The Essence of Multivariable Calculus |
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- čas přidán 19. 06. 2024
- In this video, I describe how all of the different theorems of multivariable calculus (the Fundamental Theorem of Line Integrals, Green's Theorem, Stokes' Theorem, and the Divergence Theorem, plus also the original Fundamental Theorem of Calculus in one dimension) are actually the same thing in higher math. I present this by going through each theorem conceptually step by step (no formal proofs) and then summarize a recurring idea that we observe in each theorem. Turns out, the idea I present is exactly the basis of the Generalized Stokes' Theorem, sometimes called the "Fundamental Theorem of Multivariable Calculus". Hope you enjoy!
Please note a minor error in the video:
At 9:22, inside the dot product of F with r'(t)dt, r'(t) should've been with a vector arrow on top of it to signify that it is a vector (we can't take the dot product of a scalar function and a vector field, we can only take the dot product of a vector field and another vector).
This video was made for the 2023 Summer of Math Exposition (SoME3) run by 3Blue1Brown.
Chapters:
0:00 Intro
1:28 Video Outline
2:07 Fundamental Theorem of Single-Variable Calculus
7:38 Fundamental Theorem of Line Integrals
13:05 Green's Theorem
15:29 Stokes' Theorem
18:08 Divergence Theorem
21:31 Formula Dictionary Deciphering
23:07 Generalized Stokes' Theorem
25:55 Conclusion
Subscribe for more: www.youtube.com/@FoolishChemi...
/ foolish.chemist
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#SoME3
#math
#calculus
#multivariable
#multivariablecalculus
#perspective - Věda a technologie
Professional mathematician here, who has taught various flavors of calculus a zillion times: This was magnificent.
This is Calculus III of Calculus and Analytic Geometry correct ?
@@genet.2894 well yes the generalization theorem is calculus 3, but the generalization theorem its for differential manifolds or a course on manifolds: its a course for mathematicians or physicists, on calculus 3 isnt normal to see the generalization
@JoseMedina-ug6on However, it is covered in the book itself. What's more there's a difference between a course in Calc III, aimed at those majoring in the social sciences vs. a more rigorous treatment of the subject aimed at science and engineering majors, which will cover such topics for the benefit of those students. I recall at UCLA they taught from different Calculus text books depending on major. One was geared towards the liberal arts majors and the other towards science and engineering majors
As a physics and cs double major student i liked this video :D
I second that, as a physics and cs double major. Except this time I'm taking multivariable calc in a pure maths perspective.
pilani cs?
@@rocketsandmore6505 Haha my dad did CS at BITS. I'm doing it at ANU.
@@krishyket ohh .. nice bro
The only way I see a double major CS and Physics as being remotely possible as both are hard subjects in their own light is if you know or are extremely talented in one subject area thereby able to focus more on the other area - The subjects are so disparate
My whole electromagnetism understanding has changed now . Like all these rules in school has gotten so much more colorful and fun and meaningful...
glad for you :)
Please continue making videos sir, you have immense potential for explaining complex things and more importantly for building connections and intuition, I really hope your project is recognised in the SoME3
In university I took a course about physics simulations. There reached a point where we needed to calculate the mass of an arbitrary polyhedron so we could model its forces properly. I was shown how you could use the divergence theorem to calculate the volume of a closed polyhedron by turning an integral over its volume into an integral over its surface. You could then assume constant density and use the volume to get your mass. I think that was coolest application of multi-variable calculus I've seen. Thank you for this video, it's such a good refresher!
This is so cool! I have taken multivariable calculus many years ago and you've taught me probably the coolest application of divergence theorem I never knew of.
Simple but effective. This is a dope some3 submission for sure
I swear these kind of videos should be watched at lessons in university. They explan so much better what we are studing instead of mechanically doing things and just memorizing theorems without actually understanding what they say. As an engineering student, I thank you and will show to my university mates
I don't know, I kind of think our professors want us to have these "holy shit" moments of our own volition. To go home and ponder as we fall asleep and wonder how the pieces fit.
You should know that professors don't care if you understand or not. To them this kind of math is too elementary that when you enroll in their class you are expected to take care of your self. BTW most of the good math programs in Universities want you to think not spoon feeding the students
On the subject of visualizing higher dimensional integrals, the textbook we used had a great line. Referring to triple integrals, it said something like, "You can think of this as the four dimensional volume under a three dimensional surface. This is not particularly helpful."
i love watching calculus and physics videos, and this was by far the best multivariable calculus video i've ever seen!! simple and intuitive explanations of hard topics of high-level math. never have i seen such a clear explanation of the fundamental theorem of single variable calculus it was really astonishing my jaw dropped. the fact that you're from chemistry and still make this video with that passion and beauty makes me wanna learn more about new stuff. I absolute recomend this to who's starting calculus, this is some neat material
Really great video dude! Just about to take my first multivariable calc course and this has got me all excited to unpack the levels of abstraction in more detail.
