet})(x) = \lim_0 (g_{\bullet}(x))$. Again, I see a possibility of making this better by developing the formalism for more variables so that $\Delta_\bullet$ can be interpreted as an operator from functions of $1$ variable to functions of $2$ variables. $\endgroup$
    – Pavel
    Apr 20, 2020 at 9:51

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