technically discrete math is prereq for cs related fields, calc is on a higher on the causal chain, it is required for all stem fields

It's a fine combination of algebra and geometry. It is also often first introduction to proofs, together with linear algebra.

Basic calculus is easy for those who understand math and a struggle for those who do not.

>>16904205
>What percentage of people can comprehensively learn calculus at the level of being able to solve all Spivak or East-European style textbook alternative to it?
That goes beyond calculus, that's proof based elementary analysis, "honours calculus" maybe. Many physicists and engineers are formidable users and teachers of calculus, but they never studied proof based mathematics

what would you recommend for getting up to speed with proofs in order to tackle books like spivak or apostol? I took up to multivariate calculus in college and did very well but it wasn't proof based, they just taught us to do the calculations and we did them but I feel like I barely learned anything, and looking at these textbooks now it feels like I'm walking into a class that I'm missing a very important prerequisite for, even in working through the introductory section in apostol which is specifically for catching up.

>>16904737
I recommend:
>Journey into Mathematics: An Introduction to Proofs - Joseph J. Rotman
When opening Apostol or Spivak, don't focus too much on the basics, try skipping the first chapters and focusing on the central concepts of limit, derivative, integral. Struggling with these will make your realize that the introductory chapters are the easy part, but dont let that discourage you. That being said, the part of the introductory chapters you must focus is inequalities (consequences of the order axioms). You must memorize things like the proof of the arithmetic-geometric inequality or the existence of sqrt2 from the supremum axiom, even if you dont understand them at first. Rotman, for example, teaches induction better than Spivak. But these introductory chapters are there for a reason so you'll get back to them eventually. On the other hand, you should look at two more recent books, Understanding Analysis by Abbott and Elementary Analysis by Ross. If what you want is the Apostol/Spivak level, other books like Tao's would be overkill.

>>16904958
Thanks, I appreciate the recommendations. All of those books seem reasonably approachable considering how long it's been since I studied any math.

>>16904307
>technically discrete math is prereq for cs related fields
that is indeed the case, but only because the analysis part is, on most courses, basically high school math with some of the most obvious per partes examples and some integrals you solve with preconfigured formulas. If you study say chemistry, you will endure a proof based calculus akin to the math students, that is levels more difficult than discrete math and analysis at cs combined
>>16904737
>what would you recommend for getting up to speed with proofs in order to tackle books like spivak or apostol?
grit and youtube videos if you're stuck

Usually the first chapter in calculus teaches limit, and derives the derivative as an application of limit to the average rate formula. This "tangent line to a curve at a point" is an absolute killer because 1) it's not original calculus, but a 19th century reinvention; 2) it is not well understood how to teach it. Calculus Made Easy (from 1905) teaches Newtonian/Leibniz calculus but you can't even apply it to (at least not until calculus 3 when it reconciled with the limit stuff). So people get hung up on trying to understand, which takes up weeks they don't have in a calculus course (you literally have a couple days to get a grasp on it, 4 at most) and fail to understand anything after that.

But here's the secret: right after that section you learn the Power Rule, which enables you to literally calculate derivatives in your head. It's so simple a pre-algebra student could do it if they were taught how. If you just SKIP that absolute killer of a section, you can make it through most of the class. At least until you get to integrals... integrals have the most confusing notation ever but are really all show in terms of difficulty, because if you just turn that ONE WEIRD TRICK you learned after you skipped the introduction to derivatives and its nasty algebra on its head, you can solve integrals just as easily as derivatives.

And that brings us to the real reason: conspiracy. People who have taken calculus make sure their kids understand it before they take the college courses... usually they make sure they take it in high school, where you get the 1 year you really need to get a handle on Calc 1. Or they teach it homeschool (because again, it's just ONE WEIRD TRICK). Either way their kids have the advantage while all those kids who weren't exposed to it before college have their educational trajectories completely derailed when those terribly written textbooks confuse the hell out of them. It's absolutely by design, one of the biggest scandals in history.

>>16906008
>It's absolutely by design, one of the biggest scandals in history.

Filtered by calc lol. That shit's easy.

>>16906008
>It's absolutely by design
What's by design is the lack of upwards class mobility. The phenomenom you describe has simpler explanation from this more general "by design" scheme

>>16906008
The power rule only works on polynomials. It fails for any trig or log function and if there's a nested function it's completely useless.
Meanwhile the limit definition works for every differentiable function. It can just be messy to work with. Calc 1 starts with the fundamentals first then teaches the neat little tricks because that's just how effective instruction of anything works.

It's not a conspiracy. You just got filtered.

