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I trying to find some obvious way to understand differentiability, like deffinition of continuiny gived by $\forall A\ f[CL(A)] ⊂ CL(f[A])$ - this formula can be explained to child - function preserves closed points. So, i can't find such thing for differentiation. All i found - are magical things like $d(fg) = d(f)g+fd(g)$, which can be understood algebraicly - unique linear operator, satisfying Leibniz rule, or qutient ring of continious functions over $o(x-a)$, but i can't feel geometric intution behind all of it.

So, if we take that surface is differentiable iff it's "smooth", ie there are no "sharp corners" on it, is leads us to some idea of "tangent line", but i couldn find good geometrical definition of tangent line - it's either not given, or given algebraicly by differentiation.

There are complecated fields as algebraic geometry, differential geometry, Homological algebra, sheafs theory, but i opened couple of books - Munkres, Eisenbud - definition of tangent given there by differential, not opposite way, which i search for. And i can't spend a lot of time on it without understanding that this realy will give me answer.

I found some way to define tangent (but not tangent line) topologically, but i'm not sure yet, that it is correct, and is looks realy non standard. So, may be you could suggest something?

nagvalhm
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  • What’s wrong with the usual limit definitions? They make good sense – FShrike Jul 18 '23 at 13:07
  • @FShrike Nothing wrong, but it's algebraical, and i can't find geometrical intuition behind it. I wonder if there is geometrical definition (in terms of open sets, vectors and continious and linear maps). – nagvalhm Jul 18 '23 at 13:37
  • For the case of differentiating real-real functions there’s a clear geometric intuition of finding the tangent to the graph. More generally there is a geometric intuition of the derivative being the linear map that can locally represent the change in the function. – FShrike Jul 18 '23 at 16:09
  • @FShrike So, my question - what is "tangent to the graph". Definition f(a) + f'(a)*(x-a) not working, because then definition of derivative is circular. So, i want find definition of tangent as "unique set/ map... with some properties". For me not obvious what is "change in the function", but, again, if it can be formalized as "unique set/ map... with some properties", it would answer my question, thank you. – nagvalhm Jul 18 '23 at 16:44
  • View it as a mapping, e.g. for the case of n dim real, view it as the limit of the best linear approximation. – Tim Jul 18 '23 at 16:50
  • @Tim, but i guessed, there are different best linear aproximations for some functions for different neighbourhoods of point. For $sin(x)$, tangent at 0 of sin aproximate sin in very small neighbourhoods of 0. But, it's clear that in [-pi/2, pi/2] some other line passing throut 0 best approximate sin, at least if take uniform metric for approximation. – nagvalhm Jul 18 '23 at 18:13
  • @Tim, so, may be it fair to say, that function has tangent at point, if starting with some neighbourhood, it's approximated with same line. But, it's needed to be verified, that any differentiable function has that property, or we will get situation, that function is differentiable and has no tangent. And, as u noticed, some non differentiable functions has tangent than - $|x|$ at 0, at least. – nagvalhm Jul 18 '23 at 18:24
  • @никитапичугин I am not giving a formal geometric definition of the tangent line or derivative, it is only my thoughts of the tangent space intuitively. For example, we cannot talk about "linear approximation" to a manifold since this requires a metric, and manifold is embedded in $R^n$.

    Assumption of f is differentiable is necessary so that we can define linear approximation to the graph of f. Also, to parametrize a tangent line, $\alpha(t) = tv + a$, this v comes form derivative.

     – Tim Jul 19 '23 at 06:22
  • @Tim yes, but i would answer "no" to question in topic of this post - for reasons i described in previous coments. If u mean answer, which is marked as correct, it's litteraly said in it, that being differentiable for f exatly means that there is unique line function (affine) such that f lay in same residue class with it in quotient ring $C0/o(x-a)$, and nothing said about approximation. Anyway, i hope, i found answer to my question, i'll formalize it later, and will give answer, if needed. Thanks for helping:) – nagvalhm Jul 19 '23 at 08:50
  • @Tim |x| has tangent together with first differential everywere, exept 0, and having differential <=> having tangent for real functions at least, in common sense of both words. As we know from stone-weierstrass, any continious function is aproximated with polynomials, using uniform metric - supremum of functions difference on interval, this approximation exist not only for differentiable functions and not working for defining tangent, as i shown. I did'n get, which metric u offer to use to approximate function. What do u mean by error? Integrate difference on interval or what? – nagvalhm Jul 19 '23 at 12:19
  • @Tim, i dont'n want to say, that tangent doesn't approximate differentiable function (starting with some neighbourhood), i just want to say, that it needed to be prooved (i can't find counterexample, at least), and second, it can't be used as unique property of tangent. It's not "iff", only "if" - some lines best approximate functions not being their tangents. – nagvalhm Jul 19 '23 at 13:02
  • As SE warn long comments, I only leave the references Ref1, Milo Brandt's answer may be helpful – Tim Jul 19 '23 at 15:09
  • @Tim, Milo Brandt's answer works, thanks! With metric defined by integration, but still. – nagvalhm Jul 19 '23 at 17:02
  • @nagvalhm Def of best linear approximation Cheers, looking forward to see your answer. – Tim Jul 19 '23 at 17:05
  • @Tim, finally, at the moment, i'm satisfied with defenition of tangent as of limit position of the secant line. It works for plane curves (let's say curve is function with domain $D$), because, if we define function $S(a, b)$, which takes couple of points and gives secant line passing throught it, roughly, tangent - is unique way to continiously continue $S$ to whole $D^{2}$. – nagvalhm Aug 06 '23 at 13:53
  • I camed to it considering concept of "direction", which is realy natural to consider as line going throu couple of points, and natural question is - how to generazile it to case $(x, x)$ - of single point. So, after realizing this, i found this definition in wikipedia, in a little bit other terms. The only problem, that above definition not working in 3D space directly, and i don't know yet how to generalize it to plane, tuching 3D manifolds. But, guess, it works in sence of "partial derivatives" – nagvalhm Aug 06 '23 at 14:03
  • And usual derivative formula derived from analitic formula of line passing thru couple of points on the plane - i found explanation only in russian wikipedia, but idea is realy simple. – nagvalhm Aug 06 '23 at 14:12
  • I mistaken in above comment - "and gives secant line passing throught it". Gives secant line passing throught images of the points, of course. – nagvalhm Aug 06 '23 at 14:15

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