@@BenyKarachuncurves/surfaces get themselves the fancy name of manifolds that live any arbitrary number of dimensions, and for every little patch on them, there must be a (hyper)plane a dimension less than this manifold where analytical (or holomorphic) differentiable paths are defined, i.e. it's the natural upgrade of calculus III.
@@loicboucher-dubuc4563 Idk maybe the courses are done differently in different countries Here group theory is taught as a part of discrete maths and topology is taught as a part of real analysis (and tougher calc classes), they don't get courses of their own, maybe in pure math degrees they do but not for any other engineering/cs degree
@@Immadeus real analysis is already a less theoretical / less math major oriented analysis class. Most math majors would take a class simply called "Analysis" that studies arbitrary metric spaces rather than the real numbers specifically
@@thechessplayer8328 my point was that most engineering majors would only take that with a math minor or second discipline, but yeah real analysis is definitely more useful for engineering rather than math.
Even being a mathematician, I wholeheartedly despise every aspect of it. I want to vomit every time I see the word "(iso, homo, etc.)-morphism", "functor", "group", etc... It makes my head roil.
@@spilledshelf5 i learned calc 3 from youtube's dr. trefor bazett, and then calc 3 became my favorite math subject. in reality, i never entered in a university, i just learn math from youtube
Note they said they were a major, not a professor lol, they probably just finished their 4 year degree. Even then I will admit most colleges will require more than this, this seems like the bare minimum
to be fair, a couple of those courses are considered grad-level courses. Although, real analysis is literally like the bare minimum for a bachelors, usually people get into some abstract algebra or complex analysis
@@ballisticfox9033 A professor? In Greece a Mathematics undergrad takes Calculus, Real Analysis, Topology, Geometry (includes differential), Linear Algebra, Differential Equations (includes partial), Abstract Algebra, Measure Theory, Number Theory, some history and Philosophy of math and some applied computer classes. That is the bare minimum without electives. How do you study for four years and only take the classes mentioned in this video? He even included high school math to make the list bigger. What short of degree is that?
This is joke, right? xd... right? For anyone out there who really doesn't know what courses you take in pure math, I say to you, none of them are pure math courses... they are general courses that both science and engineering tracks take. The only pure math is Discrete Math and Analysis...and this last one is actually the core of your math courses.
I did a discrete maths module in my very first semester/term of university, though for reference, i'm on a computer science course in the UK (i barely ever needed it for any other modules, so not sure why they insisted on teaching it).
How many math courses have you yet to take? I feel like unis in the US take their sweet mf time with undergrad math. Like no way you need to take calculus in uni after also taking it in high school. In Europe if you do a math degree you start out with real analysis in the very first semester lol (although thankfully we don't use Rudin that shit is torture)
If you take AP Calculus BC in high school, and make a high enough score on the exam that your college will accept it as both a calc 1 and calc 2 credit, you start with your choice in Calc 3, LA, and Diff Eq If you take AP calculus AB, most colleges will count it as a Calc 1 credit, so you start with Calc 2 Keep in mind most colleges in the US are more accepting of people who haven’t taken higher level classes than Europe, as long as they’re willing to pay for it. It’s also important to keep in mind the US has a very different ideology when it comes to classes and choices, focusing on a more broad and diverse curriculum rather than hyper-focusing on your degree. Not saying either method is better, they just have different goals.
@@ballisticfox9033 does this also apply to the top American universities like MIT or Harvard? I mean, could you get a Harvard degree in Mathematics if you did just the courses listed in this video? And can you also take the harder courses like Topology, Complex analysis, Measure theory, etc.? Is it forced/encouraged in any way?
I'd also like to add it may also depend on major. For some STEM majors at my school. like Computer Science majors for example, Calculus 3 and differential equations isn't required for them. Someone I know is in neurobiology at my school, and she only had to take Calculus 1, 2, and 3 was recommended but optional. But for someone like me (Electrical Engineering major, other engineering majors have the same math requirements I did) I had to take calculus 1, 2, 3, linear algebra and differential equations, though I was able to skip Calculus 1 because I took AP Calculus AB. But any math classes above differential equations is usually required for math or physics majors at my school, and for me it would be an optional elective class. Also, in my school there are easier variants of calculus for the premed oriented majors.
@@gregoryk_liteNo, in the top American universities such as MIT, Berkeley, Harvard, LA, etc these classes in this tier list are the entry/ requirements for the upper division classes which include complex analysis, topology, PDEs, Fourier analysis, etc. This video is about the lower divs (and the 1 entry upper div class real analysis). You cannot graduate by only taking these classes and these classes are requirements that all math majors take for their first 2 years. (But people typically get these done fast to do more stuff like complex analysis and other classes earlier)
German math bachelor just starts with Real Analysis (called Analysis I) and it involves Calculus along the way + Algebra means advanced Linear Algebra and is a 3rd semester class. Your tier list looks more like math modules for non-math students
If you're doing the quadratic formula in university, what are you even doing in high school? The first five courses you mentioned (Algebra I/II, Precalculus, Geometry, Calculus) are all things I've already studied before enrolling in university. I'm impressed that as a math major you can even choose not to take real analysis, which is a course I had in the first semester, and I don't even study mathematics; I study computer science. American universities are a joke.
