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Big Picture of Calculus (2010) [video] (youtube.com)
393 points by espeed 37 days ago | hide | past | web | 58 comments | favorite

Now I know someone is browsing my Youtube history and posting to HN. Youtube really is the math teacher I never had.

Probability Models and Axioms https://www.youtube.com/watch?v=j9WZyLZCBzs

The Exponential Function https://www.youtube.com/watch?v=oo1ZZlvT2LQ&t=100s

Vector Space https://www.youtube.com/watch?v=ozwodzD5bJM&t=36s

Tree cutting fails: https://www.youtube.com/watch?v=JHZkR6UVegY

Youtube really is the math teacher I never had.

Same here. I'm on a quest to run myself through the equivalent of a standard Calc I, Calc II, Calc III, Linear Algebra sequence, and I've managed to find great Youtube resources for all of those things (and more). It's times like these that you marvel at how awesome the Internet can be at its best. :-)

Have you seen the Essence of Linear Algebra video series posted to HN a few days ago? Seeing the geometric transformations animated gives you intuition hard to develop otherwise:

Essence of Linear Algebra (visualized) https://www.youtube.com/watch?v=kjBOesZCoqc&list=PLZHQObOWTQ...

Yep. Watched the first 3 or 4 of those right after they were posted. Good stuff. I'm not really focusing on Linear Algebra yet, but definitely looking forward to digging into those and the Gilbert Strang ones on LA.

The course that follows the MIT calculus course is "Differential Equations" (which is like applied calculus):

MIT Differential Equations: https://www.youtube.com/watch?v=ZvL88xqYSak&list=PLUl4u3cNGP...

Don't overlook mathematical analysis -- real analysis, complex analysis, etc -- that's another big leg of the mathematical stool, with geometrical foundations underpinning it all.



As you self-study and progress into upper level math, you'll come across abstract and unfamiliar topics you didn't learn about in calculus/algebra -- and sometimes it's hard to pin down what category the topic falls under -- when that happens, the topic will often be related to analysis, group theory, or topology (courses taken by math majors but not as widely known).

See http://nada.kth.se/~axelhu/mapthematics.pdf and http://space.mit.edu/home/tegmark/toe.gif

Crosslinks for all MIT courses: http://crosslinks.mit.edu/topics/?query=subject18.100

For anyone who wants to learn linear algebra, I highly recommend Gilbert Strang's book and course from MIT: https://www.youtube.com/watch?v=ZK3O402wf1c.

Hey I'd love to hear about how your learning experience is going! Care to share your experiences?

I'm about to start a self-directed learning exercise focused on CS related math. I've never used online resources to learn / brush up on complex topics before so I'm a bit unsure what to expect.

Sure, I'm at work right now, but I could write up some stuff about my experience later.

So here's how it's gone for me so far (including some background about my non-online maths education for context).

I only took through Algebra II in high-school, but honestly, I went to Algebra II class maybe 4 or 5 times all year, never did any of the homework, and took maybe 1 test. Needless to say I failed it. That was the year I was too busy being "mr rebel badass guy" to worry about studying, plus I didn't need the class to graduate, so I just didn't give a flip. OTOH, I did really well in H.S. Geometry the year before.

Anyway, when I started college I needed some maths refresher, so I took 2 semesters of College Algebra, and 2 semesters of Pre-Calculus. I also had a 1 semester Discrete Math class. I started Calc I but I dropped out about halfway through that semester.

Fast forward about 6 or 7 years, and I decided to go back to school. Having forgotten all the maths stuff I had learned before, I took College Algebra yet again, as well as Discrete Math (I don't remember now why I took Discrete again. It might be because my earlier credit was too old to transfer).

Fast forward another 7 years or so, and I've again forgotten what maths I'd learned before. But by now Khan Academy exists and that's where the online stuff kicks in.

So I've fiddled around on KA a bit over the past few years, just trying to keep up some of the really basic algebra stuff that you forget if you don't use (rationalizing denominators, factoring, etc., etc.) But that was always kind of hit or miss and piecemeal, not really a focused thing.

