WHAT COMES AFTER CALCULUS? : A Look at My Higher Level Math Courses (I Took 22 of them). 

I feel flattered that RU-vid thought I should know this

So the answer to "What comes after Calculus?" is suffering?

You know a book is expensive when it's pointed out as expensive in comparison to a bunch of college textbooks

I'm a math major as well, so far I've taken: Calculus I Calculus II Calculus III Calculus IV Discrete math I Discrete Math II Intro to programming Linear data structures Non linear data structures Geometry Linear algebra I Linear algebra II Combinatorics Probability I Probability II Statistics I Statistics II Ordinary differential equations I Ordinary Differential Equations II Mathematic Analysis I Numerical analysis I Right now I'm taking measure theory Complex analysis Numerical linear algebra Linear programming Next couple of semesters I'll be taking: Functional analysis Non linear differential equations I & II Partial differential equations I & II complex analysis II Differential Geometry Abstract algebra Numerical Analysis II Classical mechanics I've also taken some other course that aren't that math related like some C.S courses like probabilistic modeling, programming, and language courses It's been quite a ride.

As someone who never went beyond Multivariable Calculus, this is really great for giving me a glimpse of what lies beyond. Like you've taken me to the top of a mountain and showed me a whole unexplored country. You are describing all the waterfalls and forests and lakes and towns in that country, and I don't understand any of it yet really, but I get a sense that the country you visited is vast and rich, and I like gazing out and thinking about it.

Everytime a book hit the floor, my heart jolted.

It might be easier to answer: what DOESNT come after calculus? Mathematics is an incredibly vast subject, and calculus is just one of a couple starting points.

when one completely finishes calculus he/she ascends into the multiverse and loses their mortal selves.

As a physics student seeing all these fun math classes makes me want to do a math degree after physics

As a math major freshman I loved seeing what i will have to study later! Please keep the content going because is so difficult to find great math channels:)

What comes after Calculus? Nightmares. That's what.

I am equal parts fascinated and repulsed by mathematics, I absolutely loathed my courses at university and performed poorly but I love watching the videos

Y'all pure math students are on a whole different level. Double-Majored in physics and applied math (and technically minored in Chem, had enough credits for sure but our department didn't award minors. Beyond diffEQ and linear algebra many of my "applied mathematics" courses were counted within my other quantitative courses, so the usual physics major sequence and some of my chemistry sequence. I took modern algebra for a month before dropping it due to just feeling out of league with the other pure maths majors in the class. Very inspiring stuff though and makes me want to independently study much of this on my own now, so ty for the inspiration!

I made it 4 minutes in before I decided I don’t think I’m going to take any more math classes! Thank you!

Engineering physics/science background here. We did about half of these but the order was way different. Your order makes a lot more sense.

I think it is pertinent to mention that the curriculum for a maths undergrad degree is *highly* variable based upon the track and uni that you choose to study. Some of the courses discussed in this video were not even available to undergrad students at my uni (looking at you, standalone elliptic curve theory, purely for grad students), and many others were outside of my track for applied mathematics. It's important that those interested in studying maths know that this is not a definitive guide to what you will encounter and you will likely get a better guideline from looking at your uni's college of maths curriculum or speaking with an advisor.

This was soooo fun to watch. I wish I had this when I was deciding on a major. I actually had really awesome math teachers my first couple years of college since I transfered to a 4 year school. They did a lot of proofs and thoroughly explained things like what the dx is for. I taught myself Linear Algebra and wish I had taken a class so I could understand the theory better.

Man u have got to produce more & more of these videos. This is ONE of the BEST, FINEST content I have watched on Math courses & Textbooks that will be useful in undergrad & post-grad.

Just the type of video i've been looking for!

literally every class you described here, terrified me. i'm glad the last actual math class i have to take is diff eq. i was considering minoring in math, depending on how many additional math classes i have to take, but i think i will focus on taking extra mechanical engineering courses instead lol.

Thanks for the list! Very comprehensive and great descriptions. I will use this as a reference to guide my self study in mathematics (many free courses online bc degrees are very expensive!)

I am so glad this video exists. Cant tell you how helpful it was

The fact that you can remember and articulate the details of all those classes is astounding to me. I watched through the whole video even though it was WAY over my head. A funny idea for the book throws might be to add other sound effects in place of the slams. (Not sure if that's your 'brand', but it might maintain the novelty of it.)

I just finished calc III, i felt so proud of my self. But this video made me realize that I'm just in the beginning of getting absolutely shafted by math. Guess i better bite the pillow.

i didn’t understand half the things talked about in this video but i am very intrigued

Congratulations! That was a well made and interesting video. Also, a Greek I am happy to hear that you take Euclidean geometry as an undergraduate. We are very lucky in greece to have the privilege to learn Euclidean geometry in high school because it such an interesting course.