>>16904205
>dutyful
Indian coded

>>16906008
You are right about generational privilege in education. However, if you get filtered by calculus, even though you are trying, (Maybe there's an alignment issue, and I don't blame young people for that). They might need to find themselves first before really gaining an interest in academic subjects.) But if you are really trying to learn it and you just can't understand it, you should probably rethink your educational goals.

>>16906265
The power rule magically works with the series expansion of trig and log functions. That's the way Euler worked before there could possibly be a justification for such correct argument. Here's a recent book that explains the why in modern notation
>Short Book on Long Sums, A _ Infinite Series for Calculus Students - Fernando Q. Gouvêa (2023)

I learned calculus from Khan Academy in like 2013, but he only had videos and exercises for derivatives, and not for integrals, so I just never learned integrals. I don't recall thinking any of it was difficult. I've forgotten it all by now anyway.

>>16906270
If someone's having trouble with basic limits do you really think they'll have an easier time understanding a Taylor series?

>>16906008
> This "tangent line to a curve at a point" is an absolute killer because 1) it's not original calculus, but a 19th century reinvention; 2) it is not well understood how to teach it.
Did you actually read Newton's Principia?
https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)
He talks a lot about tangents. Also notice, that Newton also has a notion of a limit, unlike Leibnitz.

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38 KB JPG

>>16906280
Limits and convergence of series are conceptual, that's why people struggle. But algebraically it's easy to follow. Pic related. The hard part is proving that each intuitive step is justified and understading what could fail with different functions.

>>16904205
I never went to high school because of COVID-19, so I would not know. Take that curveball!

>>16906280
What you fail to understand is this: it's not about a person having difficulty "understanding" anything, because unless they have a severe mental handicap then given the proper attention they will understand anything PROPERLY explained to them. It is simply that most explanations on offer are poor, because again there is a conspiracy afoot. You just happened to be gifted a well articulated explanation of the material.

>>16906283
He does, but he doesn't use Pythagoras to articulate tangent lines the way Thomas and Finney do in the last section of Chapter 2 of their text, or the way the applied/business texts do, because his focus was on infintesimals (fluxions) which Weierstrass opposed. The confusing picture of calculus students are taught today-- which 99% of graduates do not understand -- was Weierstrass' conceptualization.

>>16904205
>Teachers of elementary mathematics in the U.S.A. frequently complain that all calculus books are bad. That is a case to point. Calculus books are bad because there is no such subject as calculus; it is not a subject because it is many subjects. What we call calculus nowadays is the union of a dab of logic and set theory, some axiomatic theory of complete ordered fields, analytic geometry and topology, the latter in both the “general” sense (limits and continuous functions) and the algebraic sense (orientation), real-variable theory properly so called (differentiation), the combinatoric symbol manipulation called formal integration, the first steps of low-dimensional measure theory, some differential geometry, the first steps of the classical analysis of the trigonometric, exponential, and logarithmic functions, and, depending on the space available and the personal inclinations of the author, some cook-book differential equations, elementary mechanics, and a small assortment of applied mathematics. Any one of these is hard to write a good book on; the mixture is impossible.
—Paul R. Halmos, How to write mathematics, Enseign. Math. (2) 16 (1970).

>>16906627
Halmos really did a
>I'd just like to interject for a moment. What you're referring to as calculus, is in fact, real analysis, or as I've recently taken to calling it, [math]\left( \mathbf R,\, +,\, \times,\, \leqslant,\, \left| \cdot \right|,\, \tau \,=\, \left\{ A \,\subset\, \mathbf R \mid \forall x\,\in\, A,\, \exists \varepsilon \,>\, 0,\, \left] x \,-\, \varepsilon,\, x \,+\, \varepsilon\right[ \,\subset\, A \right\},\, \bigcap_{\begin{array}{c} A \,\sigma \text{-algebra of}\, \mathbf R \\ \tau \,\subset\, A \end{array}} A,\, \ell \right)[/math]-analysis. Calculus is not a branch of mathematics unto itself, but rather another application of a fully functioning analysis made useful by topology, measure theory and vital [math]\mathbf R[/math]-related properties comprising a full number field as defined by pure mathematics.
>Many mathematics students and professors use applications of real analysis every day, without realizing it. Through a peculiar turn of events, the application of real analysis which is widely used today is often called "Calculus", and many of its users are not aware that it is merely a part of real analysis, developed by the Nicolas Bourbaki group.
>There really is a calculus, and these people are using it, but it is just a part of the field they use. Calculus is the computation process: the set of rules and formulae that allow the mathematical mind to derive numerical formulae from other numerical formulae. The computation process is an essential part of a branch of mathematics, but useless by itself; it can only function in the context of a complete number field. Calculus is normally used in combination with the real number field, its topology and its measured space: the whole system is basically real numbers with analytical methods and properties added, or real analysis. All the so-called calculus problems are really problems of real analysis.
even before the GNU/Linux pasta

