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Paul's Online Math Notes
- aliljet 6y agoFunny seeing this here, but a decade ago, this professor's differential equations explanations saved me from endlessly meandering through tough engineering lectures and dense math textbooks. I'm frankly curious to know if undergraduates and high school students still use this...
- pcbro141 6y agoI just graduated university this year and used the site in high school and a little bit in university (just freshman year calculus). I don't remember how I found the site, either I was Googling math stuff, or a friend was and sent me a link. Many friends in high school and university used the site, it's definitely still popular. A high school teacher may have recommended it as well. It's a great site.
- deleted 6y ago[deleted]
- gibspaulding 6y agoI graduated in '17 and used his site a ton once I got to classes beyond what Khan academy offered. It's an excellent resource!
- ppur 6y agoIn high school right now; my calc teacher links to this all the time when the textbook's explanation/examples are unsatisfactory
- techwizrd 6y agoI used this when I took differential equations while working on my math major (graduated in 2016). It's simply invaluable for the anxious, confused student.
- paulpauper 6y agowhy are differential equations so difficult? it would seem even smart people struggle with understanding them. Is the problem is how they are taught or is the concept inherently hard/ It should not be that hard given that we're taking 'rates of change' broadly speaking, yet it is.
- reedf1 6y agoThe concept of a differential equation is not hard. Solving differential equations can range from trivial to non-trivial.
- peterwoerner 6y agoBoth. They are taught, generally speaking, by giving a series of techniques to solve problems without the underlying mathematics (i.e why those techniques work on those problems). Calculus generally takes some time to grok, which typically doesn't have unless your really into it. Think how many people fail to grasp exponential growth (the simplest ODE).
- mhh__ 6y agoGood scientists are often terrible teachers. That and people default to being scared of mathematics rather than seeing difficulty as proof of it being rewarding to learn.
- legerdemain 6y agoEven among the engineering-oriented lower division math courses, material in differential equations courses is both antiquated and presented like a laundry list. This essay is more than 20 years old now, but the complaints are basically as relevant today.[1] [1] https://web.williams.edu/Mathematics/lg5/Rota.pdf https://web.williams.edu/Mathematics/lg5/Rota.pdf
- nso95 6y agoLearning hard things is difficult for anyone, even smart people
- CaptP 6y agoHigh school student here who self-studied multivariable calculus and differential equations. I absolutely LOVE Paul's Online Math Notes.
- MisterBiggs 6y agoIn the last year of my Aerospace Engineering degree. This site got me through calculus and I still use it any time a differential equation shows up in my studies since they still scare me.
- sq_ 6y agoCan confirm that they do. At least at the schools I’ve been at, “Paul’s Online Math Notes” is an instantly recognizable name for most STEM students.
- deleted 6y ago[deleted]
- jesvschrist 6y agoIn my senior year of undergrad now, every maths student I know shares "Paul's Notes" to people struggling.
- mkl95 6y agoClassic! I used to read these before I dropped out of Physics to make software for a living.
- matthewh806 6y agoI could write this exact sentence out and it still be true :D Seriously, a great collection of notes. These were easily the most well written, thought out, understandable & digestible notes I ever came across during my studies. Blows most textbooks and professors out of the water
- maccard 6y agoThis site was more useful to me than €300 of textbooks during my undergrad. I had to purchase two books so I could submit the homework, and in all honesty, I learned nothing from them, very little from the lectures. Almost all of my library algebra and at least my first 4 calc courses we're handled by Paul (and khan academy occasionally)
- isatty 6y agoNever thought I’d see this here. I started studying Calculus I from here before I started university. It’s really amazing how well structured and written these notes are. No video lectures needed to understand. I didn’t attend a single lecture for Calc I and II in school and still did okay all thanks to this professor.
- albrewer 6y agoWhen he teaches in person, he literally reads these notes and writes them on the board. He revises his notes based on questions he gets in class when something doesn't click for students. He's been teaching for over 20 years now so they're very polished at this point.
- virtuous_signal 6y agoIt's amazing to think how much welfare a single person's free work can add to the world. Basically FOSS but for learning math. I used this site ~12 years ago when taking calculus courses, and more recently I put it as a recommended resource on my syllabi when teaching those same courses. Apparently many others benefitted as well. A gift for exposition + selflessness + a permanent domain name (I'm guessing, due to his job at a university) = profit for everyone.
- ntsplnkv2 6y agoHonestly it's like a rite of passage to use Paul's notes to get through calculus in college :). Many thanks because without it I would've struggled much more!