Fast forward just a hair more and Coursera and the like have come into being. I started taking the Johns Hopkins sequence on Data Science, and I also took the Andrew Ng class on Machine Learning. That whole experience reminded me "I've always meant to get serious about this maths thing".

Notably I'd never taken a proper course on statistics and probability, so I wanted to fill that gap. So on Coursera I took the first 2 or 3 classes in that Duke "Statistics With R" sequence (I plan to finish the whole thing eventually) to pick up some basic stats knowledge. It also dovetailed well with the first few classes in the JHU Data Science thing, as both use R. And looking back on that, from where I am now, I highly recommend both the Duke sequence (which is more about the stats and less about R) and the JHU sequence (which is arguably more about R than stats). I think they complement each other well, and if you do both you'll learn a decent amount about basic stats.

This is also where I discovered Professor Leonard on Youtube. I was searching for some stats videos to complement the other stuff I was doing, and found his videos. I never did go through the entire Stats class he put up, but I was impressed enough with his teaching style to bookmark his channel for future reference.

Anyway, at some point I hit a lull in things and decided "now is as good a time as any to go back through Calc I with an eye towards working through all of Calc I, Calc II, Calc III and Linear Algebra". I started the MOOCulus course on Coursera, and while I like certain things about it, I feel like the content is too "choppy" since it's broken up into such short segments. That was when I went back to the Professor Leonard channel and started going through his Calc I class. And I'm just now about up to where I was at when I dropped out of school the first time. Only 20 years later...

Anyway, there was other stuff mixed in there too, including offline stuff with actual paper books. But as far as the online stuff goes, I also found Youtube channel called MathBFF that has some really good videos (at least for Calculus topics), and I also watched a chunk of the "Big Picture of Calculus" videos linked to above.

So from here out, my plan is to continue through the Professor Leonard series of Calc I, II and III (and supplementing the videos with problems from dead-tree books) and then move on to Linear Algebra, probably using a combination of Gilbert Strang's videos and the "Essence of Linear Algebra" ones that were posted on HN a couple of days ago. I might also consult the Khan Academy videos on some of this stuff.

And to go back to dead-tree books for a minute... I have this weird obsession with mathematics books. I kindof collect them. I guess because in the back of my head, all these years, I've had that "I'll get serious about maths one day" thing going on. So I'm forever buying maths texts at used bookstores, or on Amazon, or random places. I have shelves and shelves of books on all sorts of mathematics stuff to consult. So my self-learning definitely involves a heavy dose of both online resources and meatspace stuff.

If I had to summarize, I guess I'd just re-iterate that the resources you need to learn a LOT of maths is available online, mostly for free. I mean, there's a TON of stuff out there, including a lot I didn't mention above. And even though I'm mainly interested just in the stuff I need to for AI/ML, I did some poking around on Youtube out of curiosity and found that you can find videos on just about every math topic there is: abstract algebra, complex analysis, topology, group theory, measure theory, etc., etc. It really is an amazing time we live in. :-)

I'm learning linear algebra in tandem with Andrew Ng's - machine learning course on coursera. Challenging would be an apt description!

Check out Linear Algebra: Foundations to Frontiers (https://www.edx.org/course/linear-algebra-foundations-fronti...), it's being offered self paced by edx, and you can download the lecture notes for free at http://www.ulaff.net. When I took the course two years ago I liked how the lecturer will discuss basic concepts in the lectures but also gives additional material related to state of the art research being done in the field.

There also used to be a course called Coding the Matrix, I'm not sure if it is still being offered online. The lecture notes form a book of the same title, which is available for less than 10$ (Kindle).

Yeah, part of my motivation for doing a lot of maths study is exactly that I took that Andrew Ng class. You can get through the class without knowing multi-variable calculus and linear algebra and what-not, but that class made it clear to me that learning that stuff would be hugely beneficial.

Do linear algebra between Calc 2 and 3. It'll be a little easier that way, and you'll probably get more out of it.

I take more pride/pleasure by understanding through reading. Somehow I don't trust my brain not to go into a visual form of rote learning by replicating what the video said.

