How to self-study calculus? (2024)

Yes, notice I specified "working knowledge".

> there are no difficult proofs; students are only advised to have a certain level of mathematical maturity, *with* an assumption they’re exposed to XYZ

Scratching my head rn trying to understand why you’re trying to make my point for me lmfao.

Sorry, but I’m placing an undergrad with a 60 on the Putnam who hasn’t taken a course higher than Diff Eq above a graduate student who’s struggling with their topology homework in terms of mathematical maturity. It’s about the familiarity and ease with which you construct mathematical arguments, not the content knowledge, moron.

>Scratching my head rn trying to understand why you’re trying to make my point for me lmfao.
It's alright if you're stupid, not everyone needs to be smart.

Well yes but "working knowledge" doesn't mean you have some incredibly broad set of subjects that you have studied.

One of the most brilliant professors I've ever had the pleasure of working with never took a Lebesgue integration focused probability theory course and yet his 30+ years of linear algebra and convex analysis experience allowed him to solve really challenging probability problems without this "basic" set of skills you'd acquire in your first year of grad school today. Reps in the gym doesn't mean you have a broad knowledge base, it means you know how to use the knowledge base you have, regardless of its breadth.

> Didn't read the rest.

Not surprising. You got filtered by one of the most straightforward concepts in all of mathematics education.

> I wonder if you're a jeet wasting his energy on elementary geometry problems and pretending these "reps" count for maturity.

Not a pajeet. You'd be surprised how much you can learn by going through the more challenging problems in a textbook from a subject you've already taken a course on. It's not an accident that math professors really encourage things like problem books when students are trying to learn analysis/probability/algebra etc.

You could be literally Terry Tao and you'd still probably get something out of spending a few minutes going through problems you haven't seen before in something like measure/topology/algebra etc.

>schizo rambling
They could just do the same to Spivak or Apostol, re-order the questions, but they don't. For a reason.

>It's not used at top schools in the honours programs
If you were at a top school in the honors program you wouldn't be on IQfy asking how to learn calculus, but you're not. You're a mentally disabled social anxiety ridden autist online, ergo read Stewart.

>asking how to learn calculus
Do you think everyone that posts is OP? Do you know how this site works?

You cannot push Stewart over Apostol or Spivak and call others moronic.

>You cannot push Stewart over Apostol or Spivak and call others moronic.

You can when you have people specifically asking how to learn calculus for the first time. If someone is learning analysis for the first time or states that they have previous experience with rigorous math/proofs sure, but otherwise recommending rigorous calc books to people is moronic

>You can when you have people specifically asking how to learn calculus for the first time. If someone is learning analysis for the first time or states that they have previous experience with rigorous math/proofs sure, but otherwise recommending rigorous calc books to people is moronic

Spivak presents everything from the ground up though and gives sufficient intuition for what he introduces. I would agree suggesting Rudin to someone without any calc knowledge is often moronic because such analysis books presume basic intuition from calculus, but suggesting something like Spivak doesn't have this problem. Low IQ to think so tbh

>schizo rambling
Son, one day you will realize the schizos have always been right.

You're not going to learn any physics from that book, its all sympletic geometry. Buy Taylor classical mechanics used and download the solution manual on libgen.rs

Use the solutions to figure out how he solved the problems then do the problem without looking at the solutions manual from memory

No, you obviously are.

No it's not.
You haven't even defined a rotation
multiplication and imaginary numbers are not dependent ideas on angular rotations.
In fact rotations of uninteresting and meaningless without fixed information about the transform.
A complex function is literally nothing more or less than that: a complex function e.g. a mapping defined as a function of X to C
Rotations can be defined differently especially in different spaces.

Okay, so let me get this straight. You've been so poisoned by Bourbaki bullsh*t that you have forgotten that hom*omorphic transformations of complex numbers are reducable to unique scaling and rotation operations? Go back to undergrad you moron.

I'm not a physicist. I'm an engineer. Physicists are moronic and can't see the forest for the trees. You seem to take after them in that way.

You can deny the reality that multiplication of a complex number by some exp(i theta) is a rotation operation if you'd like. You'll be wrong, but you can deny that reality.

>OH MY SCIENCE!!! I DID A MULTIPLICATION IN A LINEAR SPACE AND CALLED IT A SCALE AND ROTATION I AM LITERALLY GOONING RIGHT NOW!!!
ALSO LEARN WHAT A hom*oMORPHISM YOU FRICKING VERKEKSIUM WATCHING KEKGINEER
DID I EVER SAY AN ENGINEER SUBHUMAN WAS ALLOWED TO TALK TO ME?

engineer moment

You don't know anything about what engineers do, do you? My undergrad EE degree required us to do Fourier analysis, numerical analysis, Lebesgue integration and complex integration, more linear algebra than you could even imagine etc.

