6 ms·

The Beauty of Calculus [video]

320 points by okket 7y ago

lordnacho 7y ago
By dumb luck I scrolled to around 26 minutes, where he gives a summary. Basically, the laws of nature are written in calculus, which is more than a language, and we can use this to manipulate nature.
- trevyn 7y ago
  >Basically, the laws of nature are written in calculus Calculus is a tool that powers certain specific useful predictive models of our observations of nature in certain specific domains. Having a mental model in which the laws of nature are related to calculus in some deeper fashion (or where calculus is always an appropriate tool to describe observations of nature) is likely to lead to some erroneous intuitions.
- obrit 7y ago
  Incredible, that's where the Wadsworth Constant would put you in the lecture as well.
FrankyHollywood 7y ago
Makes me think about Paulos. I read various books of him. He has an interesting view on the world as a mathematician. http://johnallenpaulos.com/booksandreviews.html http://johnallenpaulos.com/booksandreviews.html
ivan_ah 7y ago
Very nice lecture that connects calculus to the laws of nature. I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas. I encourage anyone who liked the talk to try out some of the calculations on their own using https://live.sympy.org/ https://live.sympy.org/ which is an online REPL with build in calculus function like integrate, diff, limit, summation, etc. Here is an example: >>> summation((1/2)**(2*n), [n,0,oo]) 4/3 via https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C%20%5Bn%2C0%2Coo%5D)%0A%23--%0A https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C... note the answer is an exact rational number (class 'sympy.core.numbers.Rational) and not a float approximation 1.33333... For a quick tutorial on how to use SymPy, check "Taming math and physics using SymPy" which is avail in printable format https://minireference.com/static/tutorials/sympy_tutorial.pdf#page=5 https://minireference.com/static/tutorials/sympy_tutorial.pd... or notebook https://nbviewer.jupyter.org/github/minireference/sympytut_notebooks/blob/master/notebooks/Calculus.ipynb https://nbviewer.jupyter.org/github/minireference/sympytut_n...
- mlevental 7y ago
  does sympy render latex in the repl...? also haha oo as an alias for float("inf") or whatever is clever.
- crimsonalucard 7y ago
  >I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas. I don't particularly like this dichotomy. Abstract math is connected to the real world because if it wasn't it wouldn't exist. Overall people who say this stuff are just referring to math that is only derived to work from an experimental standpoint and not from an axiomatic standpoint. Machine learning is one example of a mathematical field with no axiomatic basis and the the pythagorean theorem is an example of one that is derived from the axioms of euclid. Both exist in the real world and are therefore applicable.
  - throwawaymath 7y ago
    I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics has an axiomatic basis. When certain mathematical theories can be "applied" to real world problems, it simply means the relevant axioms and definitions are a robust approximation of reality. To speak to your example directly: machine learning absolutely has an axiomatic basis. You can conduct legitimate research in implementations and software or hardware optimizations thereof; however, fundamentally every experimental result in machine learning is an application of a variety of theorems in linear algebra, probability theory or calculus.
    - lapinot 7y ago
      Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization. On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?
nvusuvu 7y ago
The section about Usian Bolt running the 100m dash was very interesting. Usian needs 41 strides to reach 100m while other runners need 44.
mathnmusic 7y ago
Really nice! BTW, LearnAwesome's calculus topic is relatively barren: https://github.com/learn-awesome/learn-awesome/blob/master/calculus.md https://github.com/learn-awesome/learn-awesome/blob/master/c... I'd appreciate if HNers here can point us to best learning resources for Calculus. Even better if you can send a pull request.
- giu 7y ago
  Just created a pull-request; I added links to some articles and sections hosted on Paul's Online Notes page (http://tutorial.math.lamar.edu http://tutorial.math.lamar.edu). This site helped me quite a lot during my studies, and I'm sure the same holds for a lot of other students and professionals :)