You deserve much more subscribers. The level of work do on your videos is mind-blowing. Excellent content. Just subscribed

I love this video format so much. Great video.

(Edit July 31 2020): I've had a handful of requests to go through and list out the books that are in this video. For the physical books, I have gone ahead and added links in the description to where you can find them. Thanks for making this one of my most popular videos!! I really do appreciate the support 😃 === I know it took me a while to give my answer to "What comes after Calculus?" but if you want to see me talk anymore about the topics that come up in this video you can let me know here!!! (Also, really thinking about rating these classes based on my reluctance to throw them books lol :D )

"Diagonalizability" And I thought 'differentiability' was fun to say...

Very helpful for knowing what I need to study beforehand. Thanks.

I just found this and I want to tell you that this is really awesome content, really helpful! I think you should definitely make more! Thank you!

Totally unique- Love your humor haha! Can you do a ranking your math classes video?

EC Crypto relies on the Rational Points concepts as you stated. It's ridiculously complex, but oddly I recently found not very post-quantum computer resistant. I wondered what came after the difficult math, and I appreciate your sharing even though this is a little late. Kinda wish I had taken more math courses and actually really learned more complex topics of math. As I age keeping these concepts alive in my brain has become extremely difficult and try to be happy with what I can recall after sleep.

I really enjoyed watching this, the layout is great. Congratulations dude!

This video speaks to that part of me that didn't take the opportunity to challenge my potential for more advanced courses in college. Years of depression and anxiety stifled me in about every way possible in high school right through my first years in college and beyond as well. Now at 68 and retired I am beginning to explore again what I wished I had had the capacity to do all those years ago. I may be watching more of your videos in time. I suppose the book that interested me the most was fractal geometry.

Good work! I wanted to take combinatorics but it wasn’t offered my graduating semester. Neat handwriting... I could never lol

I love how you throw your books and they make a loud bang. It sort of reflects my mood when it comes to math. I sure love it when I understand it and I solve several problems in a row, but man, when the problems pile up, and you feel like you're on the verge of giving up, DOES IT EVER FEEL GOOD TO THROW THAT SHIT AWAY WHEN YOU'VE MADE IT THROUGH AND YOU'RE EXHAUSTED FROM THE WORK!

I went more heavy into Statistics, R, and computational mathematics in my undergrad than you did, and less into analysis, topology, and differential equations. Good video thanks for sharing.

It's like I'm reliving college. Yep! And now I work with teaching my daughter arithmetic, and my son how to count and group. Math is a language, and I'm ensuring my children can speak it. It's funny, but when you pull the first book off the top after the Calc book, I KNEW it was a Linear Algebra book by the...shape. Yeah, I don't know why, but they have a "look" about them. Too cool! Great video!!

I am an aspiring math major in community college but I'm barely going onto Calc 2 and linear algebra and seeing all of this made me nervous because I've heard many times that the math in university is vastly different from the one we do in say, high school, or just simply the "computational" stuff. I appreciate you showing this, as I feel like I don't spend enough time actually researching or knowing more about maths as a mathematics major, but this will make me want to look more into it all.

Real analysis for me was a real trip. Took me ages to get a grip of it but started to like it and got a decent mark in it. I do wonder why people say it's the hardest as it's only hard in its own particular way.

This video excited me to get my Masters in Math upon retirement, like I wanted to do at one point before I began tutoring part-time instead! I'm a graduate Electrical Engineer, but Math is my passion. Thanks for sharing!

You have an awesome spirit. Subscribed and looking forward to future content

Math/Science/Stat Courses I took as an undergraduate after calculus III: (* means a graduate level course) 1. Modern Physics 2. Intro. to Differential Equations 3. Introduction to Mathematical Proof 4. Mathematical Writing 5. Applied Linear Algebra 6. Linear Algebra I: A proof based linear algebra course 7. Applied Ordinary Differential Equations 8. Applied Partial Differential Equations 9. Quantum Mechanics I & II* 10. Mathematical Physics* 11. Special Topics in Fluid Dynamics* 12. Probability Theory*: It was basically a measure theory class with application to probability space and martingales. 13. Complex Analysis I* 14. Functional Analysis I* 15. Harmonic Analysis* 16. Classical Methods of Partial Differential Equations* 17. Abstract Algebra*

1:05 I still have some of my university texts, too. Though not that many.

Subscribed! Super helpful video and great breakdown.