>>16906462
kek

>>16906280
Actually, power series were used by Lagrange to ground calculus in algebra without confusing notions of limits, infinitesimals, differentials, etc. This makes the subject approachable even to 3rd year high-school students. I don't know if there are English translations, but his original works in French are very readable, and, to be honest, they feel more modern than anything on the calculus market today. I will provide a summary of the first two chapters of _Théorie des fonctions analytiques_ along with an example from the third which covers the basic calculus rules.
**Chapter 1**
Lagrange claims that a function [math]f(x)[/math] can be expanded into such power series by increasing the input of the function by an indeterminate quantity [math]i[/math]:
[math]f(x+i)=f(x)+pi+qi^{2}+ri^{3}+...[/math]
To see why this is true, begin with the first step [math]f(x+i)=f(x)+iP[/math]. Easy enough if [math]i=0[/math], then you have [math]f(x)=f(x)[/math]. Here [math]P[/math] is a function of both [math]x[/math] and [math]i[/math] - isolate [math]P[/math] to see why this is the case. Proceed in the same manner for [math]P[/math] as we did with [math]f[/math], we separate the part which is independent of [math]i[/math] (the quantity that doesn't vanish when [math]i=0[/math]).
[math]P=p+iQ[/math]. And so on, [math]Q=q+iR[/math], etc.
To avoid contradictions, the powers of [math]i[/math] must be positive integers. I also used Lagrange's notation, but if we want to be explicit and use modern notation, [math]P=p+iQ[/math] corresponds to [math]P(x,i)=p(x)+iQ(x,i)[/math]. You get the gist.

>>16909637
Chapter 2
Lagrange finds what [math]p,q,r,s,...[/math] are by taking the power series we started with and expanding it again, in two ways which should be identical, this time by an indeterminate quantity [math]o[/math]. I say two ways because you can either add [math]o[/math] to [math]x[/math] or to [math]i[/math].
First way:
[math]f(x+(i+o))=f(x)+p(i+o)+q(i+o)^{2}+r(i+o)^{3}+...[/math]
Expanding and taking only the first two terms:
[math]f(x)+pi+po+qi^{2}+2qio+ri^{3}+3ri^{2}o+si^{4}+4si^{3}o+...[/math]

Second way:
[math]f((x+o)+i)=f(x+o)+p(x+o)i+q(x+o)i^{2}+r(x+o)i^{3}+...[/math]
All we need to do now is to expand [math]f(x+o),p(x+o),q(x+o),r(x+o),...,[/math] and substitute their values above. We only focus on the first powers of [math]o[/math] (the primed functions should be thought of as the functions that appear if we expand using the same procedure).
$$
\left\{
\begin{array}{l}
f(x+o)=f(x)+f'o\\
p(x+o)=p(x)+p'o\\
q(x+o)=q(x)+q'o\\
...
\end{array}
\right.
$$
After substitution, we have:
[math]f((x+o)+i)=f(x)+f'o+pi+p'oi+qi^{2}+q'oi^{2}+ri^{3}+r'oi^{3}+...[/math]

>>16909661
Holy formatting. How do we make multiline latex?

>>16909661
>Second attempt
Chapter 2
Lagrange finds what [math]p,q,r,s,...[/math] are by taking the power series we started with and expanding it again, in two ways which should be identical, this time by an indeterminate quantity [math]o[/math]. I say two ways because you can either add [math]o[/math] to [math]x[/math] or to [math]i[/math].
First way:
[math]f(x+(i+o))=f(x)+p(i+o)+q(i+o)^{2}+r(i+o)^{3}+...[/math]
Expanding and taking only the first two terms:
[math]f(x)+pi+po+qi^{2}+2qio+ri^{3}+3ri^{2}o+si^{4}+4si^{3}o+...[/math]

Second way:
[math]f((x+o)+i)=f(x+o)+p(x+o)i+q(x+o)i^{2}+r(x+o)i^{3}+...[/math]
All we need to do now is to expand [math]f(x+o),p(x+o),q(x+o),r(x+o),...,[/math] and substitute their values above. We only focus on the first powers of [math]o[/math] (the primed functions should be thought of as the functions that appear if we expand using the same procedure).
[math]
\left\{
\begin{array}{l}
f(x+o)=f(x)+f'o\\
p(x+o)=p(x)+p'o\\
q(x+o)=q(x)+q'o\\
...
\end{array}
\right.
[/math]
After substitution, we have:
[math]f((x+o)+i)=f(x)+f'o+pi+p'oi+qi^{2}+q'oi^{2}+ri^{3}+r'oi^{3}+...[/math]

Holy shit /sci/ formatting is abysmal, maybe I'll post a pdf later.

