@@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
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
I taught myself Calculus III entirely using Professor Leonard's lectures on RUclips and doing practice problems in my college's calculus textbook. Prof Leonard does a phenomenal job explaining the intuition behind many of these key ideas, so I came into this video understanding the ideas of each theorem pretty well and understood their relationships. However, it was the end of your video that blew my mind. I honestly did not expect to see a very generalized version of all these theorems, and to see that it could be summed up so simply was super cool. Thanks for informing me about the Generalized Stokes' Theorem. I will be taking Calculus III in a month at my college, and I feel even more prepared.
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
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.
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
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.
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
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."
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).
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
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!
Incredible, incredible video. I took calculus 3 and absolutely loved it, and upon finishing it I had a vague sense that there was a connection between some of the theorems, but I never caught on to just how fundamental that connection was, and that it extended to not just between greens, stokes, and the divergence theorem, but to line integrals and the fundamental theorem of single variable calculus itself. Absolutely beautiful.
I'm studying mech engineering, but when I started first year, we could choose between learning "normal maths" or "advanced maths". My instinct told me to choose advanced maths, just cause why not. (very few people chose it because they thought that it's extra suffering for no benefits). It was one of the best descisions I've ever made. Our teachers were insanely good, and analysis 1 and 2 were some of my favourite classes I've ever taken. Sometimes I get the same feeling as you, finally properly understanding something that I couldn't grasp when we were studying it, and it's amazing. I hope that in some of my future semesters I'll have the time to retake at least one of these classes (with the same teachers hopefully!), but we'll see.
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.
Beautifully done!!! And very satisfying how it all comes together in the end!!! I was a Physics Major, studying Calculus (of course), nearly forty years ago, and these connections never occurred to me. Every minute of your video was compelling and clearly explained, and I could visualize it all (*especially* because of my familiarity with working with Maxwell's equations of Electromagnetism). I get the "spirit" of it.... the generalization... but there are still some subtleties I'll need to ponder in the coming days, which I think will put me on a firmer foundation of understanding. This video should be required viewing for every Calc-II student!
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 👑
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
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.
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
26:10 It took me 30 years 😭😭 to come to this understanding despite all the efforts spent accross the years, by a video (amazing and mind blowing 🤯) that I have landed on it by chance. Thank you ❤
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
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.
Yes. Also, the cross-product vector really is a bivector called a "pseudovector", as in n dimensions, you can have a pseudo-(n-x)-vector that is orthogonal to the x-vector.
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.
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
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!!!!!
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.
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.
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
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. 😊
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.
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 RUclips. And some 3blue1brown videos, of course. 🙂
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...
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.
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. 😁
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.
@@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.
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.
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.
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.
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.)
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
actually, it is a quotient. but you need to fix the notation. d should be an operator that does implicit differentiation, not d/dx. "[d/dx]f" is a function that applies f to d, and then divides by dx. dividing by a differential needs to be a separate step, ex: y = f[x] d[ y = f[x]] dy = df dx d[dy] = d^2y See what happens with second derivative in this notation. It is important to use implicit diff only, and divide later: d[ d[y] / d[x] ]/d[x] = d[ dy / dx ]/dx = d[ dy * dx^{-1} ]/dx = d^2y/(dx^2) + dy*(-dx^{-2}d[dx])/dx = d^2y/(dx^2) - (dy/dx)(d^2x/(dx^2)) It is a bug in the current notation in that it usually assumes that "d^2x = 0". If you make that assumption, then the algebra breaks. As long as the precedence is understood, this works for second derivative: [d/dx]^2f = [d/dx][d/dx]f = [d/dx](df/dx) = d[ df/dx ]/dx = d^2f/(dx^2) - (df/dx)(d^2x/(dx^2))
as for Integration, it's best thought of as S is an operator that cancels the operator d S[d[f]] [Sd]f = f - f_0 ie: f=x^2 df = 2x dx 2 S[x dx] = S[2x dx] = S[d[x^2]] [Sd](x^2) = x^2 - f_0
mutivariable is much easier in this noation too, because when d is implicit diff operator, differentials really are algebraic elements that can make fractions. Especially if you think of infinitesimals as objects that square to 0. ie: dz*dz=0, dz>0. Because dz>0, there is no problem in dividing by it. z^2 = x^2 + y^2 d[z^2 = x^2 + y^2] 2z dz = 2x dx + 2y dy z = x dx/dz + y dy/dz
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
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. :)
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
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 :)
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?