guru4consulting 7y ago
A while ago I saw someone posting a book about Calculus in HN forum that was written decades ago, probably 40s or 50s. I liked it and could not find it back. Does anyone have link to it?
- alfiedotwtf 7y ago
  Maybe Apostol's Calculus, or Spivak's Calculus. Both 1967 though
- throwaway55554 7y ago
  Was it Calculus Made Easy? https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480/ref=sr_1_1_sspa?keywords=calculus+made+easy&qid=1557496980&s=gateway&sr=8-1-spons&psc=1 https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/... It seems easy to find a pdf on the web, but I didn't want to post that.
  - guru4consulting 7y ago
    yes, I was looking for this book. Thank you. I wish I had read this when I was in school. I liked the book for explaining complex topic in a very simple, easily understandable way. This would be a great book to introduce calculus to your kids. I read chapter 2 (on different degrees of smallness) and found it enlightening.
    - throwaway55554 7y ago
      You're welcome. Happy reading!
  - Jtsummers 7y ago
    The original (not the Martin Gardner version) is public domain at this point. Project Gutenberg has a PDF of it. http://www.gutenberg.org/files/33283/33283-pdf.pdf http://www.gutenberg.org/files/33283/33283-pdf.pdf
    - 7y ago
molteanu 7y ago
The beauty of Calculus can only be revealed to young recruits by Herb Gross' videos and Spivak's Calculus. If those don't tickle your brain, nothing will.
- edge17 7y ago
  link? not sure what you're referring to
  - rofo1 7y ago
    https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665AB9002A https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665... There are many, many ways to study calculus, so take this as a preference. For example, this wouldn't be my first choice. In general, I find the "Definition. Axiom. Theorem." approach very dry and just doesn't fit reality. Nobody ever discovered mathematics like this. One of the best books I've read is Gausses "Disquisitiones Arithmeticae" rather than any modern Theory of Numbers book. I've had more insight in sums manipulations (and various little tricks, some of them not even justified in modern mathematics) from Euler's letters than any other book on this subject. I don't know what happened in the meantime. It sure doesn't look like 18/19 century books were so dry like the ones today.
    - learn_awesome 7y ago
      Thanks for sharing both those things. We've added these to our Calculus topic here: https://github.com/learn-awesome/learn-awesome/blob/master/calculus.md https://github.com/learn-awesome/learn-awesome/blob/master/c...
- artiste 7y ago
  I also found 3Blue1Brown's (Grant Sanderson) YouTube series "Essence of Calculus" quite enjoyable.
- less_penguiny 7y ago
  "Spivak's Calculus" is a reference to the book "Calculus" by Michael Spivak. https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/091409... Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.
0x445442 7y ago
Do they not teach why Newton invented calculus anymore? They did when I was taught physics and it all made sense. I don't recall if I considered it beautiful at that time but I do recall how impressed I was with Newton's feat; moving himself farther along with his studies in physics.
- madez 7y ago
  He wanted to calculate and predict the movement of celestial bodies. The mathematical knowledge he had wasn't capable of expressioning the relations, but he saw patterns. Then he worked out the mathematical language to express these patterns.
- tjr 7y ago
  Not sure what's in the curriculum today, but I was taught calculus (20 years ago) heavy on physics applications. I don't think I would have appreciated it as much without that.
- kodz4 7y ago
  For anyone interested, here's the BBC looking into Newton and Leibniz notebooks as they cook it up for the first time - https://www.youtube.com/watch?v=ObPg3ki9GOI https://www.youtube.com/watch?v=ObPg3ki9GOI Also Steven Strogatz is an excellent writer. His other books Sync and Joy of X can be read by anyone.
bigred100 7y ago
Steve strogatz is a pretty excellent speaker even for people who don’t like maths
adamnemecek 7y ago
Tired: Calculus Wired: automatic differentiation
75dvtwin 7y ago
