What is Jacobian? | The right way of thinking derivatives and integrals 

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 RU-vid 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 love how RU-vid is now exploding with math channels

Came for the Jacobian, stayed because - almost by accident - _you gave an intuitive explanation of the chain rule!_

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.

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

After 40 years of college, finally a good explanation .

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.

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!

vsauce narrative + 3blue1brown animation

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 🙏

I'm in last year of my Mathematics degree, and I feel I just started understanding determinants and Jacobians right now!! Thanks a lot

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

This is the only time I truly understand the Jacobian geometrically, I wish I could've bumped into this video sooner. Great stuff!

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.

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

This is one of the best, if not *the* best video on the Jacobian available on RU-vid. Wonderful job here.

wanted to just learn jacobian, but learned about linear maps and integral region mapping along the way, so cool!

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.

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!

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.

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!

I'm so happy that 3b1b created manim as it's put to good use by many

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

Before this video I only knew Jacobin and Jacobean. Now I also know Jacobian!

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!

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 is a brilliant visualisation and analysis of matrix transformations

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

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!

oh gawd, as soon as you showed matrixes I was nope. I killed it in Calculus I, II, III, and IV. I went on to do just as well in differential equations. When I hit Linear Algebra though, I was brought to my knees. It just wasn't intuitive to me. Calculus made sense. Linear Algebra was my Kryptonite. Honestly I don't think learning Linear Algebra is the approach a beginning student should take to understand differential and integral Calculus. Coming at it from an Analytical Geometry approach is much more intuitive. It's easy to see rotations around an axis, or gradients, or areas under the curve. It's not easy to visualize an abstract idea like a matrix of numbers.

One word.amazing！ I come from China.And I major in math.I feel you just did a great job！❤❤❤

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.

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.

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!

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

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 absolutely a work of art. It bridges the gap between intuition and practical notation with a splash of simple and beautiful.

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!

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 was taught to compute blindly all these nasty integrals, I feel these mysterious methods have been unlocked

6:26 why does the Jacobian Matrix for f(x) = x^2 equal 3 instead of 2?

Thank you for the amazing content. Channels like yours have been an eye-opener for me in mathematics

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.

RU-vid'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!

That 'or is it ?' just destroyed my whole life

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 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

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!

Lots of good intuition here. I would have never learn this without RU-vid.

I just finished up Vector Calculus, and this is video has very much expanded my understanding!

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. 👍

If you are still waiting for this to come out, you can drink something. stay hydrated... when you feel thirsty

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

You made a few things clicks in my head, you're a really good teacher

As soon as the narrator started talking I knew some serious learning was about to go down

The video recalls my university life of almost 50 years ago.

Clear, concise, and well-done. I wish I had this 20 years ago!

thank you for what you do. I started college with a biochemistry major, but added on math because I fell in love with calculus.

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.

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 !

Vsauce music incoming in 1...2.....3...

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

These were the last lectures on calculus at the Physics Department, 2nd year. This representation is a must for calculus.

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!

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 :)

This is fantastic. We do essentially the same in Finite Element approaches to solve many engineering problems.

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!

This is some _quality_ content! It indeed looks like a ton of effort went into making it.

This is a fantastic lecture with neat demonstration

These videos are a joy to watch. Thank you!

Two words: simply brilliant. Loved the way you explain these topics.

In Chapter 2, Derivatives in 1D, did you mean x^3 instead of x^2 if you want the scaling factor to be 3 ?

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!

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!

This is amazing man. thanks for making this. you're another 3blue1brown, Zach star in the making.

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".

Really good explanation. Especially on that linear transformation part. Thank you. ✌️

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!

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.

Thank you for the video and its subtitles!

this really helps visualizing natural coordinates in finite element method, thanks!

i took linear algebra and never learned these geometrical intuitions for linear transformations. thank you very much

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.

Amazing video!! I'm taking a Calculus III course right now. This surely is going 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.

25:45 that was the best explanation of the scale factor r and rdrd\theta. Never understood until now.

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!

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.

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!!

how have i never heard this interpretation of a determinant.. brilliant video

Nice, thanks for sharing!

Just found this channel Great content (I loved the Vsauce music)

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

Despite of other social media, youtube could be a powerful engine of culture and creativity!

Why I dropped out of school: Some subjects were like teaching a foreign language but not knowing what the words means. Math was like that. I knew how to do it but I never understood the meaning of it

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

i love when i found great math channels, i'm definitely subscribing.

Best infographic and visuals in a maths video so exciting to watch keep up