What is Jacobian? | The right way of thinking derivatives and integrals
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- Опубліковано 12 тра 2024
- Jacobian matrix and determinant are very important in multivariable calculus, but to understand them, we first need to rethink what derivatives and integrals mean. We can't think of derivatives as slopes if you want to generalise - there are four dimensions to graph the function! This video hopes to explain what the Jacobian matrix and determinant really mean, and essentially why they are actually very natural for changing variables; and also explaining something that might be glossed over when you use them - for example, we require absolute value, and the changing variables function is injective.
In the video, we have only talked about 2D transformations, but the Jacobian can be easily generalised to any number of dimensions you like - you just need to introduce linear maps in higher dimensions! Think about what that means in 3 dimensions for a start!
This video simply aims to introduce the intuition of the Jacobian, and so a lot of things said in the video is not going to be very rigorous - for example, what does approximate mean? It has a specific meaning in mathematics, but we are not getting there; and also not all functions have this nice property of looking like a linear map near a point. These belong to the realm of real analysis, which is well beyond the scope of this video. So please don't shout Fubini's theorem when you see flipping the order of integration at about 17:09.
Video chapter feature:
00:00 Introduction
01:20 Chapter 1: Linear maps
06:01 Chapter 2: Derivatives in 1D
08:08 Chapter 3: Derivatives in 2D
13:01 Chapter 4: What is integration?
17:26 Chapter 5: Changing variables in integration (1D)
19:25 Chapter 6: Changing variables in integration (2D)
22:59 Chapter 7: Cartesian to polar
If you are interested in thinking about how the formula for the determinant came about, here is it: moodle.tau.ac.il/2018/pluginf... (p. 134, 135)
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The next video will finally tackle the problem of average distance between two points in a unit disc analytically - no more simulations. I am quite proud of this video, and took almost certainly more time (I didn't keep track this time) than any other video on this channel, even though it might not perform as well in the UA-cam algorithm, but whatever, I like what I made here :)
Do leave a like, subscribe and leave a comment now, so that more people can watch this!
I can’t wait to see it! :D
@@colorfulquesadilla377 Thanks for the support! I can't wait for the video to drop as well!
If Jacobian is an cure to exploding convergence by finding the correct scaling factor using determinant and we use it for symbolic calculation but why do we use Jacobian based element in finite element analysis since that is also an integral why is Jacobian used in both numerical and symbolic calculation s
If Jacobian is an cure to exploding convergence by finding the correct scaling factor using determinant and we use it for symbolic calculation but why do we use Jacobian based element in finite element analysis since that is also an integral why is Jacobian used in both numerical and symbolic calculation s
This is really helpful...thanks alot
Came for the Jacobian, stayed because - almost by accident - _you gave an intuitive explanation of the chain rule!_
That was the whole reason I am making this video, because many people have talked about Jacobian before, and this explanation of integration by changing variables was hopefully something "new" on UA-cam.
Same here
I love how UA-cam is now exploding with math channels
which is good :)
I found a physics crank channel today, I wonder if there are math crank channels.
What are some other good ones bro?
@@joshuascholar3220 what does this mean? Someone who teaches things incorrectly as a prank?
@@nikilragav not as a prank, but because they were incapable of learning correctly, came up with their own theories and who, out of injured pride scream that everyone who isn't a crank is a fraud.
That's about the shape of the average crank. Some of them were capable of being educated and don't hate everyone - but do have grudges against some famous people and their work. Generally those people have an extreme lack of ability to put things in context like most nuts.
I have BS/MS in math, MS in statistics, and next year I'm finishing a statistics PhD, and I've never seen vector calc presented this way. Thank you for the illumination.
Your name makes me imagine a cartoon about Popeye getting radioactive powers
@@user-gs1lz2pw9v I think my then-14-year-old-self was thinking along those lines :)
Look at 3 blue two brown. A whole new level of animation of transformations.
1 blue 3 brown? 5 brown 6 blue? One of those!
@@ohgosh5892 you fucks with sacred heart geometry
After 40 years of college, finally a good explanation .
Sorry what, 40 years of College ?
@@sorvex9 oh you can't be this pedantic, he obviously meant 40 years after passing his college. God damm
@@sorvex9 🤣🤣🤣🤣
haha my first thought🙃 as i got to the explanation of the matrix via warped linear coordinates
Ah yes, the King of getting left back.
back in 2018 i spent some time learning how code animations using manim and realized how much work it requires. i became sad once i realized there was no way 3b1b was ever going to come close to animating all of maths. now i am very excited to see all of these channels coming out and tackling these concepts! thank you for your contribution to humanity
It does take a lot of work! But actually, I don't use Manim :)
As a matter of interest, what do you use? Manim is fairly good,. I have been looking at Blender for more complex animations.