Two statements/quotes that I personally found interesting through prism of current times. "... Algebra is quite sterile, Algebra is not about anything …." (at about 1:17:28 ) "... One thing I feel a little bad about, and I feel I cannot properly correct it but I just want to show you that I am woke. I mean, I really am. I am aware of this. Women are part of the story. And so are people in India, China, and Japan. And then the Mayan civilization. There are a lot of calculus being done around the world. With woman, honestly, it is only sort of , around 1800s that women were allowed to go to universities and hear lectures and stuff. …"
billfruit 7y ago
What I feel about calculus, esp school calculus, is that it seems to comprise largely of specific tricks of symbol manipulation, rather than general approaches that work in all cases. Any ways, analytic integration/differentiation seems to be full of such tricks, of which how the core intuition was first obtained remains a mystery most of the time(rather than it being a result of scientific/mathematical method or process).
- adamnemecek 7y ago
  Look into automatic differentiation. It's magic.
- throwawaymath 7y ago
  Putting aside the common issues with pedagogy, the reason you learn a "bunch of tricks" for calculus is because that's more or less all you have in many cases. If you take integration as an example, there is no single approach to solving every integral. More importantly, it's extremely common to encounter integrals for which there exists no closed form antiderivative. In fact it's technically exceptional to find an integral which can be neatly solved in the space of all possible integrals. As a direct result, solving an integral becomes an (often frustrating) exercise in transforming it into something equivalent integrals up to a negligible constant. Nonlinear optimization problems and differential equations are similar in this regard. There is something to be said for the depth of analysis, which does provide a deeper meaning and rigor to the "bag of tricks" in calculus. Outside the US it's somewhat common to skip calculus entirely and begin straight away with analysis, and I think there's merit to that. But the profundity and power of analysis doesn't provide you with any fundamentally more complete methods of solving calculus problems except insofar as they become more advanced and rigorous. Ultimately analytic mathematical work (as opposed to algebraic) is characterized by this kind of pattern-matching; this frequently results in seemingly inspired, bizarre looking proofs compared to how neat everything is in algebra.
  - billfruit 7y ago
    Perhaps for some of the students,a course that skips indefinite integrals, and focusing on definite integrals and numerical integration may be sufficient.
shashanoid 7y ago
Follow Steven on twitter for more great content. Amazing person too! https://twitter.com/stevenstrogatz https://twitter.com/stevenstrogatz
sjcsjc 7y ago
From the video 22:10 "... must have been one of the greatest aha moments in history ..." https://youtu.be/1r6893ga_So?t=1330 https://youtu.be/1r6893ga_So?t=1330
markbnj 7y ago
As a non mathematician I enjoyed David Berlinksy's "A Tour of the Calculus" when I first read it back in the late 90's. It's similar to this lecture in that it strives to provide an understanding to non-mathematical types by exploring the principles without tossing lots of notation at the reader early on.
codeproject 7y ago
Steven Strogatz is amazing. you can get lost in his lecture like Hollywood movie. it is so engaging
kwccoin 7y ago
I think whilst the video is great, the guy missed the deep question raised by the audience. Using the language in the talk, the question is may be God speak Calculus but is it the only language it/he/she speak? You have that concern whilst just like a predecessor in post Pythagoras, post Newton, post Einstein, post QM, post AI, we got a strong and strong "understanding" and manipulation of the universe. And those understanding provide us with a great physical well being and more wealth. But is the world a better place? The tradition question is still why we can send people to the moon but never able to solve the problem of ghetto. Minor issue: Even he has to admit the assumption of continuity (which mathematically different from differential but mostly do), there are many are discrete and non-differential. Hence God if it/he/she has to cover everything, he must have to speak another language. QED BTW I like the T-shirt and go to study more calculus. Great talk.
_monster_ 7y ago
Holy Sh1t ... this guy went out of his way to bash "dead white men" that developed calc at least 3 times in the "lecture"... is he a cuckold?