Awesome! Enjoy the class!
@@FoolishChemist same
This video was heat 🔥 we gotta get you more subs. I legit thought calc 3 was beyond me until I watched this and for the first time I actually get it. Keep the uploads coming king 👑
Very nice vid, one of my favorite submissions so far for #SoME3 I feel!
To add to the discussion, you might be interested in what you might view as a possible "sequel" to this: at around the 9 min mark, you mention that we "can't multiply vectors".
Tho, what if I told you that this is totally possible, and in a way where you don't have to resort to the math in general relativity, but can also take a lot of what you learned in vector calculus and extend it in higher dimensions?
(you will need to drop the cross product and replace it with something else that reproduces its properties in 3D while letting it generalize in higher dimensions, called the wedge product. You'll also need to include more "directed objects" instead of restricting yourself to just directed lines, i.e. regular vectors. For instance: directed plane segments, directed volume segments, etc, modeled by multivectors, in the same way vectors modeled directed line segments geometrically).
That subject is called Geometric/Clifford Algebra, and an associated calculus to it called 'Geometric Calculus' in a similar way vector calculus was to regular vector algebra. (There is a related area called "Clifford Analysis" that goes quite in depth with pure math formalism and rigor, but you won't need it just to extend vector calculus).
Nothing gets me going like Clifford algebra lol. God I want to meet the octopus at the depths of the math trench
or alternatively u can just learn tensors and fibre bundles and do differential geometry like the rest of the world. They are more versatile and generalize to algebraic geometry via sheaves of O_X modules
So what even is a directed plane segment and how can I think about it
@redpepper74 imagine the paralelogram formed by two vectors, sinilar to those we draw when teatching vector sum. This is the directed area. The area of the paralelogram is the area of the vector product, the direction follows the right hand rule (or the convention of the coordinate system used). This area has a sign, since inverting one of its forming vectors changes the sign of the final.area, like what happens when we calculate an integral backwards.
i can’t thank you enough for the clarity you bring to your topics! ☀️
This was beautiful, reminds me of Poincaré’s quote “Mathematics is the art of giving the same name to different things” or in this case different names to the same thing. Thanks for sharing!
Great video. As a meteorologist I enjoyed your perspective and it brought a smile to my face as you simply explained the maths i enjoy analyzing when i look at the diverging wind fields, the upward movement caused by the curvature of winds, over the different surfaces at varying pressure levels of our troposphere.
Thank you. As a recovering math-phobe, I really enjoyed this. Tremendously helpful and very instructive.
What a marvelous video. Thank you for this effort!
man i can t believe you explained it so nicely, it s the first watching one of your video, i hope you have more. congrats on you explanations, i can t believe i understood so much while i am still struggling with my PDEs and A level pure maths, etc. very big appreciation for founding your video. lots of thanks
One of best video ever❤❤
They way you summed up the whole video is just awesome 💫💫
It's really an eye opener. Mind blowing!! 🎉🎉
This was actually so beautiful
I’m absolutely astonished. I’m a dunce when it comes to mathematics generally (a dunce who is at the same time is very interested in maths); yet now, having seen your video I’m really beginning to see that calculus is within range of my understanding it. I can’t tell you how excited I am about this leap forward! Thank you so much.
please keep making more videos about math topics where you explain everything. Its very well made and helpful !
Thank you for sharing this extremely insightful simplification of an otherwise a highly complex topic (perception of complexity of multi-variable calculus). This simplified (geometric) image will likely stick in my mind for years to come. Human mind thinks differently and complex math can be translated into a human-mind-friendly format using these insightful changes in perspective.
Dude you are so underrated!!! You have only 1K subs? I can't believe it! I thought you had 1M subs, your content was that good! You truly deserve more! Please keep making content like these!!!!!
Awesome mesmerising superb. I completed my BSC in electrical and electronics engineering from most famous university in my country 23 years ago . Unfortunately I didn’t fathom anything regarding greens theorem during my fields and waves course in BSC. I wish I would have watched this video during my study. Thank awfully for this video
I'm gonna share this perspective on my college that's for sure, thanks a lot.
What a journey…great video!
Great video! I didn't understand these fundamental connections until years after I took Calc III. Good luck with #SoME3! By the way, one thing that is often left out - just as Green's theorem is a special case of Stokes, there is a 2D version of the divergence theorem, in which you're equating flux across a curved boundary in 2D to the divergence of the 2D vector field on the interior. It's actually equivalent to Green's theorem with the vector component functions rearranged.
Great work!! Very intuitive and entertaining. Thanks a lot for your efforts...
Dude,that was awesome. Nice vid. Like.
this is one of the finest videos on youtube; poetic to say the least
Amazing video. Thank you so much.
