This video took a huge amount of time and effort to produce, so if you want to and can afford to, support this channel on Patreon: www.patreon.com/mathemaniac The Google form is also linked here so that you don't have to read the description: forms.gle/QJ29hocF9uQAyZyH6 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 TH-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!
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
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 TH-cam.
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.
@@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.
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
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.
@@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 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.
I will need to watch this again, perhaps many times, and take notes, but my mind has been expanded already. Thank you so much for your generous work. God bless you.
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 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.
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.
Teaching calculus & linear algebra through the lens of analytic geometry is greatest missed opportunity in the world of teaching. Thank you for presenting these ideas in an intuitive way
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.
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
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.
Every linear algebra class should have this video as a prerequisite! Wish I had this video when I was in high school learning about matrices. Please don't stop creating these videos around linear algebra and various matrix computations!
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!
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
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.
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 :)
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'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!
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.
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.
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
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!
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!
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
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!
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!
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
TH-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!
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!
"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.
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.
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 :)
Although most professors should know about the materials, they didn't have the (animation) tool or the time to explain the Jacobian matrix in detail in class. This is an excellent video. Thanks!
my professors are literal animated corpses and their bodies are essentially immaterial. And my 32nd cousin 56 times removed is called Jacob Womb and our last common ancestor killed the last arctic tortoise. Thank you for your excellent comment which made me remenisce about the good old days. Thanks!
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 !
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!
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
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).
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)
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!
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".
This video reminds me to the Change of variables theorem. One of the longest theorems that it took to prove in my advanced calculus course. Really nice video, I like how you also slowly approach the notion of derivative in mathematical analysis and topoplogy.
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!
yeah man my mathematical intuition is engored and throbbing right now. I will force it upon whatever maths video shows up next. If only I can stop commenting under random people's comments
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).
I'm not sure if it's a difference of usage, as I'm an engineer and not a mathematician, but that's exactly how we use Jacobian determinant all the time in finite element method calculations. To me, it's mostly just a factor of a tetrahedron (for example, or whatever the shape of the element is) that will tell me if the shape of the element will give me accurate results. Meaning, the factor describes the shape, which describes roughly the distribution of integral points, in which calculations will be performed.
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 TH-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!
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.
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....
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!!
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 so much for the appreciation! Your comment was for some reason held for review by TH-cam, so I didn't see this comment. I actually feared that it could get copyright claimed, but fortunately it didn't.
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. 👍
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.
This video took a huge amount of time and effort to produce, so if you want to and can afford to, support this channel on Patreon: www.patreon.com/mathemaniac
The Google form is also linked here so that you don't have to read the description: forms.gle/QJ29hocF9uQAyZyH6
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 TH-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 TH-cam.
Same here
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.
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
@@KingAntDaProphet 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
I love how TH-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.
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
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
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!
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
I will need to watch this again, perhaps many times, and take notes, but my mind has been expanded already. Thank you so much for your generous work. God bless you.
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 is one of the best, if not *the* best video on the Jacobian available on TH-cam. Wonderful job here.
Thanks so much for the compliment!
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 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
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
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.
Teaching calculus & linear algebra through the lens of analytic geometry is greatest missed opportunity in the world of teaching. Thank you for presenting these ideas in an intuitive way
You might like the book Calculus with Analytic Geometry by Simmons
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!
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 :)
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.
Every linear algebra class should have this video as a prerequisite! Wish I had this video when I was in high school learning about matrices. Please don't stop creating these videos around linear algebra and various matrix computations!
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
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!
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
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.
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!
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 :)
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'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!
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!
One word.amazing！ I come from China.And I major in math.I feel you just did a great job！❤❤❤
I wish we had these videos during our physics undergrad 4 years ago
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!
Before this video I only knew Jacobin and Jacobean. Now I also know Jacobian!
This is a brilliant visualisation and analysis of matrix transformations
Thanks so much for the appreciation!
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!
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 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!
Lots of good intuition here. I would have never learn this without TH-cam.
Glad to hear this!
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!
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
You made a few things clicks in my head, you're a really good teacher
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!
Best explanation of the Jacobian I've ever seen.