320 points by okket 7y ago

By dumb luck I scrolled to around 26 minutes, where he gives a summary. Basically, the laws of nature are written in calculus, which is more than a language, and we can use this to manipulate nature.

trevyn 7y ago
>Basically, the laws of nature are written in calculus Calculus is a tool that powers certain specific useful predictive models of our observations of nature in certain specific domains. Having a mental model in which the laws of nature are related to calculus in some deeper fashion (or where calculus is always an appropriate tool to describe observations of nature) is likely to lead to some erroneous intuitions.
obrit 7y ago
Incredible, that's where the Wadsworth Constant would put you in the lecture as well.

FrankyHollywood 7y ago

Makes me think about Paulos. I read various books of him. He has an interesting view on the world as a mathematician. http://johnallenpaulos.com/booksandreviews.html http://johnallenpaulos.com/booksandreviews.html

ivan_ah 7y ago

Very nice lecture that connects calculus to the laws of nature. I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas. I encourage anyone who liked the talk to try out some of the calculations on their own using https://live.sympy.org/ https://live.sympy.org/ which is an online REPL with build in calculus function like integrate, diff, limit, summation, etc. Here is an example: >>> summation((1/2)**(2*n), [n,0,oo]) 4/3 via https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C%20%5Bn%2C0%2Coo%5D)%0A%23--%0A https://live.sympy.org/?evaluate=summation((1%2F2)**(2*n)%2C... note the answer is an exact rational number (class 'sympy.core.numbers.Rational) and not a float approximation 1.33333... For a quick tutorial on how to use SymPy, check "Taming math and physics using SymPy" which is avail in printable format https://minireference.com/static/tutorials/sympy_tutorial.pdf#page=5 https://minireference.com/static/tutorials/sympy_tutorial.pd... or notebook https://nbviewer.jupyter.org/github/minireference/sympytut_notebooks/blob/master/notebooks/Calculus.ipynb https://nbviewer.jupyter.org/github/minireference/sympytut_n...

mlevental 7y ago
does sympy render latex in the repl...? also haha oo as an alias for float("inf") or whatever is clever.
crimsonalucard 7y ago
>I've always thought applied math has a leg up on abstract math because of the connections to real-world ideas. I don't particularly like this dichotomy. Abstract math is connected to the real world because if it wasn't it wouldn't exist. Overall people who say this stuff are just referring to math that is only derived to work from an experimental standpoint and not from an axiomatic standpoint. Machine learning is one example of a mathematical field with no axiomatic basis and the the pythagorean theorem is an example of one that is derived from the axioms of euclid. Both exist in the real world and are therefore applicable.
- throwawaymath 7y ago
  I don't think it's worthwhile to engage in applied versus pure "math war" either, but I don't really agree with the point you've made. All applied mathematics has an axiomatic basis. When certain mathematical theories can be "applied" to real world problems, it simply means the relevant axioms and definitions are a robust approximation of reality. To speak to your example directly: machine learning absolutely has an axiomatic basis. You can conduct legitimate research in implementations and software or hardware optimizations thereof; however, fundamentally every experimental result in machine learning is an application of a variety of theorems in linear algebra, probability theory or calculus.
  - lapinot 7y ago
    Imho this is not the right way to think of it. Experimental results aren't applications of any theorem, they are just measurements. And the fact that these measurements may or may not come from machines we know how to "perfectly" measure (eg digital computers) doesn't give them any axiomatic basis. I'm not sure anyone is doing machine learning using symbolic computation, afaik it's mostly about low-precision floats--which do have some foundations themselves, but quite far away from the reals-based vector spaces upon which optimization theory is based. As addition to this point: most convergence results are much better in practice than in theory, ie we do not yet have satisfying predictive theories for experimental optimization. On this pure vs applied math thing, imho it is indeed a false dichotomy: there are symbolic objects of study and there are empiric or otherwise "preexisting" objects of study. We may use symbolic objects to approximate empiric objects and provide "applied theories" with predictive power, but we may also abstract empiric objects into new symbolic theories (both usually being done together). There is a funny phenomenon in the more abstract domains of math where at some point everyone is routinely talking about "intuition", "seeing things" and "morality" of facts. For me, building up intuition in a domain is about taking an axiomatic theory and transforming your view of it into something of the more empiric kind, one where you believe in an external understanding of how things behave. A symbolic fact being "moral" when it's consistent with the empiric counterpart you developed in your mind. My conclusion: some things are axiomatic/symbolic and some are not, but we mostly treat them the same way, ie by building empiric mental models. How else would we have proof ideas?

nvusuvu 7y ago

The section about Usian Bolt running the 100m dash was very interesting. Usian needs 41 strides to reach 100m while other runners need 44.

mathnmusic 7y ago

Really nice! BTW, LearnAwesome's calculus topic is relatively barren: https://github.com/learn-awesome/learn-awesome/blob/master/calculus.md https://github.com/learn-awesome/learn-awesome/blob/master/c... I'd appreciate if HNers here can point us to best learning resources for Calculus. Even better if you can send a pull request.