- aardvarkr 6y agoI had the honor of being a student of his. I was a high school student while attending college there thanks to a Texas program for gifted kids. As a professor he’s quite the character with his beard and his rat tail and his funny mannerisms. If you think his online notes are good, well, his lectures were an order of magnitude better those of than any other calculus class I’ve been part of. I wish he would make them available online.
- PTOB 6y agoI used to work for TAMS back in the day. Hope all is well for you!
- xelxebar 6y agoHello from a fellow TAMSter!
- aCamelCaseFan 6y agoHello from a current TAMSter, ‘21! I indeed made heavy use of these notes for Cal II last semester
- Hasz 6y agoFor math outside of calculus, I highly recommend Wolfram MathWorld. Good definitions, reasonably concise, and the primary sources are listed. https://mathworld.wolfram.com/ https://mathworld.wolfram.com/
- paulpauper 6y agothat is way too broad. mathworld does not give many examples
- mhh__ 6y agoMathworld is good to search for relatively wacky stuff (As is Wikipedia - you can sometimes find niche results squirrelled away in the depths of a big article), but as a resource for learning it's not great. It is true that outside of Calculus (through to real and complex analysis) and Linear Algebra the density of good teaching material online decreases
- gongo1 6y agoIncredible source of material for Calc 1-3
- KingEllis 6y agoSpeaking as someone ashamed at having never taken Calculus, I didn't know there were 3 of them...
- albrewer 6y agoFirst you learn basic calculus. Then you learn advanced calculus. Then you learn how to apply what you've learned to more than one dimension. The second one is actually the hardest of the three.
- gpanders 6y agoIn my experience: Calculus 1 = differential calculus Calculus 2 = integral calculus and infinite series Calculus 3 = multivariate calculus
- gibspaulding 6y agoCalculus 1 = regular calculus Calculus 2 = backward calculus Calculus 3 = n dimensional calculus Differential Equations = inside out calculus Complex Analysis = imaginary calculus
- AlanYx 6y agoDon't forget: Real Analysis = generalizable calculus That was my favorite course.
- gibspaulding 6y agoAhh yes, I enjoyed that one as well, and had a great professor for it! I remember going back over the derivation and integration proofs while taking Real II and feeling like that was when they finally clicked. I was working as a math tutor at the time and Real Analysis was actually a lot of help when trying to explain concepts in Calc I and II.
- Schoolmeister 6y agoGreat stuff. Well witten notes with clear and extensive examples. I actually used this just a few months ago to pass my calculus exam. Feel like I learnt a lot more than from my overly dense and formal textbook.
- erdaniels 6y agoThis got me right through Calculus II :)
- albrewer 6y agoDr. Dawkins was my calc. 2, 3, and differential equations professor when I went to school for engineering. He's an absolute treasure as far as teaching goes, and I don't think I'd be as successful and skillful at math had I not had him as my instructor. He's insanely energetic and positive in the classroom, and he's basically perfected the art of getting "math" into other people's heads.
- mechanicalDigit 6y agoNever thought I would see another Lamar grad here.
- PTOB 6y agoSame here. How many of us are there?
- bachmeier 6y agoI've never been to Lamar, so maybe you know something I don't, but there's no sense in which Lamar is a small school. It's probably several times the average US university.
- mechanicalDigit 6y agoMy computer science cohort was very small. That’s why I don’t expect to see many Lamar folks on these types of forums.
- PTOB 6y agoIt was small from about 1986 till the early 2000's. During that time they had the only undefeated football team in the Southland conference. Population shot up over the last two decades when they started losing games.
- cs2818 6y agoThere are dozens of us! I took Cal 2 and 3 from him.
- sartaj 6y agoGood source
- nsriv 6y agoPiling on to the praise here, but I cannot explain how much this site has helped me go from a C+ high school math student failing my first year of college calculus, to majoring in statistics, and now being a math teacher myself. Truly one of the best resources on the internet.
- programmertote 6y agoI wish I could donate something to keep this site up and running. Really admire people who spent their time creating a useful resource like this and sharing it for FREE with the public for common good. I hope I will have free time one day (when I retire) to do something like that for the subject that I'm interested in.
- mlamat 6y agoWhen I studied Calculus 1 Khan Academy didn't exist yet and I had no one to turn to for tutoring. I fell into a depression and wasted at least a year of my life.
- paulpauper 6y agoThis is a great resource. The notes should be printed up and used for textbooks instead of those huge, overpriced books
- driscoll42 6y agoSeconding many here, I used these 10+ years ago in my Calc, Diff Eq, and Linear Algebra classes. Far superior to many of the texts I had to work with. So glad people are still getting value out of it. I remember checking a couple years ago expecting it to be taken down for some reason and happy to be wrong.