PS. Great presentation - I have always been afraid of Jacobians because I didn’t understand why they existed.
@@andrewmole3355 He makes animations using a combination of Powerpoint and Geogebra; there's a video about it somewhere on his channel, I think.
@@mathemaniac you have provided more to the world than the likes of Elon
I have yet to learn multivariable calculus and area integrals, and this seems to make things a bit more digestible for me. Neat video, man!
Glad it helps!
@@mathemaniac absolutely. I never knew to think of 2d matrices as scaling the up and right vector
@@zyansheep If you’re still struggling with matrix intuition, I’d reccomend 3blue1brown’s seties on linear algebra.
Area integrals? What other type would you have learned before that?
@@orang1921 line integrals duh...
This managed to make more of an impression on me than my entire university linear algebra class. Most professors seem to just read off a PowerPoint.
Thanks so much for the appreciation!
Exactly, I end up studying most of the course content on my own. Thankfully there's great content like this that I can use in my studies
I can confirm that they do just that.
I hate powerpoints and pretty much refused to teach from them
@@johnwilson8309 do not blame the tool blame the craftsman. I love them, I like teaching from them and allows me to modify by work in real-time. sometimes someone asks an interesting question and I just markup it up right there. afterward, i decide whether it a hidden slide or something incorporated in the main class
I'm in last year of my Mathematics degree, and I feel I just started understanding determinants and Jacobians right now!!
Thanks a lot
Glad it helps understanding!
Bruh start studying man
Well, maybe you should change the studies subject then xD
Bruh
It's never too late to learn
I started learning calculus 7 years ago, and I’m still learning new perspectives of derivatives and integrals today. It’s such a fascinating subject. I actually had this intuition for 2d+ cases, but applying it back to 1d cases was what really made it click just now haha. This is very helpful for those of us who had trouble connecting u-substitution to using the Jacobians to change variables. It’s the same exact thing!
Please do one for vector calculus 🙏
Thanks for the appreciation! Glad that it helps.
I am not sure which part of vector calculus you are talking about though, but I will probably consider it.
@@mathemaniac I think he's talking about line and surface integrals. Maybe that's not what he's refering to, but what I'd like to see. I've been studying integration of differential forms, and parameterization kinda confuses me, eg., integrating a 2 form over a sphere. How does matching each coordinate plane (dx^dy, dy^dz.dz^dx) to the coordinate plane given by the parameterization (dφ^dθ) work? It's not a one to one thing like what happens to integrals over intervals.
@@canriecrystol yes, this is it. More specifically, the General Stokes’ Theorem
This is so well done. Covers a lot of intuition that many, many linear algebra classes leave out, leaving the students to decipher it on their own. Well made man, I really appreciate this video.
without exaggeration, this is the best explaining video on youtube i have ever watched. I have watched "Essense of linear algebra" playlist by 3blue1brown, but this is definetely more clear and understandable.
I am very grateful for this masterpeace.
Thank you so much for the kind words!
This is the only time I truly understand the Jacobian geometrically, I wish I could've bumped into this video sooner. Great stuff!
Glad you enjoyed it!
with a heavy heart I clicked this, having a physics degree and never knowing why we were even learning jacobians back in the day. Thanks lol
I'm in the exact same boat. Jacobians, Hermitian Operators, Hilbert Space, they all came at us so fast I didn't even have time to process them. I just went about computing what I could for a grade because that's all you can do sometimes when in University.
I had never even thought about where the extra r came from when converting integrals to polar. This video just tied all of it together fantastically
This is one of the best, if not *the* best video on the Jacobian available on UA-cam. Wonderful job here.
Thanks so much for the compliment!
vsauce narrative + 3blue1brown animation
after the iconic music I expect " Hey vsauce !......Michael here"
I was just going to write this exact comment 🤣
Yeah, same thought here. Good times
@@Cl0udEater Makes me appreciate Soothing Grant’s voice is
- the clarity of both
awesome intuitions on change of base in the context of calculus. I can see the 3b1b influence all over this content and i love that too
Thanks so much for the appreciation!
when x explain intuitions on the base of cordinate they are indicating motives and character i.e derivatives and integrals not determiners or minant i.e timelessness intent ,just only to see all in those plain of functionality coordinate in geometry or graphy
Whats 3b1b?