giu 7y ago
Just created a pull-request; I added links to some articles and sections hosted on Paul's Online Notes page (http://tutorial.math.lamar.edu http://tutorial.math.lamar.edu). This site helped me quite a lot during my studies, and I'm sure the same holds for a lot of other students and professionals :)

guru4consulting 7y ago

A while ago I saw someone posting a book about Calculus in HN forum that was written decades ago, probably 40s or 50s. I liked it and could not find it back. Does anyone have link to it?

alfiedotwtf 7y ago
Maybe Apostol's Calculus, or Spivak's Calculus. Both 1967 though
throwaway55554 7y ago
Was it Calculus Made Easy? https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480/ref=sr_1_1_sspa?keywords=calculus+made+easy&qid=1557496980&s=gateway&sr=8-1-spons&psc=1 https://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/... It seems easy to find a pdf on the web, but I didn't want to post that.
- guru4consulting 7y ago
  yes, I was looking for this book. Thank you. I wish I had read this when I was in school. I liked the book for explaining complex topic in a very simple, easily understandable way. This would be a great book to introduce calculus to your kids. I read chapter 2 (on different degrees of smallness) and found it enlightening.
  - throwaway55554 7y ago
    You're welcome. Happy reading!
- Jtsummers 7y ago
  The original (not the Martin Gardner version) is public domain at this point. Project Gutenberg has a PDF of it. http://www.gutenberg.org/files/33283/33283-pdf.pdf http://www.gutenberg.org/files/33283/33283-pdf.pdf
  - 7y ago

molteanu 7y ago

The beauty of Calculus can only be revealed to young recruits by Herb Gross' videos and Spivak's Calculus. If those don't tickle your brain, nothing will.

edge17 7y ago
link? not sure what you're referring to
- rofo1 7y ago
  https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665AB9002A https://www.youtube.com/watch?v=MFRWDuduuSw&list=PL3B08AE665... There are many, many ways to study calculus, so take this as a preference. For example, this wouldn't be my first choice. In general, I find the "Definition. Axiom. Theorem." approach very dry and just doesn't fit reality. Nobody ever discovered mathematics like this. One of the best books I've read is Gausses "Disquisitiones Arithmeticae" rather than any modern Theory of Numbers book. I've had more insight in sums manipulations (and various little tricks, some of them not even justified in modern mathematics) from Euler's letters than any other book on this subject. I don't know what happened in the meantime. It sure doesn't look like 18/19 century books were so dry like the ones today.
  - learn_awesome 7y ago
    Thanks for sharing both those things. We've added these to our Calculus topic here: https://github.com/learn-awesome/learn-awesome/blob/master/calculus.md https://github.com/learn-awesome/learn-awesome/blob/master/c...
artiste 7y ago
I also found 3Blue1Brown's (Grant Sanderson) YouTube series "Essence of Calculus" quite enjoyable.
less_penguiny 7y ago
"Spivak's Calculus" is a reference to the book "Calculus" by Michael Spivak. https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098918 https://www.amazon.com/Calculus-4th-Michael-Spivak/dp/091409... Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.

0x445442 7y ago

Do they not teach why Newton invented calculus anymore? They did when I was taught physics and it all made sense. I don't recall if I considered it beautiful at that time but I do recall how impressed I was with Newton's feat; moving himself farther along with his studies in physics.

madez 7y ago
He wanted to calculate and predict the movement of celestial bodies. The mathematical knowledge he had wasn't capable of expressioning the relations, but he saw patterns. Then he worked out the mathematical language to express these patterns.
tjr 7y ago
Not sure what's in the curriculum today, but I was taught calculus (20 years ago) heavy on physics applications. I don't think I would have appreciated it as much without that.
kodz4 7y ago
For anyone interested, here's the BBC looking into Newton and Leibniz notebooks as they cook it up for the first time - https://www.youtube.com/watch?v=ObPg3ki9GOI https://www.youtube.com/watch?v=ObPg3ki9GOI Also Steven Strogatz is an excellent writer. His other books Sync and Joy of X can be read by anyone.