Books are a bit more painful, but forces my brain to actually make sense of abstractions actively instead of passively (which is not always the case on videos, but a lot more probable).

Also, libraries are a good place to reflect.

That last vid made my day, expecting nodes and splitting algorhytms. You clever you! :)

For those of you who don't know, felling trees is incredibly dangerous. They store immense energy in a not terribly structurally safe form. Don't do it if you don't know what you are doing.

My dad always insists on cutting them himself and watching it always gave me a heart attack. That vid is basically what I play through in my head the whole time. So far nothing happened but man.

I remembering learning that the most powerful force acting on a building or tree was inertia. Watching videos like this makes that much more clear.

If you want to get into the details of Calculus

Professor Leonard https://www.youtube.com/user/professorleonard57

Herbert Gross MIT Calculus Revisited: Multivariable Calculus https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B

Herbert Gross MIT Calculus Revisited: Calculus of Complex Variables https://www.youtube.com/playlist?list=PLD971E94905A70448

Herbert Gross MIT Calculus Revisited: Single Variable Calculus https://www.youtube.com/playlist?list=PL3B08AE665AB9002A

MIT 18.02 Multivariable Calculus, Fall 2007 https://www.youtube.com/playlist?list=PL4C4C8A7D06566F38

MIT 18.02SC: Homework Help for Multivariable Calculus https://www.youtube.com/playlist?list=PLF07555F3CC669D01

MIT 18.01 Single Variable Calculus, Fall 2006 https://www.youtube.com/playlist?list=PL590CCC2BC5AF3BC1

MIT 18.01SC: Homework Help for Single Variable Calculus https://www.youtube.com/playlist?list=PL21BCE50ABFF029F1

Second the recommendation for Professor Leonard. He does a really nice job explaining things and making it accessible.

Herbert Gross MIT Calculus Revisited - is one of best lectures I have seen, and it is recorded way back in 1973.

I would also add these great lectures by Theodore Shifrin:

University of Georgia - Math 3500/10: https://www.youtube.com/playlist?list=PL5I-Eyk8l9FHdJUd9UujG...

Professor Leonard is really great if you need some extra help learning something. He moves slowly enough with the material that it's difficult to get left behind, however that might be too slow for most HNers.

I was a teaching assistant at UC Santa Barbara for 3 years, I can tell you, students might slug through the calculations , but none -- or a precious few -- really get the big picture.

Mostly, it is that calculus pedagogy is a disaster. And it hasn't been updated to include recent advances in technology (such as Data Science)

Strang is an amazing lecturer. I can tell you, that far into your Masters and PhD these same basic issues resurface. There's at least one project I can think of where the entire problem hinges on taking one derivative.

Gilbert Starng also has an excellent series on Linear Algebra on MIT opencourseware [https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...]

Honestly I couldn't slog through the video. It started off with "let's use letters" and immediately got to things like "the right letters for this case are 'df' and 'dt'". This just feels like "here's a bunch of magic that you'll just have to memorize". Reminds me of my favorite quote:

> And then Satan said "put the alphabet in math".

When I learned the basics of calculus it was via simple problems that we wanted to answer. And for a while we estimated the answers by brute forcing approximations. I still drop back to this when I can't remember how to do something and need to re-discover parts of calculus.

Thanks for sharing such an amazing thing. I am done with studies but should be helpful to understand it properly to teach my kids in future so that they just fill up pages with dy/dx without getting an idea behind it.

For kids I'm a fan of http://dragonbox.com/ and their http://www.dragonboxapp.com/ They do a pretty good job of teaching symbolic algebra to kids who don't yet know arithmetic.


I find that most people understand the general concepts of basic calculus. What's hard to wrap around is when you start getting into proofs (ie. delta-epsilon, trig, e,...and that's just some general ones).

I suppose that's how they differentiate the good students from the just okay students. Personally, I don't think youtube videos will help that much when it goes down to the nitty gritty. You really have to sit down, think, and understand.

Keisler has a introductory text book (available free here: https://www.math.wisc.edu/~keisler/calc.html) that uses non-standard analysis, which he claims is easier for beginners to understand than the notorious delta-epsilon proofs.