My favourite math unit in undergrad thus far has to be non euclidean geometries and topology. It was so interesting to look at math so different from subjects in analysis, linear algebra, math techniques etc. Complex analysis did my head in, I cant tell you how many late nights I had staring in bewilderment at the residue theorem lol

I was waiting for the textbook that you used for Axiomatic Set Theory since you've made few videos on Axiom of Choice, Zorn's Lemma....

I graduated from a lower tier university, but all of my math professors were passionate wonderful people who cared deeply about their students. My undergrad math was much lighter: * Calculus 1 - Derivatives and integrals * Calculus 2 - More integrals and infinite series * Calculus 3 - Calculus 1 with an extra variable with vector stuff on top * Linear Algebra - Matrices, transformations, systems, vectors, eigenvectors * Differential Equations - ODEs. Solving differential equation in different forms. Laplace Transforms * Mathematical Structures - Proofs, set theory, functions, and some topology. * Programming in Mathematics - Kind of like Numerical Analysis course without a lot of theory. * Real Analysis 1 - Mostly did series, more set stuff, and measure theory. * Real Analysis 2 - This felt like going back to Calculus I and proving stuff we learned from there * Geometry - 90% Euclidean geometry learned in HS but all proofs. 10% interesting non-Euclidean stuff. * Probability - Discrete probability and continuous probability. * Statistics - Confidence intervals, error, random generating functions, linear regression, and ANOVA * Combinatorics - Counting. Combinations permutations, binomial theorem, some algorithmic stuff * Graph Theory - Coloring, planarity, min/max flow, pathing, and hypergraphs. * Abstract Algebra - All groups: cyclic, subgroup, ablein groups, isomorphisms, permutations, etc Math related * Formal Languages - Regular grammars, pushdown automata, turing machine, and chomsky hierarchy * Algorithms - Asymptotic analysis, master theorem, divide & conquer, greedy method, dynamic programming. * Database - Professor was a huge theory nerd. About half course on relational algebra and theory. Graph theory was my favorite course. We were lucky enough to have a distinguished professor who made tons of advancements in graph theory. My least favorite was Geometry. Most of the course was rehashing from high schools days. The non-euclidean stuff was far more interesting, but we didn't get to it until the very end :( The most difficult for me was Real Analysis 1. It was by far the largest difficulty curve spike to overcome.

This was an AMAZING video. definitely subscribed!

Getting into calc 3 in the spring looks like a bright future ahead!

I’m a CS student and It’s all calculus : ( *cries in java*

i dont know why i’m even watching this, i’m still in high school :/

Really fun video and so informative! Thanks a lot from Philippines

I'm a kinesiology major looking to pick up a mathematics minor, and this makes me really excited!!! (I've just done calc 1 and 2, and I'm working on linear algebra at the moment). Thanks!

9:36 Harmonic oscillators are supposed to be weirdly ubiquitous in physics, right? I guess that might explain the focus. 21:20 As a (sort-of) programmer I'm glad you enjoyed Sipser! I assume it's a lot more basic than most of your other undergraduate maths classes were? 22:50 It's remarkable that fractal visualisations are so famous and so familiar by now that people talk about fractals as the maths behind the images rather than the images as visualisations of the mathematics ...

Thanks for this. I just got accepted to UC Davis for math and thinking of pursuing it. Or mechanical engineering at a maritime academy. not sure yet

Im taking both intro analysis 1 and abstract algebra 2 next term, and you have officially made me nervous. Just trying to make it through my freshman year alive 😅. Awesome video btw!

Awesome video! I really enjoyed it!

so there are like 3 or 4 classes after calc, and then 15 dedicated classes to proving the specifics of the others

Diffy Q's Kicked. My. Ass. So glad to have finished out the series with a C- (Mercy pass)

Thanks for this video! I have multiple books to purchase now :)

CHALK, I never knew of your existence. I do now. You deserve my subscription and a real hard tap at that bell icon.

Lmaooo I’m using the same linear algebra book I’m taking it this semester and honestly I needed help with understanding so I also got “Linear algebra done right” but it wasn’t required for my class

As a stats person who gets made fun of by math peers, yet usually outclasses them (at least in analysis), it sounds like Feller's probability texts would've been more your style. Thank you, and great video.

I love math and this made me to take all this courses ..thanks a lot 💚💜

So I imagine you can keep up with the MIT open courseware, damn I'm jealous AF. Love your honesty about how you tbought a class would be sum BS, but turned out to be very rewarding, however it's hard to know if the subject matter promoted that or an engaging teacher. Super thanks for the hard work in making the video.