Sure, we didn't do topology, but we did more in depth Analysis than typical math majors get until grad school.

>more linear algebra than you could even imagine
Oh wow, he took a determinant of a 3x3 matrix. Those topics were probably just chapters in the "survey" book you were asked to read. You're a formula monkey, and everyone knows it.

Of course it's a different story if you did EE at a top school.

>Of course it's a different story if you did EE at a top school.

It's no better. You can look at the courses and syllabi for EE at these top schools, its simply formula monkeying with slightly higher level math.

I did EE at a school that's competitive with MIT if that changes your perception at all. Also, linear systems theory and control theory is literally just a mix of the basics of the linear operator theory part of functional analysis (though generally a finite dimensional vector space rather than a general hillbert space) and abstract linear algebra. Take a look at modern control theory by Brogan if you want a look at our senior undergrad linear control course. Not exactly proof oriented but far less simple than you're pretending.

>Sure, we didn't do topology, but we did more in depth Analysis than typical math majors get until grad school.

Kek no you fricking didn't, you just learnt the end tools and theorems of "grad school analysis" without ever knowing how or why they are true to any level of rigor. You are simply a monkey trained to follow the instructions of your superiors (mathematicians).

The end tools being literally the entirety of signal processing (which is just a subset of Fourier analysis) and control theory (which is a subset of linear operator functional analysis and "abstract linear algebra").

I don't perceive it as a personal slight. I did a grad elective in measure theory recently with math grad students. Their background in the proof oriented side of mathematics is far stronger than mine, but they have absolutely no experience with computational mathematics and actual problem solving.

As soon as an integral actually involved some amount of computation and wasn't just "find the right sequence of functions to apply dominated convergence theorem to" they basically blew over. These are different skills for different emphasis.

This approach is fine if you are only doing numerical solutions. If you need analytical forms this approach is really doing you a disservice.

An easy example of this is within estimation theory the concept of the CRLB. You can always numerically approximate using Matlab/mathematica/python/R, but if you actually want to determine the conditions for observability of your parameter you need to stop being a lazy frick and actually do the math.

Unfortunately, most mathematicians don't actually have any experience doing proper analytical solutions for complicated problems and as a result large subsections of probability theory, control theory, optimization theory and convex analysis and information theory are left entirely to engineers.

In my experience, the engineers actually omit large pieces of the math and make very simple assumptions and approximations in their core texts.

This was my experience at least in the one time I actually stepped into engineering math for control systems. Very immature texts compared to the even core stochastic differential equations that simply touch on control such as Oksendal and Kallianpur.

Again, just my experience.

> In my experience, the engineers actually omit large pieces of the math and make very simple assumptions and approximations in their core texts.

This can certainly be the case, though I don't know where exactly the line is between a textbook approaching things from a simplified perspective as a pedagogical choice (i.e., placing emphasis in a different area) vs. out of a lack of rigor.

I wouldn't, as an example, consider Royden inferior as a measure book to Cohn's measure theory textbook just because Royden constructs the Lebesgue measure solely on the real line and doesn't leave abstract measure spaces/measurable spaces until later.

In terms of control theory texts tending to make simplifying assumptions, I agree but I think you are somewhat missing the point (especially when it comes to optimal control for potentially non-linear systems). Any system that is locally controllable can be arbitrarily well approximated by a control strategy which minimizes some strongly convex function in the same state space. The whole motto of control theory is "keep it simple stupid." If you can avoid Runga-Kutta approximation of the dynamics and get a stable and computationally tractable approximation out of a local second order expansion, that's probably a better bet even if it's simpler.

Control Theory as a discipline encourages specifically keeping things as simple as possible as it is very rare that a single controller is operating in a vacuum without being in some network. Predicting and controlling the dynamics of a system with 48 integrators and 9 separate controllers will be far more tractable if those individual blocks are quadratic approximations and those controllers are as simple as can be managed.

>Sure, we didn't do topology, but we did more in depth Analysis than typical math majors get until grad school.

Kek no you fricking didn't, you just learnt the end tools and theorems of "grad school analysis" without ever knowing how or why they are true to any level of rigor. You are simply a monkey trained to follow the instructions of your superiors (mathematicians).

I did an undergrad course in digital integrated circuits, but no, my job is not designing circuit boards. I do statistical signal processing and information theoretic statistics.