Man you're the best you cleared my all doubts
Really I get impressed by your lovely video. Great
Beautifully explained
Math Professor fr be like "if this description was confusing that's because I glossed over a few important details but don't worry about that for now" that line had me dying 10:08
That was great. My mind is blown, but yet I feel like I understand multivariable calculus much more than I did before. And I haven't even truly taken multivariable calculus yet! I've just watched about the first quarter of Professor Leonard's Calc III sequence here on CZcams. And some 3blue1brown videos, of course. 🙂
14:28 At this moment I literally was like "Dude, you really need bivectors." Learn geometric algebra, dude, this will make your math life easier. This dS really should be bivector.
Nice one mate. loved it.
Underrated video holy moly
Great work! Thanks.
I love this, I'm just an undergrad, but I'm interested in higher mathematics ❤. Much love man, make more maths videos.
amazing explanation!
Can't come up with an appropriate comment. It's very good.
Taking calc 4 rn (last chunk of calc 3 + differential equations), and this sparked my interest for math again!
Actually, derivative is a quotation, the differential devided by argument variation (you can see it if look at definition of differentiable function and differential as a linear part of function variation).
And problem with cancelling dx actually lies in that dx in integral is just a part of notation, and you actually have to prove that you can use this notation as actually multiplication by dx.
There's another problem with thinking derivatives as a quotation, when you can't prove chain rule by just cancelling out dy in (df/dy)*(dy/dx). But this problem we have because of notation duality: we have the same notation for differential and for argument variation. And in this formula dy at the bottom is a variation of argument and dy at the top is the differential.
Of course, differential is not equal to just variation, it's just a part of it. And also df that you see in two parts of this formula is a two different functions, because differential is a linear function of argument variation, and it depends on which argument function have, y or x
awesome vid, thanks!
You are genius man!!!
As a chemistry and maths double major, I'd say I loved this video!
The conclusion is very good. Certainly, well-educated friends make your life better.
Finally, a chemist that understands mathematics
I was scrolling through the scientific video suggestions, saw this one and wandered why is there a video about watching knitted woolen socks through a magnifying glass...
Excellent video
at the end HxH, love it. Also nice vid too
Great video. Thank you
5:07 i got this proof on my channel lol. it was the first thing that came to my mind when i was trying to make sense of that fundamental theorem of calculus equation like years ago in high school...thanks for pointing out that it doesn't work in all cases.would need to learn more to know more about what's going there.
i loved this video!
Great video. Title is a little misleading - maybe add ‘Stokes’ in there? The way you weaved the definitions in, and simplified for the General Stokes Theorem was magical…I guess those calculus essays helped! This is now the best video on Stokes, with Aleph O runner up! Please keep doing math videos - you have the gift! Maybe Fourier / Dot Product, linear algebra, quantum stuff? Check out ‘goldplatedgoof’ Fourier for the rest of us. Super cool - using Fourier epicycles to create equations from curves (data). Mind blowing!
Thank you! My goal for this video was make it accessible for anyone who knew even a little bit of calculus which is why I titled it as it did-I figured putting “Stokes Theorem” in there like AlephO did would have made it miss the students who haven’t taken multivar yet…but you’re definitely right, the title is a little misleading lol, albeit intentionally
My favorite explanation of why the gradient is the max rate of ascent: the more something’s increasing your result, the more of it you do; the more something’s decreasing your result, the more you do the opposite
Nice! Actually, one can already see these theorems as special cases of the divergence theorem - for this, one just needs to define the divergence operator on manifolds (lines, surfaces etc.). In a way, it's a more natural generalization than Gen. Stokes, as it doesn't require additional terminology (forms,...) and doesn't restrict our considerations to oriented manifolds.
(on the other hand, it does require Riemannian structure on the manifold, but as long as we're working with submanifolds in R^n, it's the same structure Line Integral and Stokes theorems use anyway)
My calculus textbook explicitly summed up how all the concepts of multivariable and vector calculus that were taught were extensions of the fundamental, basic concepts and theorems from the very start of calculus. It made me appreciate the courses more once I saw how seemingly unrelated concepts were simply logical extensions of earlier concepts that carried with them incredibly broad and far reaching conclusions and applications.
name your calculus textbook
@@vvvvvvvvvvv631 I actually had three different books I used since I took each calculus course online from 3 different campuses, but the book that this comment is referring to is Calculus for Scientists and Engineers, Early Transcendentals by Briggs, Cochran, and Gillet. In the final chapter, (15.8) before the exercises, there is a couple paragraphs and a table showing how line integrals, green’s theorem, stoke’s theorem, and divergence theorem are just extensions of the fundamental theorem of calculus.
really really good explanations
Thank you itachi ❤
Amazing
I havent yet learnt multivariable calculus yet. But I think this was a really good introduction video. Now I am really excited to this in my college. 😊
BEAUTIFUL
23:14 a more general one is the fundamental theorem of geometric calculus, it gobbles up the ones from complex analysis as well!