Hi there, I agree American education is a bit scuffed. But for my high school we took calc3+linalg but had to test out in college so that was {nice?} And analysis is required. But probably not needed for like 95% of the population
@@zeegy99 Okay, sorry if I was harsh. I presume that somebody who has studied Calculus III already knows Precalculus, so why waste time studying things you already know?
It's insane to me how much different Americans' (I assume you're American) maths programs are from Europe. In Germany, this is the order we did in a mahtematics bachelor degree. A semester is around 4-5 months Semester 1: Linear Algebra 1 Real Analysis 1 (In one dimension) Semester 2: Linear Algebra 2 (Topics: Bilinear forms, Jordan decomposition, dual spaces) Real Analysis 2 (Topics: Generalizing derivatives in arbitrary dimensions and ODEs) Semester 3: Real Analysis 3 (Topics: Measure theory, Lebesgue Integration, Vector Analysis) Stochastics Semester 4: Complex Analysis Abstract Algebra (Endgoal: Galois theory) Numerical Analysis Starting from the fifth semester, you then choose what topics you want to specialize in (some options are Probability Theory, Stochastical Processes, Functional Analysis, Group Theory, Category Theory, ODEs or PDEs, Numerics of ODEs or PDEs, Optimization etc.). But the classes I listed are all obligatory and are expected to be done in two years. There is no distinction between analysis and calculus
I wouldn't take this video too seriously. Not that i know much about education in Germany, but I think you guys are able to start specialising earlier than most students in the US.
@@walter274 Yeah, our equivalent of high school teaches some very watered down calculus among some other things, so people can start sooner. Another aspect is that university is almost free compared and the requirements for studying things like math are very low - but, and this is the important part, it is accepted and expected that most people drop after the first year. So starting with the terror that is real analysis right from the beginning serves both to get students up to speed and also to filter them out. Also, dont US bachelor STEM programmes involve you taking up 1-2 humanities courses along the way? Or at the very least that the philosophy is to have students take up a wider array of courses, which might even drift a bit from the major.
as a physics student (also german), I have more math than this guy😂. Im in the first month of my second semester and we had the introduction to complex analysis today, after doing everything till calc. 3 in one semester.
@@kerr354 It's way more than that, I had 2 semesters of history, the two required english classes, and I took a poetry class to fulfil some sort of requirement, i took a philosopy class (logic), i had to take an ethics and technology class, and two social science classes (Human development, and abnormal psyc). I also took 4 semesters of physics and a semester of comp sci. I went to a liberal arts school, not a tech school, so i ended up having to take a broader range of stuff than if i went to a tech school. Even then you still have to take a bunch of electives.
When I was in highschool (during the pandemic) pre calc was essentially our Algebra 2 class, so if nothing else it was very important for the students who had their math classes compromised by Covid
Real analysis is where you have the seemingly most obvious stuff, and proving it is harder than anything you ever had to do before. One time I had to prove limsup a_n = maxAP(a_n), where (a_n) is a convergent series for n->infinity and AP(a_n) stands for the set of accumlation points of (a_n). I spent more than 4 hours on it and gave up.
Ironically Real Analysis was the best result I achieved in any module in university. People talk about its difficulty but it's more about being ok with the epsilon-delta formulation and then just applying it logically to the different areas of analysis. The multivariable section of Real Analysis (multivariable Taylors theorem etc) scared the shit out of me though. Diff Geom is a beautiful subject (and follows from Real Analysis/Manifolds) but a bit of a beast to truly understand.
No to be that guy, but daamn the list is feels soo incomplete, like basically there's nothing there. I have read that a lot of US universities basically have courses specifically designed to regularize their students since the education at a high school level can be very bad. Additionally, where i study my calc 1 and calc 2 are just basically real analysis of the US, and they are in the first year, additionally we have up to calc 4, with the same approach and we divide calc 3 in two with a rigoruous approach. Sometimes i feel like the US really need to get their shit together in terms of education because some of their unis just seem soo lacking
In America you only need to go up to Linear Algebra to be a math teacher, and unless your family's rich enough to afford 2 grand+ college programs over the summer or are lucky enough to have a connection with a professor, you can't go up further than AP Calc BC. (And even then only 53% of American high schools offer it, mine didn't)
algebra, pre calc, and geometry in college?? For a *math* major? I was considered behind for starting at Calculus 3, most math majors at my college started at linear algebra.... Almost everything listed, up to discrete math (proofs) or even real analysis, are freshmen year classes. WHERES non euclidian geometry, abstract algebra, graph and network theory, combinatorics, topology etc which would be taken sophmore thru senior year...