@@diulaylomochohai 3blue1brown, another maths channel :)
Thank you. Really educational. I came to this because I was reviewing my vector calculus course and I'm very confused about why is the definition of line integral and so on. This video gives me insights about the essence of derivatives & integration.
I was able to solve these questions mathematically as taught by college profs., but never actually got the intuition of how things are flowing geometrically. Thanks a lot for explaining in such an intuitive way!
Glad that you can see the intuition now!
Thanks for the time spent in creating and sharing this video with meaningful insights of linear algebra, calculus, etc. Math is amazing and I’m glad we’re living in the time where deep math concepts can be explained clearly with aid of animations. Cannot judge all math professors for not having these tools decades ago and have to explain these concepts. But man , it really does a big difference.
Thanks so much for the compliment!
wanted to just learn jacobian, but learned about linear maps and integral region mapping along the way, so cool!
Thanks so much for the appreciation! They are related concepts, so you would more or less have to understand all those concepts at the same time anyway.
I did a course called dynamical systems and chaos in my second year of undergrad, and the ideas were extremely impactful but I had very few opportunities too apply them through the rest of a pure maths degree. In particular, linear approximations of non-linear approximations to inspect critical points for stability, bifurcation etc. This was the the method though
Thank you for this masterpiece. I think that is the best maths video i've seen so far. The amount of understanding that you provided me with this video🤯. Keep doing this amazing work!
I love watch those type of videos. I remember when I took Calculus II in my undergrad in Statistics and had to use these jacobians to change the coordinates. This link with linear map was awesome!
As a bonus, your explanation at 19:00 also provides a nice demonstration of *why* the chain rule works. That is something I only truly figured out (beyond memorizing it and knowing *that* I had to use it) over the past year or so of casual thinking about math, after the end of my formal education!
Yes, that's also something that I have to think a lot more before making this video, because I never came across this explanation before, and had to think of this myself :)
It’s been so long since I took multivariate calculus and linear algebra, so I definitely appreciated the little refresher on determinates and linear maps
Your video series on complex calculus and this one has now given me an amazing visual understanding of derivation and integration and the connection between complex and real derivatives. THANK YOU
Great to hear!
This is crucial in understanding how to develop boundary fitted coordinate systems and grid transformation metrics in the field of computational fluid dynamics. When implementing a finite difference discretization on a non-rectangular physical grid it is necessary to transform the irregular physical grid to a rectangular grid in computational space. The transformations require the Jacobian! Excellent explanation ! Thank you
What a brainful
I'm going into my second year of undergrad in a few weeks. I can almost guarantee I will be referring back to this video once I get into the weeds of my courses. Thank you for making such digestible (and entertaining) videos, dude!
Hopefully it will be helpful!
I was smiling with resentment the whole video.. after aquiring master degree in theoretical mathematics, I realized I never really understood the concepts I know how to calculate the minute I woke up. That goes to the quality of my university, professors (with some exceptions) and my own will to go to the bottom of rabbit hole.
Thanks for the appreciation!
I myself was not taught with this intuition either, so it just really takes a lot of time to actually think it through and thoroughly understand it, and to come up with a good intuition.
@@mathemaniac i can understand
This is such a nice way of thinking about u-substitution. I only knew the usual proof using the product rule, but that barely gives any geometric insight. Thank you for this visual intuition for something I thought was a purely analytic concept!
I assume you mean chain rule? Nonetheless, thanks so much for the appreciation! I myself didn't know about this insight before this video either, and I actually thought hard about it and came up with this explanation. Glad that you enjoyed it!
6:26 why does the Jacobian Matrix for f(x) = x^2 equal 3 instead of 2?
It is actually (2a) at the point x = a. And so the Jacobian matrix changes from location to location. The animation just shows that the point is x = 1.5, and so the Jacobian matrix there is (3).
@@mathemaniac Thank you so much for taking the time to reply. Your explanation is exactly what had me confused but now I understand! Thanks again.
This is absolutely a work of art. It bridges the gap between intuition and practical notation with a splash of simple and beautiful.
Thanks so much for the compliment!
I learned about this at the end of last semester. It was a lecture about applying higher mathematics (I'm in first semster) in a physical context and so we spent quite a lot of time on multidimensional integrals. At the end I had an intuition where these formulas came from but it would be much quicker I think if I had found this video earlier. So good stuff, your explanations are really helpful I think
Well shit, here I was suffering through the Wikipedia definition for ages, when you come along and tell me that the Jacobian is just the best linear approximation for a function at a given point... So much more intuitive!! Thank you!
