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Paul's Online Math Notes

297 points by rfreytag 11y ago

rfreytag 11y ago
Taken from this Reddit thread with many other fine mathematics resources: https://www.reddit.com/r/AskReddit/comments/4e6cim/what_is_the_best_free_way_to_catch_up_on_advanced/ https://www.reddit.com/r/AskReddit/comments/4e6cim/what_is_t...
medymed 11y ago
Fantastic resource for learning and as a quick reference. Coincidentally was browsing it before I stumbled upon this post.
sna1l 11y ago
This is how I made it through all of college. This man is a god.
IndianAstronaut 11y ago
Another good online math resource. http://faculty.atu.edu/mfinan/nnotes.html http://faculty.atu.edu/mfinan/nnotes.html
JamilD 11y ago
After three years so far of undergraduate engineering, very few days go by where I don't have this site open in at least one tab!
- Zombieball 11y ago
  Paul's notes got me through many of my engineering courses as well (+4 years ago).
jammett 11y ago
Good to see this site still alive, I used it back in 2006 and I got distinctions thanks to the site.
deleted 11y ago
[deleted]
avianbc 11y ago
Thanks to these and patrickJMT on youtube, I was able to ace all my college math courses.
nimos 11y ago
Maybe a sign I should get back to studying calc..... Another resource is https://www.youtube.com/user/patrickJMT/ https://www.youtube.com/user/patrickJMT/ Khan Academy is good if you don't know anything but I like PatrickJMT's videos better if you're reviewing.
markwaldron 11y ago
Paul's Math Notes got me through all of my undergrad math courses. My professor was fantastic, but having these to reference when the professor wasn't around was a much-needed asset to learning and finishing homework. I recommend this site to everyone I know taking Calculus or higher courses.
tarheeljason 11y ago
Really nice, succinct notes here as well: https://www.math.hmc.edu/calculus/tutorials/ https://www.math.hmc.edu/calculus/tutorials/ (includes some linear algebra, despite the URL)
- kaitai 11y ago
  Yep. One benefit of the HMC notes over Paul's is that the HMC notes have more visualizations. I thought the linear algebra portion of the HMC notes (change of basis, eigenstuff) was especially useful for learners as a counterpoint to most textbook presentations.
arvinsim 11y ago
I wish I had these resources a decade ago when I was still in college. It would have helped immensely.
- Zombieball 11y ago
  This site may have been around! I didn't hit up the way back machine, but I graduated +4 years ago and used this site throughout my undergrad degree.
- nadinno78 11y ago
  Me too. Was browsing this one site recently http://betterexplained.com/ http://betterexplained.com/
- canistr 11y ago
  This was probably around back then since that's when I roughly started my undergrad and discovered the site. It definitely got me through engineering.
ageofwant 11y ago
Very good. Every so often I get despondent about the state of the world; then I'm reminded that somewhere some bright Indian girl is figuring out how to build a fusion reactor from pigshit and string with knowledge she gained from sites like this. One can hope.
krob 11y ago
I didn't take Eigen Values & Vectors until I took Linear Algebra. In his notes he went over Eigen Values in Diff Eq. Not Vectors, but rather functions. Also I never learned Fourier Series in Diff Eq. We did go over Laplace transformations. Where ever he took this material, they were insanely thorough.
code5fun 11y ago
This is amazing! Thanks for the effort!
sillyquiet 11y ago
I had Prof. Dawkins for Calculus a good decade and a half ago. He was a fantastic teacher, and the reason I finally 'got' a lot of the concepts I had struggled with up to then.
Pingviini 11y ago
I highly recommend Professor Leonard's videos to anyone struggling with Calculus. https://www.youtube.com/user/professorleonard57 https://www.youtube.com/user/professorleonard57 I must also confess that his ability in helping me understand and fall in love with Calculus two years ago was the main impetus for me to select Mathematics as my major. I'm now focusing my attention on Number Theory with high hopes of one day becoming a Theoretical Mathematician. I should also add that Princeton Companion book to Mathematics is a valuable resource for learning what is out there in Pure Mathematics.
- wrong_variable 11y ago
  > 'Theoretical Mathematician' what does that even mean ?
  - LanceH 11y ago
    In math, this means that we've established that he exists, but we haven't found him yet. Last I checked, they established a lower bound on his street address of 6325.
  - todd8 11y ago
    My first degree was in "Theoretical Math". I took classes in subjects like Number Theory, Topology, Analysis, Non-commutative Ring Theory. The subjects were about Math not how to do Math to solve other problems. We studied proofs not applying math to solve word problems. As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math. An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.