I can't say how effective it is at that goal, but it is all mathematically legitimate, and has a nice intuitive appeal to it.

Looks similar to this book, which I bought after a recommendation here: https://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Ma...

Having had a non-traditional route through learning various math topics, I'd like to advocate the teaching of calculus (the understanding of the accumulation of infinitesimally-small changes) before the teaching of trigonometry (the relationships between different geometric shapes, and the resulting association with waves).

In my (humbly biased) opinion, it's easier to learn how trigonometric concepts operate with a calculus background, than with merely an algebra-and-geometry background. Trigonometry complicates calculus education - knowing trigonometry may facilitate better calculus understanding to some, but not knowing trigonometry doesn't necessarily complicate learning calculus concepts. The trigonometry-first tradition in math education reflects a "number-phile" bias, not the most optimal route for humans to ingest and retain concepts.

Edit: The "number-phile" bias is clear - we want our teachers to be people with passion on the topic, as we're more likely to get a better understanding of the topic. Understanding sine waves as the result of an integral was much easier for me to understand than as the result of a complicated calculation.

It’s somewhat misleading/unfortunate to use angle measures as the basic characterization of an angle/rotation, when just trying to solve basic geometry / triangle measurement problems. It’s almost always easier to use a vector representation, written in terms of explicit coordinates if necessary (e.g. as a complex number). http://www.shapeoperator.com/2016/12/12/sunset-geometry/

Where angle measures and sines/cosines start to be useful is with uniform circular motion, which as you say is a calculus problem.

The key to understanding the form of trigonometry courses is to understand the context for their creation. Namely, all calculations (e.g. for astronomy, navigation, engineering, mapmaking, ...) used to be done by hand by humans, which was very expensive. It was important to have someone with fluent knowledge of trigonometric identities simplify formulas to a form with as few arithmetic operations and table lookups as possible to save money, or just to match the available function tables. Today in a computer age, extensive memorization of trigonometric identities is an anachronism, and spending lots of time on practicing their manipulation is okay algebra practice but not anything directly useful per se.

> it's easier to learn how trigonometric concepts operate with a calculus background

I can't imagine what this means, could you give an example? Which trigonometric concepts?

I love Gilbert Strang, but _by far_ my favorite book on calculus is "Calculus Made Easy" by Silvanus Thompson. It supplies a very elegant and powerful set of tools to derive many of the "fundamentals" of calculus.

What would be some practical "daily life" applications of calculus? I've been discussing a focus on teaching with applicable examples in a lot of subjects, but this is one where I've been struggling a bit.

In a book review from 1988 [1, p. 892] the mathematician Underwood Dudley claimed:

> First-semester calculus has NO applications.

That said, the central ideas of calculus are firmly grounded in daily life (as Strang points out in the video): Differential calculus is about investigating speed (or, more generally, rate of change), and integral calculus is about investigating area.

[1]: http://www.maa.org/sites/default/files/0002989051112.di99173...

Throwing things is estimating calculus: you pick the velocity to counteract the accelerations the object experiences to get the displacement you desire. Acceleration from air resistance is a function of velocity.

Also, you can measure boobs with integrals and model their jiggle with differential equations.

...I may have been a teenager when I was first learning. But modeling boob motion is apparently serious science and actually gets extremely complex when you try to model the interactions of the different constituent tissues.

"The greatest shortcoming of the human race is our inability to understand the exponential function." - Dr. Albert A. Bartlett

That's the thesis of Dr. Bartlett's famous lecture "Arithmetic, Population and Energy": https://www.youtube.com/watch?v=sI1C9DyIi_8

Calculus is foundational for understanding our rapidly changing, data-driven world. It gives us the ability to see.

You can apply calculus anytime you have a quantity that's changing over time, and you want to see where things are going, how things are changing, and when something is going to happen based on its rate of change -- e.g. stocks, startup growth metrics, how an idea is spreading, mass-market adoption, etc -- anytime you have time-series data and you want to see what it means -- make sense out of it, and understand its implications.