For the scene from Good Will Hunting, where Will solves the problem on the blackboard in the hallway, would that problem be from Fractal Geometry?

im not sure why why YT is showing me this after graduating from economics when all i learned was i am not as good in math as i thought i was

Ten years ago, I was a very precocious child. I was the only one of my peers so interested in physics or anatomy or higher math and other abstract thought in elementary school. My academic zeal may have faded slightly, but I'm glad to see my youtube recommendations age with me. I especially appreciate the covering of complex topics that I will either not delve deeper into, or not study deeply for many years. I'm not so presumptuous as to call myself a modern Renaissance man, but I definitely have such vast, disparate interests, and such a hunger for an accurate depth of knowledge and experiences, that it would be impossible for me to satisfy my curiosity. That can't stop me from trying.

I’m in community college in Calc 2 right now. I was always wondering this question so thanks for this video. (I have linear algebra and differential equations next semester but that’s the highest math my college has)

So what advice can you give for a undergrad when proving proofs becomes a dominant part of the course?

Being Italian, there is one thing that I always find strange: we have no calculus. If you need calculus, you'll get real analysis. Also, maths is always proof-based for people who really need it (physicists as myself, mathematicians, engineers, etc...). Also, scientific subjects practically never have required readings.

My discrete math 2 course at my community college was basically just graph theory and a little bit of algorithms. Very much geared for CS people, but I wish I learned more about it. I might take another class while I get my bachelors in CS. I'm debating about getting the math BS because after calc 3 and linear algebra I knew it was only going to get harder from here and for me those classes were pretty hard. Almost all my classes for my associates degree was just me learning from teacher notes and not really using a textbook. Those classes were also heavily proof based and extremely rigorous. I had a single test for linear algebra that would have been about 80 pages of work had I chosen to do all of it but I couldn't.... I managed 56 I think. I like the way you described things so I subscribed. Thanks for sharing. I do hope you take graph theory because for me it was my most interesting subject in math.

I understood everything until 2:45, I've got a journey ahead of me! great video :)

Man/Woman does a doctorate in both Physics and Maths. The power is over 9000

What's the book you used for Differential forms and Vector Calculus? I'm studying physics in grad school (year 1) and I feel like that would be an extremely useful topic to become very familiar with.

Cool, I have that same abstract algebra book for that class. Not saying I finished and understood the course, it was just neat seeing that book.

There ya go, stayed right to the end of your course. Like running a marathon. You can post my diploma, after the results come in.

Lol I had Otto Bretscher as my linear algebra professor...I will reserve my opinions about him 😂

I want to know how much math "after calculus" could come "before calculus." In other words, if calculus were not "universally" used as a flunk-out filter, a masochistic masticator, and a pupil purifier for professorially-preferred progeny, what would a math curriculum look like if the courses that can be taken without calculus first were encouraged to be taken and enjoyed?

Love the chalkboard format. I did Euclidian geometry freshman year in the Greek, though I'm not a math major but a philosophy major. Finished my senior year with some topology and non-euclidian geometry: Gauss and Lobachevsky. Can't wait to see more of your videos. Subbed.

Great! You're my new advisor!

dude stop throwing books ,respect them geeeezz!

#4 suprised me, in Germany "real analysis" is the way calculus is approached in Uni from the first Semester onward ... and in my opinion rightly so. Sounds like you learn a lot of stuff heuristically over there.

I am back listening to you and taking notes.

I really enjoyed partial differential equations and fourier series, also learned about perturbation theory in that class which was very fun!

How were you able to learn greek that fast?You know the correct orthography of the name eucledian and how to put the correct emphasis and diacretics in ancient greek!?! All that in 4 months?

depression come after calculus

thank you for sharing!!!

Lovely video. Thank you. It honestly feels like food :)

Why would you throw your books like that. I even wrapped my books in a plastic cover....anyways, nice content.

How did you self-study for Multivariable Calculus??

Great video. Thanks for posting this.....2 years ago !

GREAT VID- Kudos

Bruh you shouldn't be throwing books like that. Books and knowledge are a privilege.

Me when he pulled out the calculus textbook and it wasn’t Stewart’s 🥺🥺

I live in england and I just graduated with my BSc in mathematics. We share most of the fundamental courses like linear algebra, differebtial equations and complex analysis (real analysis wasn’t available for some reason lmao). I didn’t expected most of our modules to not overlap which was a nice surprise. I got to do a lot more applied modules such as fluid dynamics, quantum mechanics, tensor analysis. I even had modules related to coding, stochastic processes and metaphysics. I’m rather surprised you didn’t study group theory but i envy the fact you got to do topology since it wasn’t an option where i studied.

My first brick wall was after breezing thru the calc series, facing Halmos Finite Dimensional Vector Spaces for my first look at Linear Albebra. I had nowhere to hide.