tekproxy 11y ago
I had this guy as my math teacher. He was an absolute machine. Always came in at exactly the same time, put his bag down, took his watch off, and started transmitting knowledge. Loved his classes. All homework problems were really realistic with messy answers and the test questions were easy. Only a few problems for both. Once, when talking about higher dimensional math, a student asked if string theory said there are 13 dimensions. He quickly replied "yeah" and I'm sure he knew it was more complicated than that but it was an irrelevant question. Without thinking I blurted out it was consistent with up to 13 dimensions and the whole class turned around to look at this kid who corrected The Machine. He quickly admitted that was true and moved on. Highlight of my education.
vzip 11y ago
He used to have Linear Algebra material as well. I wonder why he took it down.
Nickersf 11y ago
This site mixed with Patrick jmt's videos tends to help me more than lectures.
zxcvvcxz 11y ago
Great notes, I remember using these years ago for calculus and differential equations. Very straight-forward and example-based. Remember for most of us math is about solving problems!
amjaeger 11y ago
This site is how I learned calculus & the only reason I never failed a math class. But I wish he had some notes for linear algebra
- Undertow_ 11y ago
  MIT Courseware must have some linear algebra notes.
- lmitchell 11y ago
  Basically what I came in here to say. I swear half of my uni calculus class would have failed without Paul's notes :D
- kaitai 11y ago
  He did have linear algebra notes at one time but took them down, I believe. Agreed -- I'd love to have those!
Undertow_ 11y ago
I've got an AP test coming up. This should come in handy.
Someone 11y ago
Those cheat sheets baffle me. I cannot imagine that someone who needs help remembering, to pick a few examples, that a^m a^n = a^(m+n) or that y = mx + b is a line will be able to use those facts in a real-world problem, with the cheat sheet in hand. For those who used these and found them useful: did you really use them as cheat sheets, that is, to look up things while working on a problem, or did you use them more as a checklist before entering an exam, to check that you remembered most of them?
- kaitai 11y ago
  I suggest big parts of Paul's Online Notes to students who need some brush-up or remediation. When I was teaching college precalc, for instance, people usually had a reasonable grasp of y=mx+b but difficulties with exponentiation were almost universal. Some of these "cheat sheets" are useful for such students just to tape above their workspace so that while working through the rote mechanical practice problems they must do they can use the "cheat sheet" as a checklist while they work. The review of complex numbers, on the other hand (not a cheat sheet but a condensed review) I assign to some masters' students who have not used complex numbers for 2 years and need to recall what they once learned. There it's just a concise but reasonably comprehensive refresher list.
  - Someone 11y ago
    That makes waaay more sense to me; the first uses them more as training wheels than as cheat sheet; the second is more akin to the "check that I know it" that I envisioned. Thanks.
marpstar 11y ago
After passing Calculus I with a C in college about 8-9 years ago, I found this site and started working with his examples. I ended up passing Calculus II (regarded at the "hardest" math class at my college) with a B+. Highly recommended!
joeyrobert 11y ago
This got me through undergraduate calculus I, II, III and differential equations. Can't recommend it enough, it's a free text book. One of those hidden gems you can occasionally find on professor's websites.
bthornbury 11y ago
These are the best online math notes around. I used them as a textbook replacement for many math classes in college (from Diff EQ to Triple Integrals) and never had a time where it didn't cover something I needed. Really great resource here.
DarkContinent 11y ago
These saved my Honors Calc III grade this semester :)
curiousDog 11y ago
He has great notes. I have to thank him sometime for helping me test out of Linear Algebra and help skip a class.
bradlys 11y ago
Yep, love this guy. His notes helped me through my calculus finals in high school eight years ago and then through college as well. If you notice an error be sure to correct it though. I did once while studying for my high school finals while staying up to 3AM. He responded quickly and fixed it immediately. Great to see someone who pays attention. Just checked my email account. Apparently, I've emailed him corrections three times over the period of 4 years. He always emailed back within 24 hours.