Thanks so much for the appreciation! Wikipedia does have this kind of intuition, just not in the Jacobian page, which is kind of strange actually: en.wikipedia.org/wiki/Derivative#Total_derivative,_total_differential_and_Jacobian_matrix
Yeah. Reading intuition off of equations really is an Art. It's a way of seeing beyond the formalism, which kind of is what makes maths so strong, but also very difficult to digest.
@@carl6167 It really is a valuable and pretty rare skill. For any important equation, I try to understand it by asking "How would I explain this to a child? To a high-school graduate? To an upper undergrad?" Pretty much just anyone who knows less than I do about it. That helps me get an intuition for the equation - where it comes from, how to use it.
@@joelcurtis562 the Feynmann method is quite cool because of that.
Thank you to much for this setting :It helps me to have a good representation of the concept of jacobian and now i understand it deeply!Thousands thanks again !
Glad that it helps so much!
Thank you for the amazing content. Channels like yours have been an eye-opener for me in mathematics
Thanks so much for the appreciation!
and how many degrees have your eyes opened my friend?
That last section of the video blew my mind. I always understood the concept behind polar coordinates, especially their necessity for easing problems. But I don't think my college classes ever delved into the linear algebra explanation for it. Really cool stuff!
Thanks for the effort in creating these animated videos. They make math infinite times more enjoyable. I believe every math class should be like this. 👍
Excellent work! Just curious, are you using 3B1B's graphics framework for your visualizations?
Regardless, love your videos and can't wait to see more!
Thanks for the appreciation!
Not really - as said in the description, I will probably do a reveal of how I make these videos in the future :)
I feel that about 11 min in, the material rushes forward more rapidly than the preceding material. I stopped the video there and will return to it at a future time. I guess I missed the part where why I would want to think about this as a map.
If I understand it correctly -> linear maps are easy, its just simple transformation and scaling. You can look at any 2D object as linear map as long as you zoom in enough. So you can forget about curves and just use simple transformation and scaling. So if you transform to polar coordinates you keep them information about diameter and angle in the limits but you look at dr*dtheta as rectangle instead of wedge.
This makes so much more sense then 2 first years on my faculty through mathematics 1,2 and strength of materials. These transformations are very important in engineering science and using a dull textbook is not hettinf it close to students. I only came to understand at 28 through little trial and error at work what i was learning with no reference at 22. Only then it made sense and i was lucky to have come across it again.
Glad that you like the video!
thank you for what you do. I started college with a biochemistry major, but added on math because I fell in love with calculus.
UA-cam's algo is getting good lately. This was a term/topic that has been coming up in other studies of mine recently and your explanation was thorough and illuminating. I can think of many applications for this new knowledge. Thank you!
Thanks for the appreciation! Glad that it helps!
Same here!!! Studying for my Calc 3 test.
This is a brilliant visualisation and analysis of matrix transformations
Thanks so much for the appreciation!
Great content. The concept of "linear map approximation" connects dots... now I know how to identify the "skeleton" of the linear map, which leads to the Jacobian... no more confusion on which indices run horizontal/vertical ... there are 4 possible combinations, and I was never able to remember it... Thanks!
Glad to help! It is confusing to memorise which variable to differentiate with respect to, but this hopefully helps!
You made a few things clicks in my head, you're a really good teacher
Before this video I only knew Jacobin and Jacobean. Now I also know Jacobian!
Clear, concise, and well-done. I wish I had this 20 years ago!
Thanks so much for the appreciation!
"clear, conscience and well done steak" used to be my racist great uncle's motto. I wish he could have read your comment 20 years ago before he succumbed to a wasp sting.
Sir, you are a GENIUS. Thank you so much for your time and effort, this video clarifies many topics all at once. It was really a profound explanation that clarified many doubts regarding numerous topics. Thank you so much again for this video and keep up!!
Glad to help!
This is one of the best videos on this topic. All it was missing was a really cool example of how you can change the integral to something easier. Polar to Cartesian and vis versa is easy to teach, but there are some cool ones that I have no idea how they were done that deal with non standard shapes.
Thanks for the appreciation!
I don't know of any "cool" example for other changes of variables though. The reason why I specifically covered Cartesian to polar is simply that this is what we are going to need for the next video.
I remember when you first made the announcement that you were starting a channel on Quora. I had been reading you for a while then so I’m very happy that you are getting some attention now :)
Thanks! You are an actually OG fan haha :)
One word.amazing！ I come from China.And I major in math.I feel you just did a great job！❤❤❤
Two words: simply brilliant.