LMAO, what? I assure you that as a general rule, it is exceedingly rare to see a freshman taking real analysis. An introductory proof-writing class is what's expected of the most advanced first year students, along with linear algebra and ODEs.
Wtf I did all this and more the first year of university, are you sure you are not in high school? Also saying Taylor series are "sometimes" useful is wild and plain wrong.
I think this is highly dependent on your interests and what else you're taking. I can't imagine surviving deep learning without Calc III, for instance. So Calc III is definitely at least interesting + fun (if not legendary) for me
I liked the reference to the formula from Rudin's second page. I made a video explaining how you could come up with that formula instead of accepting it. Anyways: "great professors can make a mediocre subject excellent, and bad professors can make an epic subject lame" However, universities tend to be more interested in hiring doesn't focus too much on teaching so you get what you get
I feel like it isnt too hard to reason about right? Huerestically going along with how we expect real arithmetic to work: p - (p^2 - 2)/(p +2) = p - (p - sqrt(2)) (p + sqrt(2))/(p+2) = p - (p -sqrt(2))*h where h is some num in (0,1). So thus it takes us to a number closer to sqrt(2).
@@mlgswagman6002 Cool perspective or how it works! Thanks for sharing it! It would be cool if Rudin gave such explanation instead of throwing just the formula. I used this in my video as an excuse to explore some other math as well though. But I like your explanation a lot!
I took a math analysis class once. I felt like I needed to already have a math degree to understand the mysterious runes the professor was writing. I thought I was learning a new language for a second.
If you don't learn mathematical notation, then you can't be successful in mathematics. Mathematicians -- of which you aren't, but I am -- use mathematical notation all the time, big whoop. What else would they do? Write their equations, results, etc. in words and poems like the old times? No. We're a way more civilized, way more developed society now by several orders of magnitude greater now than we were back then. Mathematicians don't do that anymore.
@@Gordy-io8sb Way to miss the point of my comment. I don't know what kinda insecurities you're dealing with to cause you to project this hard and aggressively on me, but I hope you get over them. Because you look like an idiot behaving like this.
Idk, compared to other undergrad courses, real analysis is pretty easy, at least in my experience. Stuff like commutative algebra are a lot more difficult imo (mostly due to the fact that they are much more abstract)
I showed up to the first class of number theory and asked the professor what a number is. He just shook his head and said I should ask a logician or something.
Only course that belongs in "forced to take" is pre-calc; and MAYBE algebra I, and II (since you can basically learn all of this easily on Khan academy on your own) .. everything else belongs up above.
Hey. Can you explain the differences between undergrad math and physics? I am conflicted because I like both. I would've double majored but it is not possible.
What I've heard is that physics is the application of math. Generally physics is easy if you understand the math, but no clue how hard it ends up getting. But I chose math b/c I wanted to do quant (and it's an easier pathway). At least for math it started getting harder during the analysis --> probability theory --> measure theory path since there weren't tricks from comp math to save me
Physics is entirely different than math when it comes to it's approach. You can work with basic concepts of calculus/linear algebra and be very difficult, that's the difference. So it's not simply a dumbed down version of math, it's a different beast in itself.@@zeegy99
Hey man great video! I have some constructive criticism tho. I thought the pacing of the video was great but half way through I realized I was watching it at 1.5x speed. When I started watching it at normal speed it instantly became more boring. I would advise you to work on your pacing, speak faster and with more energy. Otherwise great content imo.
If you fear calculus, then you're just incompetent. It was easy for me when I actually put in the effort to learn it, and didn't just sit there and whine about how hard Cauchy sequences, limits, integrals, higher-order derivatives, and the such etc. are.
@Supercatzs Some will teach things like epsilon delta definitions of limits. But that is baby real analysis. I don't think anything mentioned in this video, perhaps with the exception of baby real analysis, is outside the scope of what a high school student might learn.
@@Supercatzs 1. I said most of this, not all of this 2. Looking at some programs online we definitely did some of what you call real analysis in high school. Also 100% of the courses in the video is studied between high school and the first year of almost every STEM degree course of university here, including engineering and chemistry.
@@fipillo4658 They really arent and if they are taught in engineering and chemistry, they are taught at a more semplicistic level, this is literally how any uni works. Chemistry guys get to study some calculus, but not Analysis which are different things because it would be useless for them to study In-depth math instead of useful math since Chemistry is not about studying deep mathematical models from a mathematical point of view, but from a chemical one. Same goes for Engineering. Some unis have more in depth courses than others but overall you wont be studying differential geometry or abstract algebra in chemistry or engineering because they literally dont need it. I could understand physics guys who need basically almost as much math knowledge as math guys themselves, but not the others
How about just being an autodidact? I taught myself all of the mathematics I know, with only like a handful of exceptions. Taking courses and getting a "muh degree" will just hinder you.
Looking back, yeah intro stats was basically a taster but not satisfying on its own. But going deeper? It's like playing Dark Souls. Find the average longest streak of heads in K fair coin tosses.