297 points by rfreytag 11y ago

Taken from this Reddit thread with many other fine mathematics resources: https://www.reddit.com/r/AskReddit/comments/4e6cim/what_is_the_best_free_way_to_catch_up_on_advanced/ https://www.reddit.com/r/AskReddit/comments/4e6cim/what_is_t...

medymed 11y ago

Fantastic resource for learning and as a quick reference. Coincidentally was browsing it before I stumbled upon this post.

sna1l 11y ago

This is how I made it through all of college. This man is a god.

IndianAstronaut 11y ago

Another good online math resource. http://faculty.atu.edu/mfinan/nnotes.html http://faculty.atu.edu/mfinan/nnotes.html

JamilD 11y ago

After three years so far of undergraduate engineering, very few days go by where I don't have this site open in at least one tab!

Zombieball 11y ago
Paul's notes got me through many of my engineering courses as well (+4 years ago).

jammett 11y ago

Good to see this site still alive, I used it back in 2006 and I got distinctions thanks to the site.

deleted 11y ago

[deleted]

avianbc 11y ago

Thanks to these and patrickJMT on youtube, I was able to ace all my college math courses.

nimos 11y ago

Maybe a sign I should get back to studying calc..... Another resource is https://www.youtube.com/user/patrickJMT/ https://www.youtube.com/user/patrickJMT/ Khan Academy is good if you don't know anything but I like PatrickJMT's videos better if you're reviewing.

markwaldron 11y ago

Paul's Math Notes got me through all of my undergrad math courses. My professor was fantastic, but having these to reference when the professor wasn't around was a much-needed asset to learning and finishing homework. I recommend this site to everyone I know taking Calculus or higher courses.

tarheeljason 11y ago

Really nice, succinct notes here as well: https://www.math.hmc.edu/calculus/tutorials/ https://www.math.hmc.edu/calculus/tutorials/ (includes some linear algebra, despite the URL)

kaitai 11y ago
Yep. One benefit of the HMC notes over Paul's is that the HMC notes have more visualizations. I thought the linear algebra portion of the HMC notes (change of basis, eigenstuff) was especially useful for learners as a counterpoint to most textbook presentations.

arvinsim 11y ago

I wish I had these resources a decade ago when I was still in college. It would have helped immensely.

Zombieball 11y ago
This site may have been around! I didn't hit up the way back machine, but I graduated +4 years ago and used this site throughout my undergrad degree.
nadinno78 11y ago
Me too. Was browsing this one site recently http://betterexplained.com/ http://betterexplained.com/
canistr 11y ago
This was probably around back then since that's when I roughly started my undergrad and discovered the site. It definitely got me through engineering.

ageofwant 11y ago

Very good. Every so often I get despondent about the state of the world; then I'm reminded that somewhere some bright Indian girl is figuring out how to build a fusion reactor from pigshit and string with knowledge she gained from sites like this. One can hope.

krob 11y ago

I didn't take Eigen Values & Vectors until I took Linear Algebra. In his notes he went over Eigen Values in Diff Eq. Not Vectors, but rather functions. Also I never learned Fourier Series in Diff Eq. We did go over Laplace transformations. Where ever he took this material, they were insanely thorough.

code5fun 11y ago

This is amazing! Thanks for the effort!

sillyquiet 11y ago

I had Prof. Dawkins for Calculus a good decade and a half ago. He was a fantastic teacher, and the reason I finally 'got' a lot of the concepts I had struggled with up to then.

Pingviini 11y ago

I highly recommend Professor Leonard's videos to anyone struggling with Calculus. https://www.youtube.com/user/professorleonard57 https://www.youtube.com/user/professorleonard57 I must also confess that his ability in helping me understand and fall in love with Calculus two years ago was the main impetus for me to select Mathematics as my major. I'm now focusing my attention on Number Theory with high hopes of one day becoming a Theoretical Mathematician. I should also add that Princeton Companion book to Mathematics is a valuable resource for learning what is out there in Pure Mathematics.

wrong_variable 11y ago
> 'Theoretical Mathematician' what does that even mean ?
- LanceH 11y ago
  In math, this means that we've established that he exists, but we haven't found him yet. Last I checked, they established a lower bound on his street address of 6325.
- todd8 11y ago
  My first degree was in "Theoretical Math". I took classes in subjects like Number Theory, Topology, Analysis, Non-commutative Ring Theory. The subjects were about Math not how to do Math to solve other problems. We studied proofs not applying math to solve word problems. As an example, I wasn't very good at solving differential equations, a very important part of math used to solve many real-life problems. Instead I studied things like the Lebesgue Integration which extends the notion of integration to a larger set of functions for which the more familiar Riemann integral wouldn't be defined. That was taught in my second semester real analysis course that had these prerequisites: Real Analysis I, Complex Analysis I, Differential Equations, Calc I and Calc II. A lot of work to get to an interesting subject, but a subject of interest to Mathematicians not engineers using math. An undergraduate degree in Applied math, in contrast to theoretical math, would probably have involved learning more about, say, differential equations.