Loved the way you explain these topics.
Thanks!
These videos are a joy to watch. Thank you!
In chapter 2, the scaling is 3. When the function is f(x)=x squared. Do the derivate is 2x. I couldn't figure out why the scaling is 3 instead of 2.
It took reading a lot of comments to realise:
1) The red point is 0, so the test value is 3. So the derivative here is 6.
2) BUT The points are apparently only 0.5 apart, so the scaling is 3.
Nope, I'm still not getting it. I'll come back after lunch.
The function f(x) = x squared has the derivative 2x for all x. So, when x=3, the derivative is 6. What does this mean? Consider x=3.1. f(3.1) = 3.1 * 3.1 = 9.61. Here, a change in x of 0.1 from 3 to 3.1 causes a change in y of 0.61. So, f is stretching the distances between these two points, 3 and 3.1, on the x axis by a factor of approximately 6. I think this is what the author here should be mentioning. Now consider x=--2. f(-2) = 4. f(-2.1) = 4.41. A change in x of -0.1 from -2 to -2.1 causes a change in y of 0.41. So, f is stretching distances in x here by approximately -4. Note that the derivative of f(x) at x = -2 is exactly -4.
At 6:28 the value of a is 1.5.
1- The function is f(x) = x squared.
2- Near the point 1.5, the function is aproximately a line(the tangent line) --> g(x) = 3x-2.25
He choses points next to 1.5 with a distance d between each other and calculate their value in the line:
Points: 1.5-d , 1.5 and 1.5+d
Values: 2.25-3d , 2.25 and 2.25+3d
6:28-6:35 After mapping a to 2.25 through the function f(x) or g(x), its neighbours 1.5-d and 1.5+d are mapped through g(x). Its neighbours were at a distance d, and now are at a distance 3d.
I'm sure a=1.5 because at 7:39 he makes a=-1 and f(a) = 1
If a was -2, f(a) would be 4 and would be to the right of its actual value.
f(1)=1 which is the only point that doesn't change (6:07).
The yellow mark before one approximates 0(because 0.5 squared is smaller than 0.5)
That's a lot of information and a lot of great insights for under 30minutes of video. With this high density, you can easily map this video to 50 pages worth of algebra textbook :-) Well done and thanks!
Thanks so much for the compliment!
These were the last lectures on calculus at the Physics Department, 2nd year. This representation is a must for calculus.
I just finished up Vector Calculus, and this is video has very much expanded my understanding!
The derivative in XY(2D) plane can be seen as "slope" and in the XYZ(3D) plane it is seen as "area". The same analogy is applicable for integrals: XY plane represent "area" and in XYZ plane "volume".
Amazing video!!
I'm taking a Calculus III course right now. This surely is going to help :).
Thanks! Hope it does help in your course!
Of course I enjoy this nice video .
your effort in preparing this video and explaining this complex math topic deserves our full gratitude. therefore I subscribe, I like and I share !! WELL DONE. Good luck for other videos !!
Wow thanks so much for your kind words!
Mathemaniac you are welcome sir. you sincerely deserve them.
Wonderful piece of explanation. I remember performing the computations in multivariable calculus at university without understanding the concept of the Jacobian. I guess content like this requires lots of preparation so thanks a lot
Thanks so much for the kind words! This video did take a lot of time and effort to make, so thanks for recognizing this!
In Chapter 2, Derivatives in 1D, did you mean x^3 instead of x^2 if you want the scaling factor to be 3 ?
x^2 will have a derivative of 3 at 3/2, so you can imagine that he's chosen a = 3/2 when showing a scaling factor of 3, in the same way that it's implied a = -1 when the scaling factor turns out to be -2.
@@kikones34 Indeed, but this is like reverse engineering to give meaning to something that made no sense. The portion of the video talking about the neighbors of a point a being mapped to 3 times the distance under the mapping f(a)=a^2 is super sloppy and can only create confusion. Either choose a generic point a and map it to 2a or illustrate the concept by choosing the specific cases 1.5 and -1 (which are never mentioned).
@@italnsd Yeah, I'm not sure why he didn't include the values in the number line, or chose more straightforward examples.
This is a fantastic lecture with neat demonstration
Thanks so much for the appreciation!
Best infographic and visuals in a maths video so exciting to watch keep up
Thanks so much for the appreciation! Glad that you like it!
Really good explanation. Especially on that linear transformation part. Thank you. ✌️
This is some _quality_ content! It indeed looks like a ton of effort went into making it.