Those are very elementary topics, even at the supposed research level (save for number theory). I don't think Zeegy should waste his breath on those, even by his standards.
This video is soooo cute. I'm gonna make my own rankings for my stuff I took for undergrad (◑‿◐) S: lin alg, adv stats A: Calc 1, abstract alg 1, Real Anal B: Alg 2, Complex Anal, abstract alg 2, diff eq C: Alg 1, financial math, stats F: Geomertry (fuck that shit)
I have taken primarily prob classes! but i couldn’t really compare to the other classes at the same level so I just left it out. After seeing the comments, it seems like people care more about the harder topics. I’ll try to accommodate (but it will be hard)
I'm a mathematician, and I think linear algebra is completely pointless on it's own. You need to accompany it with ring theory, group theory, etc. for it to be useful.
I mean, if you major in mathematics/have majored in mathematics for a math PhD or the such, you're a math major....buuut, it doesn't mean you've done anything of note/ever will do of.
topology? differential geometry? complex analysis? topology? group theory? where is itttttt
Topology?
Group theory is in discrete maths, topology is in real analysis/calc.
Idk wtf is differential geometry though
@@BenyKarachun no???
@@BenyKarachuncurves/surfaces get themselves the fancy name of manifolds that live any arbitrary number of dimensions, and for every little patch on them, there must be a (hyper)plane a dimension less than this manifold where analytical (or holomorphic) differentiable paths are defined, i.e. it's the natural upgrade of calculus III.
@@loicboucher-dubuc4563 Idk maybe the courses are done differently in different countries
Here group theory is taught as a part of discrete maths and topology is taught as a part of real analysis (and tougher calc classes), they don't get courses of their own, maybe in pure math degrees they do but not for any other engineering/cs degree
seems more like a high school + math classes that are required for engineers tier list
Ah yes my favorite class in engineering undergrad. REAL ANALYSIS (the rest were all engineering math tho)
@@Immadeus real analysis is already a less theoretical / less math major oriented analysis class. Most math majors would take a class simply called "Analysis" that studies arbitrary metric spaces rather than the real numbers specifically
@@thechessplayer8328 my point was that most engineering majors would only take that with a math minor or second discipline, but yeah real analysis is definitely more useful for engineering rather than math.
@@Immadeus agree
Ahahahah
real analysis in No is crazy, of course this is not a math graduation
abstract algebra, differential geometry, and complex analysis enjoyers kicking and screaming (are these not undergrad (pure) math major at most unis)?
ABSTRACT ALGEBRA IS THE BANE OF MY EXISTENCE.
Even being a mathematician, I wholeheartedly despise every aspect of it. I want to vomit every time I see the word "(iso, homo, etc.)-morphism", "functor", "group", etc... It makes my head roil.
@@Gordy-io8sb boil+roll? lmao
edit: i just looked it up and that's an actual word... wut
@@MrStuffs??? You didn't respond to a single thing I said other than a very specific word.
@@MrStuffsAre you going to respond to the actual substance of my replies or no?
The only thing I would change is Calc 3. I thought it was fun
Calc 3 is lit
Depends on the teacher. Learning Calc 3 from TH-cam's Prof. Leonardo is the best. Learning it from my class is an absolute 180. Terrible
@@spilledshelf5 i learned calc 3 from youtube's dr. trefor bazett, and then calc 3 became my favorite math subject. in reality, i never entered in a university, i just learn math from youtube
@@spilledshelf5You could say it's π radians.
(Or τ/2.) What, who said that?
@@spilledshelf5professor leonard used to teach at the JC im currently going too
Differential geometry? Complex analysis? Abstract algebra? Group theory? Topology? Measure theory? What kind of university is this
measure theory is sometimes in real analysis courses
Note they said they were a major, not a professor lol, they probably just finished their 4 year degree.
Even then I will admit most colleges will require more than this, this seems like the bare minimum
to be fair, a couple of those courses are considered grad-level courses. Although, real analysis is literally like the bare minimum for a bachelors, usually people get into some abstract algebra or complex analysis
A high school
@@ballisticfox9033 A professor? In Greece a Mathematics undergrad takes Calculus, Real Analysis, Topology, Geometry (includes differential), Linear Algebra, Differential Equations (includes partial), Abstract Algebra, Measure Theory, Number Theory, some history and Philosophy of math and some applied computer classes. That is the bare minimum without electives.
How do you study for four years and only take the classes mentioned in this video? He even included high school math to make the list bigger. What short of degree is that?
This is joke, right? xd... right?
For anyone out there who really doesn't know what courses you take in pure math, I say to you, none of them are pure math courses... they are general courses that both science and engineering tracks take. The only pure math is Discrete Math and Analysis...and this last one is actually the core of your math courses.
He could be just a second year, which is why he hasn't taken algebra, topology, complex analysis, etc
I did a discrete maths module in my very first semester/term of university, though for reference, i'm on a computer science course in the UK (i barely ever needed it for any other modules, so not sure why they insisted on teaching it).