Just like mathematicians don't spend all day writing out log-tables anymore, not every EE designs circuitboards. I knew soldering before I went to school for EE anyways, but that's most because I liked working on audio equipment as a hobby and would terminate my own cables.

Upon a very brief review of the subjects covered, this looks like a very useful book to a beginning Engineer or Physicist. I was impressed to see that right away they start talking about group theory.

By the amount it was shilled, I expected to see some flashy American style text, but the two authors teach in German universities, so that gives me some confidence that my positive first impression will be correct.

Just when I was saying nice things, I decided to read the preface... Oh no.

Hah yeah soon after posting I confirmed it.

The book contains half the solutions, while the manual contains the other half

Right, and also something I'd have noticed if I'd finished the preface before posting.

4 years of doing proofs in mathematics then reading a math methods books, that's pretty fricking stupid. The supermsjority of physicists never use anything in real analysis.

The 4 years part was a joke you sperg(s). Yet even without it being analysis, a student should at least see a more thorough treatment of single variable calculus, otherwise things will be missed. For example, again from Altland/Delft

P1 in the derivative chapter is to take the derivative of a polynomial. Simple right? Well they haven't discussed the linearity of the derivative, or even more simply, the "sum rule". If a student is reading this without any background, how could 'she' expect that she has any right to just assume it?

Linear approximation != linearity property... physicists man, I'll tell you what.

In the linear algebra section yes, there is a discussion of linearity and differential operators. That section L9 depends on later sections of the calculus part C6 (which implicitly implies C1). So no, it hasn't been covered prior to our student's reading of her calculus 1 chapter. All they needed was a simple
d(f+g)=df+dg, d cf = c [or combine them, applicability to polynomials follow by induction]

This is omitted, because it's assumed she has studied calculus in high school. There's explicit mentions to this, I'm not even sure what there is to dispute.

>d(f+g)=df+dg
how could this not make sense, it's just algebra, i don't see what the big deal is you're making it out to be, a person doesn't need a proof to understand that d(f+g)=df+dg.

This is why the physicist can learn in one semester what the mathematician takes 4 years to learn, and they still get paid more than mathematicians. Not everything needs a proof, I don't understand why you would need a proof for that honestly, it's pretty obvious to me.

I didn't even say it needed a proof (even though it could be done in a few lines), it's literally just not even mentioned.

Despite sneaky notation, it's not algebra in the elementary sense. d is of course a function (operator if you will), and being able to write that requires the property of linearity. For example, sine obviously doesn't have that property.
[math]sin(pi+pi/2) neq sin(pi)+sin(pi/2)[/math]

It should have been stated, it's a fact used all the time. It's not like I'm arguing he needs to define a polynomial as an element of a formal power series over some arbitrary ring. If you're one of his students, let him know.

At least your shilling resulted in a book sale (hardcopy naturally). I've been looking for something like this for some time to give to software "engineers" whose math skills (whatever they were) have atrophied since university and something like this can serve as a good refresher. There's an Amazon review from some other mathgay who liked it for similar reasons. So far so good, minor gripes notwithstanding.

This part made me laugh though, because occasionally you see a level of rigour you don't expect to see. The inner German always comes out I guess.

I'll just include books which either start from zero or have very primitive requirements, and those which cover the topics an American calls Calculus even though some of these are called Analysis.

Unironically, the book by Altand/Delft the other anon has been shilling in this thread is very good. Zorich is another nice choice if you want something more rigorous (as in proof based), and Amann/Escher is an option for well prepared students who want to study math for its own sake (more abstract than Zorich).

Other notable classics are:
Spivak
Apostol
Courant/John
Piskunov

Why do you consider that a meme list? What would you suggest then?

I'll just include books which either start from zero or have very primitive requirements, and those which cover the topics an American calls Calculus even though some of these are called Analysis.

Unironically, the book by Altand/Delft the other anon has been shilling in this thread is very good. Zorich is another nice choice if you want something more rigorous (as in proof based), and Amann/Escher is an option for well prepared students who want to study math for its own sake (more abstract than Zorich).

Other notable classics are:
Spivak
Apostol
Courant/John
Piskunov

Do you know of a rigorous calculus/analysis book that doesn’t overwhelm the reader with too many exercises?

Zorich doesn't have too many exercises, I just think they go beyond the material in the book on occasion, almost as though they've been designed to challenge students in a room with a TA to work through them. You could always read Zorich, do any questions that interest you (there are some good ones) and then work through an analysis problem book with worked solutions (and not too many unsolved) like pic rel.