Thanks so much for the appreciation!
I'm so happy that 3b1b created manim as it's put to good use by many
Glad you enjoyed the video! Actually I didn't use Manim - will reveal how I make all these videos in the future.
@@mathemaniac thanks for clearing
@@mathemaniac I'm already crazy wanting to know it...
@@mathemaniac wait you have your own visualization library? Looking forward to that
I never really understood determinant until I watched your video. This is amazing. Why can't schools teach it this way? Nobody mentions that determinant is the scaling factor in linear maps!
Vídeo incrível! Tudo o que eu precisava!
I think you survey could be improved a lot. I don’t think I’m the only one who have learned a lot of math through the math community (for example by watching math videos like these instead of taking classes). Therefore, I can’t really tell which subjects I have mastered and which I have learned all the basics of (perhaps I’ve missed something crucial). You could perhaps ask about what terminology we are familiar with, which concept we understand, and what impression a problem gives us (easy, solvable or scary). It would be easier to interpret, more useful and fun.
Thank you for the video! I’m eager to see the final of this series.
You are not the only one in thinking that! I did originally want to make the form like what you said, but there are way too many concepts within each option, and I felt that it wouldn't gather as many responses because the form would be far too long, just for a UA-cam channel, even though it is more useful to me. This is why you would have the option to tell me in more details what you actually know later on in the form. In that case, tell me that you have mastered the basics, but perhaps not completely.
Maybe I could rephrase the options a little bit so that it becomes a bit clearer.
Thanks for the appreciation of the video though!
@@mathemaniac just post a google form then you can make multiple questions
Very good video! I had several years of calculus as an undergrad and learned all this stuff years ago, but I still like how you presented it. Linear maps are indeed a useful way to think about differentiation and integration, even in 1D.
Glad that you enjoyed the video! Well, derivatives *are* linear maps in higher dimensions, so it is probably the way to learn about calculus anyway :)
And is the Base for the Math Language of Physics and Engineering of the Future ....The Differential Forms or The " Algebraic Geometry" or Cliffor Calculus....
david terr or istanbul. Dont do it david
INTEGRALS AS MASS MAKES SO MUCH SENSE, I’VE NEVER THOUGHT OF IT THAT WAY. YOU ARE A GENIUSSS
Also thank you so much for this vid, Jacobians have been tripping me up
this really helps visualizing natural coordinates in finite element method, thanks!
This is amazing man. thanks for making this. you're another 3blue1brown, Zach star in the making.
Wow, thanks!
That 'or is it ?' just destroyed my whole life
This really is a great video. I am only understanding it now, on my third watch. I watched it the first two times in high school, and now I am watching it again after learning basic linear algebra, partial derivatives, directional derivatives etc.
Great video, halfway I was wondering what could the intuition be for transforming areas into each other. And the best one I found was that if you want to calculate areas using a Mercator projection of the earth you have to use a different factor at different distance from the equator.
Thanks for the appreciation! I would not say that the Mercator projection is the best intuition though - the globe is (roughly) spherical, so there is some sort of 3d structure to it. What I am trying to say is that you are mapping a region in the 2-dimensional plane to a patch of surface in a 3-dimensional space, which is not the best representation of what we have here. The more honest intuition is really just thinking about transformations of the plane, even though the Mercator projection is a bit more interesting.
Lots of good intuition here. I would have never learn this without UA-cam.
Glad to hear this!
This is incredible !
I really want to know how we can use that to get the average distance between 2 points on a disk !
Thanks for the appreciation! The next video is probably out in a week or so, so stay tuned!
Really great - you have a real talent for explaining complex ideas. Liked and subscribed.
Thanks so much for the appreciation!
I would have loved your videos in my University time. Excelent material, congratulations!
Thank you for the video and its subtitles!
Thanks for the appreciation!
I'm only going to say Amazing dude!, I'm an undergraduate student in maths and the books some times are really hard to digest, have a picture of the sightseen helps a lot in the abstraction. Thanks!
Thanks for the appreciation!
I always knew I missed this information, but never knew where to find it. Thank you
Glad it helps!
Nice! Its been a while since i seen linear algebra and multivariable calculus but this video is so good that i remembered all the stuff i needed justo to understand better the visualization your brought.
Also, you said you were not using Manim. Which software you are using then? Would help me a lot my students use this kind of visual.
Again, well done!
Thank you so much!
As for the software, I will probably reveal it in the future - but if you are not patient enough, you can also email me about it :)
This is actually the first time that I've heard linear transformations being called linear map.