- disabled 6y agoThis is effectively the best reference notes on the internet for calculus 1, calculus 2, calculus 3, and differential equations. You can definitely become proficient in these subjects on those notes alone, with practice. Paul’s Online Math Notes were my main reference material back in 2008-2009 in university. I am glad to see that it is still very popular. The absolute best Calculus book for undergraduates majored in the formal engineering disciplines is Calculus, 6th edition, by Swokowski, Olinick, and Pence. ISBN-10: 0534936245 ISBN-13: 978-0534936242 My (fraternal) twin brother used this book to teach himself calculus 2 in preparation for another course he needed to take. There are lots of supplementary materials that you can get for the book, like solutions manuals, study guides, and even a linear algebra supplement (although the latter is hard to find—I should probably scan all of the materials)
- colmvp 6y agoThe thing I appreciated the most when I was self-learning math was the answers where he explained the steps cogently, which was incredibly helpful in seeing where I went off the rails or got stuck.
- Drblessing 6y agoThis website helped me so much!
- bigasscoffee 6y agobeen around for awhile, I used these for Calc 2 in 2008, and linear algebra and doffeq in 2010/11!
- spike021 6y agoIn college I used a mix of these and PatrickJMT's videos on Youtube. The awesome thing about Patrick's YT channel is that he walks through a plethora of examples for all kinds of concepts. So if you're more of a hands-on learner who doesn't absorb as much when your instructor just lectures without showing/demonstrating a concept, then the channel is a great supplementary tool. I wouldn't have made it to my degree in CS without his content.
- robviren 6y ago10/10, would have not made it through diff EQ without this site. Had to self teach because of constant professor changes. So lucky I found this site in 2012.
- therein 6y agoWhile I realize I am just repeating the sentiment that others have already shared, I would like to do so in case Paul sees this and realizes how many people he has helped. Best content online about Calc I, II, III and DiffEq. Hands down.
- uptheroots 6y agoThis is an amazing resource! Used it in college a lot
- siraben 6y agoNever thought I'd see Paul's Notes on the front of HN. My classmates and I used it so often when we learned calculus in high school, and it still is useful at the university level for multivariable, vector calculus and differential equations. Huge kudos to Paul.
- lordnacho 6y agoGreat notes. Now my question is whether there's some sort of standard for what's in Calc 1, 2, 3... I've seen a number of people on the internet refer to them, but it doesn't seem clear to me what the numbers mean. Wouldn't this be different depending on where you were? And yet people talk casually about the numbers instead of the topics (first order diff eqs). I'm European, did my degree in the UK, and never came across any standardized numbering.
- throwaway2245 6y agoI have tutored people taking Calc 1, 2 and 3 at several universities in the US. There's no exact standard that I'm aware of, but they tend to closely align at different universities. Dividing James Stewart's Calculus into three parts is a reasonable guide: you can see the contents on Amazon preview. I'd expect solving first order linear differential equations would be solidly in Calc 2. * https://www.amazon.co.uk/Calculus-Transcendentals-International-Metric-Version/dp/1305272374/ https://www.amazon.co.uk/Calculus-Transcendentals-Internatio...
- starky 6y agoFor the most part people talk about calc in the following way: Calc 1 is differential calculus (and maybe basic integrals), Calc 2 is integral calculus, and Calc 3 is multivariate calculus (plus potentially vector calc). Differential equations are usually a separate topic split into ODEs and PDEs.
- Scarbutt 6y agoThanks for this! Sequences, Probability and Counting Theory seems like a rare omission.
- mirrorlake 6y agoWhen I graduated college, I e-mailed him a thank you note for all the help his site gave me. It truly is a great resource.
- mttyng 6y agoI adore this site. I’ve taken some time off from Math and have been using this as a refresher for some Linear Algebra stuff I’ve forgotten. What a treasure.
- caseyscottmckay 6y agoPaul's Online Math Notes were an invaluable resource in relearning math as a lawyer trying to start a career as a software developer. Relearning math seemed a lost cause until I discovered Paul's notes. Thanks so much!
- Thorentis 6y agoWow, haven't seen this for years! Used this during my CS and ENG undergrad all the time for math courses. A fantastic resource.
- turnerdhooch 6y agoI enjoyed math in high school and some of college, but I could never understand, and it wasn't explained, how to apply it to anything in my life, so I thought of it as basically just fun puzzles and after finishing the required classes I majored in something else. Decades later, I'd like to start learning it again but still want to know how to apply it for practical purposes. Where can I find this kind of approach?