@@skiplangly6591 he is indeed a 2nd year
How many math courses have you yet to take? I feel like unis in the US take their sweet mf time with undergrad math. Like no way you need to take calculus in uni after also taking it in high school. In Europe if you do a math degree you start out with real analysis in the very first semester lol (although thankfully we don't use Rudin that shit is torture)
If you take AP Calculus BC in high school, and make a high enough score on the exam that your college will accept it as both a calc 1 and calc 2 credit, you start with your choice in Calc 3, LA, and Diff Eq
If you take AP calculus AB, most colleges will count it as a Calc 1 credit, so you start with Calc 2
Keep in mind most colleges in the US are more accepting of people who haven’t taken higher level classes than Europe, as long as they’re willing to pay for it.
It’s also important to keep in mind the US has a very different ideology when it comes to classes and choices, focusing on a more broad and diverse curriculum rather than hyper-focusing on your degree. Not saying either method is better, they just have different goals.
@@ballisticfox9033 does this also apply to the top American universities like MIT or Harvard? I mean, could you get a Harvard degree in Mathematics if you did just the courses listed in this video?
And can you also take the harder courses like Topology, Complex analysis, Measure theory, etc.? Is it forced/encouraged in any way?
I'd also like to add it may also depend on major. For some STEM majors at my school. like Computer Science majors for example, Calculus 3 and differential equations isn't required for them. Someone I know is in neurobiology at my school, and she only had to take Calculus 1, 2, and 3 was recommended but optional. But for someone like me (Electrical Engineering major, other engineering majors have the same math requirements I did) I had to take calculus 1, 2, 3, linear algebra and differential equations, though I was able to skip Calculus 1 because I took AP Calculus AB. But any math classes above differential equations is usually required for math or physics majors at my school, and for me it would be an optional elective class. Also, in my school there are easier variants of calculus for the premed oriented majors.
@@gregoryk_liteNo, in the top American universities such as MIT, Berkeley, Harvard, LA, etc these classes in this tier list are the entry/ requirements for the upper division classes which include complex analysis, topology, PDEs, Fourier analysis, etc. This video is about the lower divs (and the 1 entry upper div class real analysis). You cannot graduate by only taking these classes and these classes are requirements that all math majors take for their first 2 years. (But people typically get these done fast to do more stuff like complex analysis and other classes earlier)
Yet that terse style of rudin's is what fits theorems most in people's heads rather than the overly long walls of texts of Abbott's
German math bachelor just starts with Real Analysis (called Analysis I) and it involves Calculus along the way + Algebra means advanced Linear Algebra and is a 3rd semester class. Your tier list looks more like math modules for non-math students
If you're doing the quadratic formula in university, what are you even doing in high school? The first five courses you mentioned (Algebra I/II, Precalculus, Geometry, Calculus) are all things I've already studied before enrolling in university. I'm impressed that as a math major you can even choose not to take real analysis, which is a course I had in the first semester, and I don't even study mathematics; I study computer science. American universities are a joke.
Hi there, I agree American education is a bit scuffed. But for my high school we took calc3+linalg but had to test out in college so that was {nice?}
And analysis is required. But probably not needed for like 95% of the population
@@zeegy99 Okay, sorry if I was harsh. I presume that somebody who has studied Calculus III already knows Precalculus, so why waste time studying things you already know?
@@zeegy99 So are you done with your math major?
It's insane to me how much different Americans' (I assume you're American) maths programs are from Europe. In Germany, this is the order we did in a mahtematics bachelor degree. A semester is around 4-5 months
Semester 1:
Linear Algebra 1
Real Analysis 1 (In one dimension)
Semester 2:
Linear Algebra 2 (Topics: Bilinear forms, Jordan decomposition, dual spaces)
Real Analysis 2 (Topics: Generalizing derivatives in arbitrary dimensions and ODEs)
Semester 3:
Real Analysis 3 (Topics: Measure theory, Lebesgue Integration, Vector Analysis)
Stochastics
Semester 4:
Complex Analysis
Abstract Algebra (Endgoal: Galois theory)
Numerical Analysis
Starting from the fifth semester, you then choose what topics you want to specialize in (some options are Probability Theory, Stochastical Processes, Functional Analysis, Group Theory, Category Theory, ODEs or PDEs, Numerics of ODEs or PDEs, Optimization etc.). But the classes I listed are all obligatory and are expected to be done in two years.
There is no distinction between analysis and calculus
I wouldn't take this video too seriously. Not that i know much about education in Germany, but I think you guys are able to start specialising earlier than most students in the US.
@@walter274 Yeah, our equivalent of high school teaches some very watered down calculus among some other things, so people can start sooner.
Another aspect is that university is almost free compared and the requirements for studying things like math are very low - but, and this is the important part, it is accepted and expected that most people drop after the first year. So starting with the terror that is real analysis right from the beginning serves both to get students up to speed and also to filter them out.