>>16904205
How do you get better at it? I found as an engineer that I can easily perform mechanical operations and understand complex systems. But when I try to come up with a solution to a problem I've never seen or where there are many different paths, I struggle to find a point to begin.
I've long been interested in pursuing an academic career, how do you get better?

>>16910579
There's no royal road. Grab a problem book and start solving. Keep going back to your favorite textbook for reference. Hell, if you haven't solved all the interesting problems from your favorite textbook, it's not a surprise at all that you feel stuck. This progress stagnation also happens to people learning programming or playing the guitar

>>16910677
Yeah been spending a whole year solving each and every problem from Stewart. Feel a lot smarter than when I began, but its still a struggle when I don't know when to begin. Worst part is being a wagie I have a limited amount of time to dedicate to math.
Planning to go after Differential Equations after I'm done with Stewart, and since I already have a steady job, doing it more for fun and in the hopes of one day applying to Academia.

>>16910691
>each and every problem from Stewart
You need to be smarter about it. Some people really need the practice, but there's a lot of repetition in Stewart's exercises. If only we had a tool that could recognize redundancy and apply a clustering algorithm, bringing out the best one or two "representatives" from each "equivalence class" of problems. On the other hand, even before the exercises, Stewart exhibits examples below many definitions and theorems, you should try to solve them before reading his solution while studying the theory.

Calculus is EASY AS FUCK.
There's a reason it's a FRESHMAN course, not a SENIOR one.
It's BASIC. The BARE minimum.
>Why is calculus always the filtering agent in studies like computer science (the few that still teach it properly), physics and chemistry?
Because people are becoming increasingly retarded. We have all the information we can ask for at our fingertips, and yet your average zoomer decides to spend all that precious time on TikTok.
It's not even an education issue, it's a cultural/socioeconomic one.

>>16909661
[math] f((x+o)+i)=f(x)+f'o+pi+p'oi+qi^{2}+q'oi^{2}+ri^{3}+r'oi^{3}+... [/math]

>>16904205
Because you're supposed to take physics and calculus in high school so that it rolls out the basics over a slow digestable period. Collegiate calculus and physics are paced as if you've already had some exposure in high school.

>>16904205
Linear Algebra is the great filter, Calculus isn’t that bad.

>>16912837
>Linear Algebra is the great filter
What part of it is supposed to filter anyone with an IQ over 110?

>>16910579
>I can easily understand complex systems
>But I can't solve problems involving them
Then you don't understand them.

>>16912839
Jordan decomposition filters most students.

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25 KB GIF

>>16904205
Because is what separates cattle from humans.

>>16912419
Thanks anon. I left the last part out, comparing the coefficients of [math]o,io,i^{2}o,...,[/math] of both expansions.
[math]
\left\{
\begin{array}{l}
p=f'(x)\rightarrow p'=f''(x)\\
2q=p'\leftrightarrow q=\frac{f''(x)}{2}\rightarrow q'=\frac{f'''(x)}{2}\\
3r=q'\leftrightarrow r=\frac{f'''(x)}{2\cdot 3}\rightarrow r'=\frac{f^{1V}(x)}{2\cdot 3}\\
4s=r'\leftrightarrow s=\frac{f^{1V}(x)}{2\cdot 3\cdot 4}\\...
\end{array}
\right.
[/math]
When you substitute these values of [math]p,q,r,s,...,[/math] into the the original equation here >>16909637
then you obtain the Taylor series
[math]f(x+i)=f(x)+f'(x)i+\frac{f''(x)}{2}i^{2}+\frac{f'''(x)}{2\cdot 3}i^{3}+...[/math]
That's how Lagrange defines derivatives, as coefficients of power series. No infinitesimals, limits or differentials in sight. Just high-school algebra

>>16912861
For some reason I do when they're physical systems, but whenever I see it in abstract form like coming up with the equation on my own, I can't figure which rule would be good to start.

>>16913537
>I can't figure which rule would be good to start.
The only rule, or guiding principle the universe follows is its constant search for equilibrium. If you want to come up with the equations yourself, you need to look into variational calculus. In theory you can derive all the physical laws this way since they're consequences of the universe wanting to extremize some quantity. This is the unifying theme of physics, it reduces mechanics, EM, QM, etc. to the same problem.

>>16913525
nice