tekproxy 11y ago

I had this guy as my math teacher. He was an absolute machine. Always came in at exactly the same time, put his bag down, took his watch off, and started transmitting knowledge. Loved his classes. All homework problems were really realistic with messy answers and the test questions were easy. Only a few problems for both. Once, when talking about higher dimensional math, a student asked if string theory said there are 13 dimensions. He quickly replied "yeah" and I'm sure he knew it was more complicated than that but it was an irrelevant question. Without thinking I blurted out it was consistent with up to 13 dimensions and the whole class turned around to look at this kid who corrected The Machine. He quickly admitted that was true and moved on. Highlight of my education.

vzip 11y ago

He used to have Linear Algebra material as well. I wonder why he took it down.

Nickersf 11y ago

This site mixed with Patrick jmt's videos tends to help me more than lectures.

zxcvvcxz 11y ago

Great notes, I remember using these years ago for calculus and differential equations. Very straight-forward and example-based. Remember for most of us math is about solving problems!

amjaeger 11y ago

This site is how I learned calculus & the only reason I never failed a math class. But I wish he had some notes for linear algebra

Undertow_ 11y ago
MIT Courseware must have some linear algebra notes.
lmitchell 11y ago
Basically what I came in here to say. I swear half of my uni calculus class would have failed without Paul's notes :D
kaitai 11y ago
He did have linear algebra notes at one time but took them down, I believe. Agreed -- I'd love to have those!

Undertow_ 11y ago

I've got an AP test coming up. This should come in handy.

Someone 11y ago

Those cheat sheets baffle me. I cannot imagine that someone who needs help remembering, to pick a few examples, that a^m a^n = a^(m+n) or that y = mx + b is a line will be able to use those facts in a real-world problem, with the cheat sheet in hand. For those who used these and found them useful: did you really use them as cheat sheets, that is, to look up things while working on a problem, or did you use them more as a checklist before entering an exam, to check that you remembered most of them?

kaitai 11y ago
I suggest big parts of Paul's Online Notes to students who need some brush-up or remediation. When I was teaching college precalc, for instance, people usually had a reasonable grasp of y=mx+b but difficulties with exponentiation were almost universal. Some of these "cheat sheets" are useful for such students just to tape above their workspace so that while working through the rote mechanical practice problems they must do they can use the "cheat sheet" as a checklist while they work. The review of complex numbers, on the other hand (not a cheat sheet but a condensed review) I assign to some masters' students who have not used complex numbers for 2 years and need to recall what they once learned. There it's just a concise but reasonably comprehensive refresher list.
- Someone 11y ago
  That makes waaay more sense to me; the first uses them more as training wheels than as cheat sheet; the second is more akin to the "check that I know it" that I envisioned. Thanks.

marpstar 11y ago

After passing Calculus I with a C in college about 8-9 years ago, I found this site and started working with his examples. I ended up passing Calculus II (regarded at the "hardest" math class at my college) with a B+. Highly recommended!

joeyrobert 11y ago

This got me through undergraduate calculus I, II, III and differential equations. Can't recommend it enough, it's a free text book. One of those hidden gems you can occasionally find on professor's websites.

bthornbury 11y ago

These are the best online math notes around. I used them as a textbook replacement for many math classes in college (from Diff EQ to Triple Integrals) and never had a time where it didn't cover something I needed. Really great resource here.

DarkContinent 11y ago

These saved my Honors Calc III grade this semester :)

curiousDog 11y ago

He has great notes. I have to thank him sometime for helping me test out of Linear Algebra and help skip a class.

bradlys 11y ago

Yep, love this guy. His notes helped me through my calculus finals in high school eight years ago and then through college as well. If you notice an error be sure to correct it though. I did once while studying for my high school finals while staying up to 3AM. He responded quickly and fixed it immediately. Great to see someone who pays attention. Just checked my email account. Apparently, I've emailed him corrections three times over the period of 4 years. He always emailed back within 24 hours.