Also, dont US bachelor STEM programmes involve you taking up 1-2 humanities courses along the way? Or at the very least that the philosophy is to have students take up a wider array of courses, which might even drift a bit from the major.
as a physics student (also german), I have more math than this guy😂. Im in the first month of my second semester and we had the introduction to complex analysis today, after doing everything till calc. 3 in one semester.
@@kerr354 It's way more than that, I had 2 semesters of history, the two required english classes, and I took a poetry class to fulfil some sort of requirement, i took a philosopy class (logic), i had to take an ethics and technology class, and two social science classes (Human development, and abnormal psyc). I also took 4 semesters of physics and a semester of comp sci.
I went to a liberal arts school, not a tech school, so i ended up having to take a broader range of stuff than if i went to a tech school. Even then you still have to take a bunch of electives.
@@walter274 that explains it. So I was right, (up to positive scalar scaling)
This guy ain't a math major bro hates proofs
Proofs are extremely tedious and slow to write. So yes, he is allowed to hate them.
Real analysis is the best part of maths, it shows who understands and who doesnt
When I was in highschool (during the pandemic) pre calc was essentially our Algebra 2 class, so if nothing else it was very important for the students who had their math classes compromised by Covid
same here
U must have done more than that in undergrad.. what about group theory? complex analysis? algebraic topology? variational calculus?
Real analysis is where you have the seemingly most obvious stuff, and proving it is harder than anything you ever had to do before. One time I had to prove limsup a_n = maxAP(a_n), where (a_n) is a convergent series for n->infinity and AP(a_n) stands for the set of accumlation points of (a_n). I spent more than 4 hours on it and gave up.
These are like 1st or maybe 2nd year classes, where is topology, measure theory, logic and Galois theory?
i was gagged when you said you don't use taylor expansions
Ironically Real Analysis was the best result I achieved in any module in university. People talk about its difficulty but it's more about being ok with the epsilon-delta formulation and then just applying it logically to the different areas of analysis. The multivariable section of Real Analysis (multivariable Taylors theorem etc) scared the shit out of me though.
Diff Geom is a beautiful subject (and follows from Real Analysis/Manifolds) but a bit of a beast to truly understand.
No to be that guy, but daamn the list is feels soo incomplete, like basically there's nothing there. I have read that a lot of US universities basically have courses specifically designed to regularize their students since the education at a high school level can be very bad. Additionally, where i study my calc 1 and calc 2 are just basically real analysis of the US, and they are in the first year, additionally we have up to calc 4, with the same approach and we divide calc 3 in two with a rigoruous approach. Sometimes i feel like the US really need to get their shit together in terms of education because some of their unis just seem soo lacking
In America you only need to go up to Linear Algebra to be a math teacher, and unless your family's rich enough to afford 2 grand+ college programs over the summer or are lucky enough to have a connection with a professor, you can't go up further than AP Calc BC. (And even then only 53% of American high schools offer it, mine didn't)
Come on, how about applied math? PDEs, Dynamical Systems, Numerical Analysis, Optimization, Fourier Analysis, Functional Analysis?
It depends on the size of the school he's at and his interest.
Bro the calc 1 and calc 2 were the two first math classes i took in uni, and both were literally real analysis lmfao.
algebra, pre calc, and geometry in college?? For a *math* major? I was considered behind for starting at Calculus 3, most math majors at my college started at linear algebra.... Almost everything listed, up to discrete math (proofs) or even real analysis, are freshmen year classes. WHERES non euclidian geometry, abstract algebra, graph and network theory, combinatorics, topology etc which would be taken sophmore thru senior year...
LMAO, what? I assure you that as a general rule, it is exceedingly rare to see a freshman taking real analysis. An introductory proof-writing class is what's expected of the most advanced first year students, along with linear algebra and ODEs.
Currently going through real analysis and i must know: did you guys proof your theorems for all metric spaces or just R^n ?
Yes, it's nightmare
Wtf I did all this and more the first year of university, are you sure you are not in high school? Also saying Taylor series are "sometimes" useful is wild and plain wrong.
I still have scars from Real Analysis 1 and 2 and I took it in 07-08.
My university has a course where the first part of the title is "introduction to chaos"
US education moment
I think this is highly dependent on your interests and what else you're taking. I can't imagine surviving deep learning without Calc III, for instance. So Calc III is definitely at least interesting + fun (if not legendary) for me
I liked the reference to the formula from Rudin's second page. I made a video explaining how you could come up with that formula instead of accepting it. Anyways:
"great professors can make a mediocre subject excellent, and bad professors can make an epic subject lame" However, universities tend to be more interested in hiring doesn't focus too much on teaching so you get what you get
I feel like it isnt too hard to reason about right? Huerestically going along with how we expect real arithmetic to work:
p - (p^2 - 2)/(p +2) = p - (p - sqrt(2)) (p + sqrt(2))/(p+2) = p - (p -sqrt(2))*h
where h is some num in (0,1). So thus it takes us to a number closer to sqrt(2).