@@RocketsNRovers 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
I taught myself Calculus III entirely using Professor Leonard's lectures on RUclips and doing practice problems in my college's calculus textbook. Prof Leonard does a phenomenal job explaining the intuition behind many of these key ideas, so I came into this video understanding the ideas of each theorem pretty well and understood their relationships. However, it was the end of your video that blew my mind. I honestly did not expect to see a very generalized version of all these theorems, and to see that it could be summed up so simply was super cool. Thanks for informing me about the Generalized Stokes' Theorem. I will be taking Calculus III in a month at my college, and I feel even more prepared.
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
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.
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
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
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
Simple but effective. This is a dope some3 submission for sure
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."
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
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!
Thank you. As a recovering math-phobe, I really enjoyed this. Tremendously helpful and very instructive.
Incredible, incredible video. I took calculus 3 and absolutely loved it, and upon finishing it I had a vague sense that there was a connection between some of the theorems, but I never caught on to just how fundamental that connection was, and that it extended to not just between greens, stokes, and the divergence theorem, but to line integrals and the fundamental theorem of single variable calculus itself. Absolutely beautiful.
I've never taken MVC, but the instruction in this video was so clear that I have working proficiency in it.
I'm studying mech engineering, but when I started first year, we could choose between learning "normal maths" or "advanced maths". My instinct told me to choose advanced maths, just cause why not. (very few people chose it because they thought that it's extra suffering for no benefits).
It was one of the best descisions I've ever made. Our teachers were insanely good, and analysis 1 and 2 were some of my favourite classes I've ever taken.
Sometimes I get the same feeling as you, finally properly understanding something that I couldn't grasp when we were studying it, and it's amazing.
I hope that in some of my future semesters I'll have the time to retake at least one of these classes (with the same teachers hopefully!), but we'll see.
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.
Beautifully done!!! And very satisfying how it all comes together in the end!!!
I was a Physics Major, studying Calculus (of course), nearly forty years ago, and these connections never occurred to me.
Every minute of your video was compelling and clearly explained, and I could visualize it all (*especially* because of my familiarity with working with Maxwell's equations of Electromagnetism).
I get the "spirit" of it.... the generalization... but there are still some subtleties I'll need to ponder in the coming days, which I think will put me on a firmer foundation of understanding.
This video should be required viewing for every Calc-II student!
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 👑
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
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.
This was actually so beautiful
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
26:10 It took me 30 years 😭😭 to come to this understanding despite all the efforts spent accross the years, by a video (amazing and mind blowing 🤯) that I have landed on it by chance.
Thank you ❤
i can’t thank you enough for the clarity you bring to your topics! ☀️
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
please keep making more videos about math topics where you explain everything. Its very well made and helpful !
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.
Can I ask how you learned geometric algebra? I've been wanting to take a course but my university doesn't seem to offer any in the subject
@@FoolishChemist, I recommend you start with the «Swift introduction to geometric algebra» video on RUclips.
Yes. Also, the cross-product vector really is a bivector called a "pseudovector", as in n dimensions, you can have a pseudo-(n-x)-vector that is orthogonal to the x-vector.
I'm gonna share this perspective on my college that's for sure, thanks a lot.
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.
What a marvelous video. Thank you for this effort!
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
Finally, a chemist that understands mathematics
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!!!!!
It's really an eye opener. Mind blowing!! 🎉🎉
Can't come up with an appropriate comment. It's very good.
The conclusion is very good. Certainly, well-educated friends make your life better.
One of best video ever❤❤
They way you summed up the whole video is just awesome 💫💫
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 love this, I'm just an undergrad, but I'm interested in higher mathematics ❤. Much love man, make more maths videos.