See https://en.wikipedia.org/wiki/Calculus#Applications

And Calculus gives you the ability to calculate how lifelong learning can impact your growth over time -- the significance it can have throughout your life/career -- and realize it's never too late to learn anything ;-)

In the words of Prof John Ousterhout: "A little bit of slope makes up for a lot of y-intercept."


I'm going to expand the definition of "daily life application" from things you actually use daily to things you experience daily. These could be engaging too!

For example... what are sunspots? How does the plumbing around your house work? How does a car battery work? How does a microwave oven work?

Another thought. If we are unable to find applications for a technique, should we place such an emphasis on teaching it? I look around and see all the very practical (and interesting!) math and science that is not being taught, e.g. signal processing, just because they are nontraditional. It's a shame to me.

That's kind've where I was going with it. Basic understanding of the building blocks of society type stuff: power, water, gas, construction, radio, biology, growing food, vehicles, finances, etc.

Thinking back, I just realized the main reason that I was bored to tears in high school wasn't so much that the subjects weren't worth knowing...just that at the time I was being taught formulas without any clear understanding of why it was useful to know it. If you don't know why it's useful to know it, there's very little reason to remember it beyond passing a test.

You might enjoy this. I actually think this is well done.


Obviously, it could use more work, but the kernel of something really good is there.

Been thinking of doing the same.

Additionally, I think game development is a compelling way to teach math and science.

"Calculus" by Tom Apostol has lots and lots of simple physics applications.

I cant think of any obvious applications of calculus to the average non-technical person's daily life. If that's what you mean, you're going to have a bad time, as I think it's a misconception of what math is supposed to be for.

Well, there's plenty of applications that may not be "daily life" but rather "you're likely to encounter this at some point". Take the relation between speed and distance traveled, or the difference between marginal cost and per-unit cost.

Approximately zero people will ever need to answer questions like "if I drive t^2 miles per hour for k minutes, how far did I go?"

Say you want to measure how much energy is hitting the top of our atmosphere in a given day? Triple integral from infinity (the sun) to top of the atmosphere (call it 0).

Examples with classic physics problems are good too. How about F=ma => F=m*dv/dt. Everytime you apply a force to an object calculus magic is there to mathematically describe the process.

I'm only part way though this book, but I think you might find it interesting, vis-a-vis the above question:


The average person is never going to have to use anything but basic arithmetic in 'daily life', but calculus is essentially the mathematics of change, which touches on a huge number of subjects. You can't do any sort of useful physics without it, or machine learning, or statistics, etc, etc.

The average person is never going to have to use anything but basic arithmetic in 'daily life',

The average person may only have to use basic arithmetic in daily life, but I'd argue that pretty much everybody could benefit from using more math (including calculus) in everyday life. I mean, just think of anything in life that can be counted/measured and can be described by a function that can be optimized. Boom, there's an application for calculus - minimization and maximization.

This is why I disagree with the recently popular narrative about how "we don't need to teach all kids calculus" and "math isn't all that important", etc. Me, I wish I'd studied more math when I was younger. I feel like you almost can't teach too much math. If anything, I'd say the problem is a combination of pedagogy and not spending enough time on applications when teaching math.

None, save for maybe dealing w money. But you don't have to learn a ton of calc before you will have the tools to really get that, like, everything is waves, man... and that's pretty rad

When you look at the GPS estimated time of arrival, or check the current MPG or mile left for your car.

Thank you for this video, is there a platform that you could aggregate all this resources mapping them to certain knowledge tree ? So that you could have the big picture and then zoom in for specific topics videos. Something like Khan Academy's knowledge map[1], that you could customise to add you own contents, this video for example.

[1] https://www.khanacademy.org/exercisedashboard

I worked pretty hard on something like this many years ago. I am not a coder so it was a formidable challenge and I never really got it polished or complete. The goal was to create a blank canvas where anyone could organize instructional content from the web (i.e. KhanAcademy, YouTube, etc) into their own class. Each class topic was structured like a subreddit for each topic, with the best results hopefully filtering to the top

I just had that idea this weekend - also like a github branch-diagram of math/physics history

https://arxiv.org/abs/gr-qc/9704009 p2 (fig1) is pretty nice

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