Great video.
so good
Just a note as a viewer: at 18min I was unsure about your distinction between surface and curve when describing Stokes theorem, it's clear now that C is a 1d curve embedded in 3d, and S is a hull of that curve, a 2d surface embedded in 3d. For a while there I thought C was a 2d surface in 3d, which caused the confusion.
I still engage with math pretty heavily but I haven't touched multivariate calculus specifically for a while.
I wish I would have discovered this video when I so desperately wanted to understand what integrals really were. I tried learning from 3blue1brown, and though they are awesome, I couldn't understand what they were telling me. This video however explained it perfectly. 😁
Awesome video!
Which app are you using on the iPad?
awesome...
Great video! One question: could you ask your physics/CS friend for me if the generalised Stokes theorem has anything to do with the Holographic Principle? Seems like there should be some connection..
Hi, do you have a reference list of sources you used to make this video? Where did you learn about this kind of motivation for the Fundamental theorem of calculus for single variable calculus?
I don't really have a concretelist of references sorry-I pretty much made this video based off of what I learned in my multivar class in college. I used Wikipedia and some other sites (like Paul's Online Notes, which I highly recommend, see here for an example: tutorial.math.lamar.edu/classes/calciii/GreensTheorem.aspx) to refresh my knowledge and clear up the nuances, but that was pretty much it
And the generalized stokes theorem is just a weaker version of de rhams theorem :) (at least for smooth manifolds).
Amazing video, thank you.
Congratulations sir, you have just beaten my earlier university professors. I learned more in your 30 min than their 30 weeks 😂
(Not entirely fair, as I learned a LOT since those sophomore weeks. Much of it from 3B1B, obviously, but let's not bash those profs to death.)
I didn't take calc 3, (but I did watch some video courses on youtube). I first encountered the ++real++ fundamental theorem of calculus in the context of geometric algebra/calculus. I have no idea what it means, though. :)
The essence of multi-variable calculus, IMO, is: hey, here's what all this stuff you learned in two dimensions [x, f(x)] looks like in three or more dimensions of which there are only a handful of analytic solutions. So, don't get too excited unless you are prepared for some really deep and complex rabbit holes trying to solve what look like trivial equations on the surface (pun intended). However, if you are an engineering major, you will need multi-variable calculus because the real world is 3D.
do you think that zeta(3) could have a closed form and why ?
In a way, the bounday it´s acting like de derivative of their shape. Really a crazy conection when you notice that this just happends in geometry as well.
thinking of a circule, it´s area is equal to πr² while it´s perimetrer (the bounday of the area) it´s equal to 2πr, the derivative of the area respect of its radius r.
In Ari´s words, its just amaizing.
You blew my mind
Gradient points to the direction of greatest ascent, as the opposite direction of the gradient can still be the greatest change
>"mathematically, the derivative is not a quotient"
Hyperreal numbers & non-standard analysis have entered the chat.
I was trying to understand why if the integral of the derivative is equal to the function of the boundary is the case, then for the single variable case we have it equal to f(a) - f(b) and not f(a) + f(b). Then it occurred to me. It's because if I treat those two points as an enclosing boundary then they will be pointing in different directions.
great video! what software did you use for writing?
Penbook on the iPad!
@@FoolishChemist thanks
YO OMG LMFAO I KNOW YOU IRL 😭 Math 53 with Sethian must have been crazy man
If df/dx is an operator and not a quotient, then how can we solve d(anything) when using u-substitutions or move d(anything) around the equation in differential equations?
If ive learnt anything from what ive seen in physics, df/dx cant be treated as a fraction until it can.
What Tablet is this?
I recommend Spivak's Calculus on manifolds if you liked this video
What does a essay for calculus involve? Really not looking forward to writing any in my math degree 😭
I'm surprised to hear that as well. Maybe it was something like researching the history of calculus, and writing a paper on a historical figure.
2:53 why are differential expressions not mathematically quotients?
Is this the "Functions are vectors" video?
If I had one criticism, I’d say that calling some of those sentences “plain english” is… a bit of a stretch. Since it’s basically just a direct substitution of the symbols with their names.
I think it would be more enlightening to use language like “accumulations” and “small contributions” instead of integrals and derivatives.
This is an excellent idea! Lwk should've done that lol
You guys focused on the video??? I was terrified at the idea that you ought to write essays for calculus
How is this free?? I’m a mechanical engineering major and what a wonderful video! Thank you
i love you
As an English college student, we learn this at a level. Do Americans not learn it in high school?
dude I would've gotten a 100% in calc III if I knew the generalized stokes' theorem. They gotta teach that earlier