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?
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.