bigred100 7y ago

Steve strogatz is a pretty excellent speaker even for people who don’t like maths

adamnemecek 7y ago

Tired: Calculus Wired: automatic differentiation

75dvtwin 7y ago

Two statements/quotes that I personally found interesting through prism of current times. "... Algebra is quite sterile, Algebra is not about anything …." (at about 1:17:28 ) "... One thing I feel a little bad about, and I feel I cannot properly correct it but I just want to show you that I am woke. I mean, I really am. I am aware of this. Women are part of the story. And so are people in India, China, and Japan. And then the Mayan civilization. There are a lot of calculus being done around the world. With woman, honestly, it is only sort of , around 1800s that women were allowed to go to universities and hear lectures and stuff. …"

billfruit 7y ago

What I feel about calculus, esp school calculus, is that it seems to comprise largely of specific tricks of symbol manipulation, rather than general approaches that work in all cases. Any ways, analytic integration/differentiation seems to be full of such tricks, of which how the core intuition was first obtained remains a mystery most of the time(rather than it being a result of scientific/mathematical method or process).

adamnemecek 7y ago
Look into automatic differentiation. It's magic.
throwawaymath 7y ago
Putting aside the common issues with pedagogy, the reason you learn a "bunch of tricks" for calculus is because that's more or less all you have in many cases. If you take integration as an example, there is no single approach to solving every integral. More importantly, it's extremely common to encounter integrals for which there exists no closed form antiderivative. In fact it's technically exceptional to find an integral which can be neatly solved in the space of all possible integrals. As a direct result, solving an integral becomes an (often frustrating) exercise in transforming it into something equivalent integrals up to a negligible constant. Nonlinear optimization problems and differential equations are similar in this regard. There is something to be said for the depth of analysis, which does provide a deeper meaning and rigor to the "bag of tricks" in calculus. Outside the US it's somewhat common to skip calculus entirely and begin straight away with analysis, and I think there's merit to that. But the profundity and power of analysis doesn't provide you with any fundamentally more complete methods of solving calculus problems except insofar as they become more advanced and rigorous. Ultimately analytic mathematical work (as opposed to algebraic) is characterized by this kind of pattern-matching; this frequently results in seemingly inspired, bizarre looking proofs compared to how neat everything is in algebra.
- billfruit 7y ago
  Perhaps for some of the students,a course that skips indefinite integrals, and focusing on definite integrals and numerical integration may be sufficient.

shashanoid 7y ago

Follow Steven on twitter for more great content. Amazing person too! https://twitter.com/stevenstrogatz https://twitter.com/stevenstrogatz

sjcsjc 7y ago

From the video 22:10 "... must have been one of the greatest aha moments in history ..." https://youtu.be/1r6893ga_So?t=1330 https://youtu.be/1r6893ga_So?t=1330

markbnj 7y ago

As a non mathematician I enjoyed David Berlinksy's "A Tour of the Calculus" when I first read it back in the late 90's. It's similar to this lecture in that it strives to provide an understanding to non-mathematical types by exploring the principles without tossing lots of notation at the reader early on.

codeproject 7y ago

Steven Strogatz is amazing. you can get lost in his lecture like Hollywood movie. it is so engaging

kwccoin 7y ago

I think whilst the video is great, the guy missed the deep question raised by the audience. Using the language in the talk, the question is may be God speak Calculus but is it the only language it/he/she speak? You have that concern whilst just like a predecessor in post Pythagoras, post Newton, post Einstein, post QM, post AI, we got a strong and strong "understanding" and manipulation of the universe. And those understanding provide us with a great physical well being and more wealth. But is the world a better place? The tradition question is still why we can send people to the moon but never able to solve the problem of ghetto. Minor issue: Even he has to admit the assumption of continuity (which mathematically different from differential but mostly do), there are many are discrete and non-differential. Hence God if it/he/she has to cover everything, he must have to speak another language. QED BTW I like the T-shirt and go to study more calculus. Great talk.

_monster_ 7y ago

Holy Sh1t ... this guy went out of his way to bash "dead white men" that developed calc at least 3 times in the "lecture"... is he a cuckold?