Pretty good lecture though.
Do you have any recommended path for studying linear algebra alone(from dummy to pro)?(courses, playlists, books anything)
Thanks for the appreciation.
3Blue1Brown's series on linear algebra is quite good for intuition.
@@mathemaniac After that I suggest MIT OpenCourseWare 18.01
The video recalls my university life of almost 50 years ago.
My thoughts exactly!
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his is stunningly good. Really makes simple a sometime baffling subject. I found this very helpful. Thank you!
how have i never heard this interpretation of a determinant.. brilliant video
I was taught to compute blindly all these nasty integrals, I feel these mysterious methods have been unlocked
Glad to help! Nobody should be taught to blindly apply something if they don't understand!
Very good explanation of the Jacobian, differentials/derivatives and integrals! 😊
Except it is inelegant to define 2-d integrals with the ugly absolute value built in, they should be allowed to be negative, just like the 1-d integrals, using signed bivectors to describe the density change as a linear approximate function of change in the plane.
f(x,y)*dx*dy is a bivector dependent on đ_2(đ_1f(x,y)/đx)(x,y)/đy = đ_1(đ_2f(x,y)/đy)(x,y)/đx whenever this identity holds, which i think it does when we define f from any of these, using integration. đ means partial derivative.
I don't know how to reason about integrals related to derivatives whenever these two partial derivatives do not commute.
Also you conflate explicit functions with expressions having implicit functional dependencies on free variables, but almost every other mathematician does that also, even though it is completely wrong.
Like for example f(x,k) is not the same as f, nor is f(x,y) the same as f. And of course g(x) is not the same as g.
So we can't correctly write đg/đx for (g(x),h(x)) = f(x,k). We can write đg(x)/đx though.
Instead we should write đ_1f(x,y)/đx = đg(x)/đx and đ_2f(x,y)/đx = đh(x)/đx, where đ1 refers to the linear change in the first coordinate, and đ2 refers to the linear change in the second coordinate of a two dimensional expression implicitly (functionally) dependent on some variables we differentiate with regards to.
It is correct though to say that f = (x |-> f(x)) = (y |-> f(y)) = (k |-> f(k)), as long as these variables (x, y, k etc) are not free in f. If say g(x) is defined as f(x,k) that is g = (x |-> f(x,k)), then k is free in g, so we are not allowed to say g = (k |-> g(k)), since this would change g(x) to f(x,x) instead of f(x,k) for a fixed k.
The symbol "|->" denotes function abstraction here, it is what changes an expression with implicit functional dependencies on a particular free variable into a function (or function expression) not dependent on this variable, because it is now bound.
In lambda calculus, this concept is denoted by a lambda letter, but i think the "maps to" symbol is far more "intuitive"/obvious in its meaning to most mathematicians.
f(x,y) may be abstracted into (x,y) |-> f(x,y) for a function with two paired arguments, or into x |-> y |-> f(x,y) for a function with two arguments taken one by one, that is a function of one argument, returning as result a new function of one argument. This latter form is sometimes called "currying" after the mathematician Haskell Curry, famous (together with logician William Alvin Howard) for the Curry-Howard isomorphism between pure functional computer programs and mathematical proofs of intuitionistic (constructive) logic.
We may also abstract f(x,y) to just x |-> f(x,y), giving us a function of one variable, this has the same effect as considering y a constant, compared to the function (x,y) |-> f(x,y) or its variants. Not abstracting either x or y in f(x,y) is the same as considering both as constants. At least until we start differentiating or integrating which effectively rebinds some of our variables to new expressions, through abstraction, function manipulation, and application of functions to the old variables' expressions, with their old implicit dependencies intact.
Whether a variable is constant or not depends on whether we manipulate expressions containing it inside or outside of the scope of its binding.
And which way we bind it: through functional abstraction (of itself, or of one of its dependent variables), or through setting it to a specific constant, like 0 or 1.
So we should not ask whether a variable is constant or not, but where it is constant and where it is not constant.
I also would like to see integrals defined for functions with multiple results, like f where (u,v) = f(x,y).
Also integrals in two variables ought to be able to be integrated in steps, first with respect to say x, then with respect to say y. Using definite integrals, this should reduce the number of arguments of the new function we receive from the old function each time, until we are left with a "fixed" value. Writing S for integral, we should thus be able to define S_(x=a,y=c)^(x=b,y=d):f(x,y)*dx*dy as
S_x=a^x=b:(S_y=C(x)^y=D(x):f(x,y)*dy)*dx =
S_y=c^y=d:(S_x=A(y)^x=B(y):f(x,y)*dx)*dy
provided this identity holds.