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.
this is one of the finest videos on youtube; poetic to say the least
Underrated video holy moly
Awesome video!
Which app are you using on the iPad?
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
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. 😊
Really I get impressed by your lovely video. Great
Dude,that was awesome. Nice vid. Like.
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.
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 RUclips. And some 3blue1brown videos, of course. 🙂
Taking calc 4 rn (last chunk of calc 3 + differential equations), and this sparked my interest for math again!
As a chemistry and maths double major, I'd say I loved this video!
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...
Beautifully explained
Amazing video. Thank you so much.
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
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. 😁
Nice one mate. loved it.
What a journey…great video!
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)
Great work!! Very intuitive and entertaining. Thanks a lot for your efforts...
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.
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
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.
If ive learnt anything from what ive seen in physics, df/dx cant be treated as a fraction until it can.
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.
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.)
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
You are genius man!!!
amazing explanation!
7:15 glad the basicallyexplained guy made a cameo in this!
also thanks for making this video im tryna re-learn math after a long time so this helps.
23:14 a more general one is the fundamental theorem of geometric calculus, it gobbles up the ones from complex analysis as well!
Great work! Thanks.
YO OMG LMFAO I KNOW YOU IRL 😭 Math 53 with Sethian must have been crazy man
Amazing
Excellent video
Gradient points to the direction of greatest ascent, as the opposite direction of the gradient can still be the greatest change
really really good explanations
at the end HxH, love it. Also nice vid too
Math is beauty.
>"mathematically, the derivative is not a quotient"
Hyperreal numbers & non-standard analysis have entered the chat.
i loved this video!
awesome vid, thanks!
actually, it is a quotient. but you need to fix the notation. d should be an operator that does implicit differentiation, not d/dx. "[d/dx]f" is a function that applies f to d, and then divides by dx. dividing by a differential needs to be a separate step, ex:
y = f[x]
d[ y = f[x]]
dy = df dx
d[dy] = d^2y
See what happens with second derivative in this notation. It is important to use implicit diff only, and divide later:
d[ d[y] / d[x] ]/d[x] =
d[ dy / dx ]/dx =
d[ dy * dx^{-1} ]/dx =
d^2y/(dx^2) + dy*(-dx^{-2}d[dx])/dx =
d^2y/(dx^2) - (dy/dx)(d^2x/(dx^2))
It is a bug in the current notation in that it usually assumes that "d^2x = 0". If you make that assumption, then the algebra breaks.
As long as the precedence is understood, this works for second derivative:
[d/dx]^2f
=
[d/dx][d/dx]f
=
[d/dx](df/dx)
=
d[ df/dx ]/dx
=
d^2f/(dx^2) - (df/dx)(d^2x/(dx^2))
as for Integration, it's best thought of as S is an operator that cancels the operator d
S[d[f]]
[Sd]f
= f - f_0
ie:
f=x^2
df = 2x dx
2 S[x dx]
=
S[2x dx]
=
S[d[x^2]]
[Sd](x^2)
= x^2 - f_0
mutivariable is much easier in this noation too, because when d is implicit diff operator, differentials really are algebraic elements that can make fractions. Especially if you think of infinitesimals as objects that square to 0. ie: dz*dz=0, dz>0. Because dz>0, there is no problem in dividing by it.
z^2 = x^2 + y^2
d[z^2 = x^2 + y^2]
2z dz = 2x dx + 2y dy
z = x dx/dz + y dy/dz
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
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. :)
so good
Making sure your getting paid, lol never had so many commercials for 1, 25 minute video
awesome...
How is this free?? I’m a mechanical engineering major and what a wonderful video! Thank you
Great video. Thank you
I recommend Spivak's Calculus on manifolds if you liked this video
As an English college student, we learn this at a level. Do Americans not learn it in high school?
Great video
Thank you itachi ❤
You guys focused on the video??? I was terrified at the idea that you ought to write essays for calculus
great video! what software did you use for writing?
Penbook on the iPad!
@@FoolishChemist thanks
BEAUTIFUL