Great animation along with it too.
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!
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
derivative is scaling factor near f(x), then how the scaling factor was written to be 3 when derivative at 3 for f(x)=x^2=>f'(x)=2x=6
哇，太棒了，虽然没有完全解释所有东西，但给了我最最重要的灵感。我刚好被这个问题折磨好多天了。
謝謝你的讚賞！
This is amazing man. thanks for making this. you're another 3blue1brown, Zach star in the making.
Wow, thanks!
i love when i found great math channels, i'm definitely subscribing.
Welcome aboard!
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?
thank you for what you do. I started college with a biochemistry major, but added on math because I fell in love with calculus.
TH-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.
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!
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.
Top notch quality right here, extremely underrated amazing job on this video.
This is a fantastic lecture with neat demonstration
Thanks so much for the appreciation!
You are a math maniac! I will support the channel its amazing your work. Thanks!
Much appreciated!
The video recalls my university life of almost 50 years ago.
My thoughts exactly!
I just finished up Vector Calculus, and this is video has very much expanded my understanding!
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.
Thanks ...from 🇦🇷....your efforts will be remembered always...
Thanks!
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 :)
Although most professors should know about the materials, they didn't have the (animation) tool or the time to explain the Jacobian matrix in detail in class. This is an excellent video. Thanks!
my professors are literal animated corpses and their bodies are essentially immaterial. And my 32nd cousin 56 times removed is called Jacob Womb and our last common ancestor killed the last arctic tortoise. Thank you for your excellent comment which made me remenisce about the good old days. Thanks!
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!
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!
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!
These videos are a joy to watch. Thank you!
It cannot be overstated how the density of intuition is uniformly high over the course of this video.
Thanks!
my man mustered up quite the sentence there. whoo
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!
This is some _quality_ content! It indeed looks like a ton of effort went into making it.
Thanks so much for the appreciation!
i took linear algebra and never learned these geometrical intuitions for linear transformations. thank you very much
If you are still waiting for this to come out, you can drink something. stay hydrated... when you feel thirsty
Haven't needed this in over two years yet here I am watching this on a saturday
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.
Really good explanation. Especially on that linear transformation part. 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)
agree this is confusing...
As soon as the narrator started talking I knew some serious learning was about to go down
Vsauce music incoming in 1...2.....3...
Right after a dramatic "Or is it?"
It got a good laugh out of me.
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!
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".
This video reminds me to the Change of variables theorem. One of the longest theorems that it took to prove in my advanced calculus course.
Really nice video, I like how you also slowly approach the notion of derivative in mathematical analysis and topoplogy.
Maybe I am only silly one here. Matrix (3) or scaler 3 came out from x^2? Doesn’t slope depend on 2x ? @7:22
T
his is stunningly good. Really makes simple a sometime baffling subject. I found this very helpful. Thank you!
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!
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!
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 :)
this seems way better than just reading the theory. thanks man it really helps a lot.
yeah man my mathematical intuition is engored and throbbing right now. I will force it upon whatever maths video shows up next. If only I can stop commenting under random people's comments
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.
I'm not sure if it's a difference of usage, as I'm an engineer and not a mathematician, but that's exactly how we use Jacobian determinant all the time in finite element method calculations. To me, it's mostly just a factor of a tetrahedron (for example, or whatever the shape of the element is) that will tell me if the shape of the element will give me accurate results. Meaning, the factor describes the shape, which describes roughly the distribution of integral points, in which calculations will be performed.
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 TH-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
These were the last lectures on calculus at the Physics Department, 2nd year. This representation is a must for calculus.
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
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!
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!
This is awesome. Two years of calculus summarized clearly in 30minutes
Glad that it helps! I wouldn't think this is actually summarising everything in calculus, but hopefully it helps understanding!
Just found this channel Great content (I loved the Vsauce music)
Thanks so much for the appreciation! Your comment was for some reason held for review by TH-cam, so I didn't see this comment. I actually feared that it could get copyright claimed, but fortunately it didn't.
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. 👍
Thank you for the video and its subtitles!
Thanks for the appreciation!
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.