This would be perfectly analogous of differentiating a function two times, first with regards to one argument variable, then to the other argument variable.
Finding and formulating proper theorems relating these differentials and integrals in multivariate calculus is an important task, although i think most of the work is already done, it just needs to be expressed correctly with non-confusing syntax and clear, rigour semantics.
Lastly it would be nice to generalise these concepts (Jacobian, differential, derivative, integral) into functionals a la functional analysis, where we have higher order functions taking (say analytic) functions as arguments, and we can derive the higher order functional with regards to a variable representing an explicit function itself, rather than merely its value at some point dependent on its argument.
dF(f)/df is thus something completely different from dF(f(x))/dx, or dF(y)/dy where y = f(x). In the first case F has type (|R -> |R) -> |R, while in the second case F has type |R -> (|R -> |R) which is similar to |R² -> |R.
In this context it is absolutely clear that we can NOT say that y = f, because dF(f)/df is completely different from dF(y)/dy when y = f(x), the latter y in F(y) might be interpreted as the constant function g = z |-> g(z) = y where y is independent of z, in this case g is dependent of x, because y is, but x is independent of z, thus g is a different function from f, g is constant in its argument while f is generally not constant. And F evaluated at f and g respectively may yield different values. More importantly F' = (h |-> đF(h)/đh) may yield different values when evaluated at h = f, compared to h = g, for g dependent on f and x in the sense of g(z) = f(x) for all z. Provided F' is defined, that is there is a well defined total, or directional derivative of F.
Comparing with ordinary multivariate functions, let's say we have a function f where we may say z = f(x,y) and thus f has type |R² -> |R. If we set v = (x,y), we may express z as z = f(v). Now we can define the directional derivative f'(v) = df(v)/dv as = (Nabla^T)(f)(v). Which may then be evaluated at any particular vector w, giving the linear approximation of the difference between f(v+w) and f(v), thus the derivative in the direction of w at the point v. This generalises naturally and effortlessly to the Jacobian, without needing to fix any particular inner product, while the gradient Nabla demands an inner product, in order to change this vector function f'(v) into a vector Nabla(f)(v) describing the direction of fastest change, and its rate. I suggest using the notation f' for the Jacobian J(f). Of course we need to use the correct syntax here as well and differentiate (pun unintended) between say J(f) and J(f)(v) = J(f)(x,y). Usually people write J(v) when they actually mean J(f)(v), since the Jacobian of f depends on f, and we may simultaneously use both J(f) and J(g) for different functions f and g. Remember that J(f)(v) is a matrix, which need to be multiplied with a vector w, like J(f)(v)*w in order to get the rate of change of f(v) in the direction and rate of w, near the point v. If we want rate of change to make sense, we usually want to normalise the length of w, which requires an inner product, or equivalently a norm. We might also write J(f)(v) as df(v)/dv, if we understand the derivative division to actually mean linear abstraction, that is df(v) = J(f)(v)*dv (matrix multiplication being linear application), where dv = w is the direction of change from v that we are interested in.
holy shit I wasn't expecting a whole essay
I think it’s worth mentioning the limitations but the point of the video is to give visual intuition. Perhaps you can make a corresponding video giving visual intuition using bivectors instead.
It is very common and natural to over-simplify in math to make it more accessible to students and then later add levels of complexity (and what one considers complex really varies) but we usually state the assumptions at the beginning, yeah?
I definitely saw this during the University days, I totally forgot it too. A couple of years ago I was looking at some video game physics engine article and they do mention that, paraphrasing, “to solve the collision and final orientation between to objects, we need to find the ‘Jacobian’ (…)”, I stopped there as I had no clue what were they talking about.
Now I think maybe they were referring to derivatives style, “what is the best solution at this iteration time”, as with everything video game engines the whole thing advances in tiny amounts of time dt (an ever going integration the easy way numerically speaking). And when some precision is needed they do several smaller iterations within that dt that just occurred on a single frame
This was wonderful and very well explained. Thank you so much!
Thanks for the kind words!
If you are still waiting for this to come out, you can drink something. stay hydrated... when you feel thirsty
Nice, thanks for sharing!
Thanks for the appreciation!
25:45 that was the best explanation of the scale factor r and rdrd\theta. Never understood until now.
Thanks!
Hi, What is the software that you are using? Loving the content and presentation. Always found DifferentialGeometry and Integration theory so beautiful for exactly that reason how you can visualize/imagine what's happening