@@mlgswagman6002 Cool perspective or how it works! Thanks for sharing it!
It would be cool if Rudin gave such explanation instead of throwing just the formula.
I used this in my video as an excuse to explore some other math as well though. But I like your explanation a lot!
I took a math analysis class once. I felt like I needed to already have a math degree to understand the mysterious runes the professor was writing. I thought I was learning a new language for a second.
If you don't learn mathematical notation, then you can't be successful in mathematics. Mathematicians -- of which you aren't, but I am -- use mathematical notation all the time, big whoop. What else would they do? Write their equations, results, etc. in words and poems like the old times? No. We're a way more civilized, way more developed society now by several orders of magnitude greater now than we were back then. Mathematicians don't do that anymore.
@@Gordy-io8sb Way to miss the point of my comment. I don't know what kinda insecurities you're dealing with to cause you to project this hard and aggressively on me, but I hope you get over them. Because you look like an idiot behaving like this.
@@panqueque445Did I not mention I'm a mathematician, dearie -- a real one? Also, I think I hit the point straight on.
i think real analysis is a very difficult, but very important tool for learning stuff like topology and functional analysis.
Idk, compared to other undergrad courses, real analysis is pretty easy, at least in my experience. Stuff like commutative algebra are a lot more difficult imo (mostly due to the fact that they are much more abstract)
Calc 3 is acc useful for machine learning tf are u saying
me studying math entrance on yt then seeing this banger come up in my feed right in time to give me existential crisis
Calc 3 is when physics and other math started to make sense, linear was alright and I know it’s used for a lot of stuff.
Where's the godly NUMBER THEORY!?
why did you only do the first half of undergrad?
I showed up to the first class of number theory and asked the professor what a number is. He just shook his head and said I should ask a logician or something.
real analysis is like understanding dark memes
Tf? Calc 3 is peak. Maybe it doesn't have as much utility as linear alg or smth, but it's mathematically beautiful
Only course that belongs in "forced to take" is pre-calc; and MAYBE algebra I, and II (since you can basically learn all of this easily on Khan academy on your own) .. everything else belongs up above.
based linear algebra take
Imagine being a math major and not liking analysis. They really just let in anyone these days.
Hey. Can you explain the differences between undergrad math and physics? I am conflicted because I like both. I would've double majored but it is not possible.
What I've heard is that physics is the application of math. Generally physics is easy if you understand the math, but no clue how hard it ends up getting.
But I chose math b/c I wanted to do quant (and it's an easier pathway). At least for math it started getting harder during the analysis --> probability theory --> measure theory path since there weren't tricks from comp math to save me
@zeegy99 By Quant do you mean quantum mechanics?
@@hasibulazamsehab866 Quant finance (like Rentec, Janestreet)
@@zeegy99 did you do competition math in high school? did you go to the IMO?
Physics is entirely different than math when it comes to it's approach. You can work with basic concepts of calculus/linear algebra and be very difficult, that's the difference. So it's not simply a dumbed down version of math, it's a different beast in itself.@@zeegy99
Hey man great video! I have some constructive criticism tho. I thought the pacing of the video was great but half way through I realized I was watching it at 1.5x speed. When I started watching it at normal speed it instantly became more boring. I would advise you to work on your pacing, speak faster and with more energy. Otherwise great content imo.
Thanks for the feedback!
a math major not enjoying real analysis? *there is an imposter among us*
dude sounds like an engineer the way he describes math classes lol
Haven’t seen the vid yet, but I quite enjoyed real analysis!
divided by countries united by fear of calculus
If you fear calculus, then you're just incompetent. It was easy for me when I actually put in the effort to learn it, and didn't just sit there and whine about how hard Cauchy sequences, limits, integrals, higher-order derivatives, and the such etc. are.
Excuse me are these university courses in America? In my country most of this is made in high school
Calc 1 and 2 are optional as ap classesin america which i think are equivalent to A level maths or ib not sure
no highschool is going to be teaching real analysis lmfao
@Supercatzs Some will teach things like epsilon delta definitions of limits. But that is baby real analysis. I don't think anything mentioned in this video, perhaps with the exception of baby real analysis, is outside the scope of what a high school student might learn.
@@Supercatzs 1. I said most of this, not all of this 2. Looking at some programs online we definitely did some of what you call real analysis in high school.
Also 100% of the courses in the video is studied between high school and the first year of almost every STEM degree course of university here, including engineering and chemistry.
@@fipillo4658 They really arent and if they are taught in engineering and chemistry, they are taught at a more semplicistic level, this is literally how any uni works. Chemistry guys get to study some calculus, but not Analysis which are different things because it would be useless for them to study In-depth math instead of useful math since Chemistry is not about studying deep mathematical models from a mathematical point of view, but from a chemical one. Same goes for Engineering. Some unis have more in depth courses than others but overall you wont be studying differential geometry or abstract algebra in chemistry or engineering because they literally dont need it. I could understand physics guys who need basically almost as much math knowledge as math guys themselves, but not the others
putting diff eq and calc 3 so low is heinous. they are my world
Poor calc 3 and real analysis.
Also , where's diff geo? Topology? Diff topology? Complex analysis? Calc 4 and 5? That's the fun part of mathematics.
from this video i can just tell you're into applied math more than pure math lmao
What about PDE?
Real analysis is ez. Metric spaces and measure theory are super intuitive, its just fourier analysis that is hard.
Putting linear algebra in s tier for usefulness but not diff eq. and multivariable calculus is crazy. physics? Hello?
Group and Category theory is like being punched in the face but fun
Point-Set Topology, Differential Geometry, Differential Topology, Analysis, and Algebra are fine, but… Algebraic Topology -.-
Noice! Calculus is just the prologue of math
UC Berkeley has a class on C* Algebras, where can I find that on the tier list?
I just took my discrete math final and that segment of the vid made me burst out laughing 😂
Functional analysis or nothing
I get nightmares from real analysis
How about just being an autodidact? I taught myself all of the mathematics I know, with only like a handful of exceptions. Taking courses and getting a "muh degree" will just hinder you.
Somehow I was able to get an A+ in my real analysis course lol
Analysis is fun even though my brain is better at Algebra. Proof based math is the reason I bothered to get a math degree.
Thank you zeegy99 ❤
our glorious king
descret and calc 3 are the most fun math classes
bruh where is cirnos math class, that one should be in a tier of its own
Looking back, yeah intro stats was basically a taster but not satisfying on its own.
But going deeper?
It's like playing Dark Souls.
Find the average longest streak of heads in K fair coin tosses.
Hmm sounds like an avg interview question - I’ll try it for an hour and see how it goes
@@zeegy99 It's basically impossible to calculate it analytically.
You have to use Montecarlo sims. Better boot up R!
diff>>linear for me. I did not enjoyed linear algebra.
new zeegy just dropped
PDE, number theory, set theory, functional analysis... ?
Those are very elementary topics, even at the supposed research level (save for number theory). I don't think Zeegy should waste his breath on those, even by his standards.
ermm what the sigma
This is only year 1.
When you said Algebra 1 I thought you meant fields and galois theory stuff ngl
Galois theory is absolutely pointless.
Discrete math was my favorite
This video is soooo cute. I'm gonna make my own rankings for my stuff I took for undergrad (◑‿◐)
S: lin alg, adv stats
A: Calc 1, abstract alg 1, Real Anal
B: Alg 2, Complex Anal, abstract alg 2, diff eq
C: Alg 1, financial math, stats
F: Geomertry (fuck that shit)
You never had to take a probability class? What kind of math major were you?? Did you even math, bro?
I have taken primarily prob classes! but i couldn’t really compare to the other classes at the same level so I just left it out.
After seeing the comments, it seems like people care more about the harder topics. I’ll try to accommodate (but it will be hard)
@@zeegy99 You'll fold like a house of cards against an actual mathematician -- someone like me -- if you think this way. I dare you to debate me.
I feel like this guy just doesn’t really tv like his major lol
What's the song at 2:00?
Bonetrousle - Undertale
@@zeegy99 thank you zeegy very cool 👍
but analysis is cool like complex analysis
Bro's a first year
I have an Analysis 2 test in 9 hours, it's 2 AM :)
o7
@@zeegy99 o7 🫡
thoughts on set theory?
LINEAR ALGEBRA, MY QUEEN!!!!!!!!!!!!!!! (I literally did the moistcritical meme I got excited, I love linear algebra, cool ass shit)
I'm a mathematician, and I think linear algebra is completely pointless on it's own. You need to accompany it with ring theory, group theory, etc. for it to be useful.
Bro since when is Algebra about adding numbers wth. Its about Galois theory
Quite snarky. But not snarky enough to be a troll. Next.
Your name is "Gauß". Lol. Do you seriously think you're a genius at the level of Gauss? Seriously? Do you really think that?
No, I don't
@@Leo-io4bq "My delusions of grandeur are totally real!" "Source: Trust me bro"
@@Leo-io4bq I wonder if you could stand in a debate with an actual mathematician. You'd fold instantly.
what about the others
whats the music during the ode segment at 3:28
Another Medium - Toby Fox
Most of the songs in the video are from Undertale
This dude is NOT a math major
I mean, if you major in mathematics/have majored in mathematics for a math PhD or the such, you're a math major....buuut, it doesn't mean you've done anything of note/ever will do of.
I liked the integration parts of calc 2 but yeah series suck
Series are hilariously easy -- the arithmetic isn't even that tedious.
Undertale music 😀
stats is the most awful class why would you put it in ok i hate stats i hate stats i hate statas i haTES STATS I HATE STATS
Filtered
ZEEGY99 CONTENT
You should try topology before yoloing your life savings in crypto!
Ignore the supporters
arithmetic :c
He DID rank number theory.
I guess this is supposed to be funny...?
Linear algebra sucked
delete math