The teaching techniques used in this video are vastly superior to classical math education. The animations are a massive improvement. And the open why and how approach is refreshing.
This is the fantastic visualization of how Jacobian works. It may not be a very secrets concept for those who attended the higher level math course. But for those student new to advance calculus, this is definitely a precious gem --- Quickly guide them to the Multivariable mapping world.
I remember my cal prof at the time saying "this is the formula for the jacobian, no need to know what it is just compute it" and i never even realized the areas of the shapes don't match when going from cartesian to polar ... you just blew my mind. Thank you
I don't expect to learn everything from university. The teachers don't give me the rigor of detail that makes me satisfied. I learn more from self-study and these external sources. I just think of university as being there to give me an outline of what to learn and to hand me my diploma. To be fair, the laboratory facilities and libraries also facilitate my learning, but never the professors.
@@frieden6298I’m sorry you and many others have had this experience. This is a common/standard undergraduate experience. During my degree obtained at my university, I was very fortunate to dive into the “why”, and that motivated me to pursue graduate studies. My standard undergraduate multivariable calculus course went over the conversions from Cartesian to polar-and spherical-and discussed the Jacobian and its importance at length
A great refresher on stuff I've forgotten over 20 years. After you've been out of school for a while, you just robotically insert "r dr dθ" for "dx dy" without even considering why you are doing it. Jacobians are just things you vaguely remember when trying to code up some algorithm you found in a paper and cursing how much linear algebra you have forgotten.
As a disclaimer I am a graduate of A&M college of engineering (graduated 20 years ago and never looked back). I came across this video skeptically, and to my surprise, I'm very impressed.
There is a simpler way to think of this concept without having to understand why the jacobian is neccessary. dy and dx are graphically the deriviatives of lengths. The corresponding derivatives of lengths in polar coordinates is dS and dr, not d(theta) and dr, where S is arc length. dS = rd(theta). dA then = dy dx = dr dS = r dr d(theta). It works better with visuals, but I always found this a helpful way to understand this concept.
Understanding the Jacobian helps to generalize whatever transformation you'd like to do, regardless how much you can "visualize" the new integral. However, the geometric example using dS, as r dθ, helps to visualize "what's going on" when beginning in integral change of variables.
Brazilian book of calculus from the mathematician Hamilton Guidorizzi is the best book of calculus! He explained every detail of the Jacobian with high class!
This is the best explanation I found on YT even much more popular channels just come up with the transformation matrix, but fail to come up with a good example.
What I do not think is explained is why the determinant of this is precisely what you need to make the conversion. Maybe it is based on top of a prior video where this meaning is explained. Anyway, good video and well explained. I also like the comment about how this can be understood without Jacobians. I always have used that way of thinking, but thinking about the same topic in multiple ways is helpful
This is too good!! Why didn't I come across this video back when I was taking Calc 2 class in early 2022?!?! Thank you sooo much for this videoo! I might need it for a future encounter with the Jacobian
@@pianodan1608 I think every math department classifies it differently. With the current curriculum, my uni has four calculus classes: - Calc 1 (single variable calculus): limits, derivatives, integrals, integration techniques, FToC, IVT, MVT, trigonometric and hyperbolic functions, surfaces of revolution - Calc 2 (multivariable calculus): partial derivatives, iterated/double/triple integrals, polar/cylindrical/spherical coordinates - Calc 3 (sequences and series): improper integrals, Taylor series, sequence/series convergence (Dirichlet's test etc), orthogonality conditions, Fourier series - Vector calculus: divergence, curl, Gauss' divergence theorem, Stokes' theorem
This is a great video! Your explanation is very calm and clear, which makes it easy enough to follow what is happening in what can sometimes be a little overwhelmingly confusing math. The video really helped me understand the topic better, thank you!
Oh my god this video is amazing, such an intuitive explanation of it! It clicked to me right as you said the areas change and the jacobian is the determinant of the matrix (which represents the area of the thing, and since things are in derivatives, I’m guessing it translates to the change in area). Definitely subscribed
Excellent explanation of the Jacobian! If I'd been able to watch this 40 years ago when I first learned this, I could have learned in 15 minutes what took me a month or so to learn!
Thank you!!! This makes me finally understand a concept I was confused about for quite a while now. The visual interpretations in this video are really helpful!
Good question. The tl;dr is that the typical 2D integral is wrong and is missing a wedge product. If you include the wedge product, then deriving the Jacobian is simple algebra and nothing mystical at all. A (very long) but more complete explanation is as follows. Honestly you could make an entire video on properly explaining this this idea... First, note that the double integral we're familiar with is missing something crucial. We know from 1D integration that integrals are signed; they carry information about orientation. This is still present in 2D with swapping bounds, but it isn't present when swapping the order of integration. For example, calculate the volume of a cube, given by f(x, y) = 1 from 0 to 1 ∫₀¹∫₀¹ f(x, y)dxdy = ∫₀¹∫₀¹dxdy= 1 This is positively oriented: you're tracing out the 2D square in the x-y plane in the anti-clockwise direction (we arbitrarily call this "positive") However, if we swap our order of integration we should get something negatively oriented, but we don't ∫₀¹∫₀¹dydx = 1 This *should* be negative because it traces out the square in the opposite orientation (clockwise), but it's not. What gives? The answer is that dxdy was the incorrect area form, and should instead be dx∧dy. This wedge operation behaves like the cross product in 2D and 3D (but it also works in arbitrary dimension), so it has the rule dy∧dx = -dx∧dy. This alone fixes the problem and recovers orientation. You could also arrive at the same conclusion by trying to integrate along axes that aren't perpendicular (i.e. using infinitesimal parallelograms as the fundamental unit instead of infinitesimal squares, as both can tile the plane both should be valid in principle). The standard dxdy has no nice way of handling anything other than perpendicular axes natively, but with the wedge product dx∧dy = dx×dy = sin(ϕ)dxdy we can handle it easily. This is the same answer you'd get through the Jacobian, but presupposing that would somewhat be circular logic. Now, with that established, deriving the Jacobian is a straightforward exercise in algebra. The entire reason that the Jacobian feels like a leap is because this entire backstory I presented here is swept under the rug and only implicit in the Jacobian. This means that you are indeed taking a leap as none of this is ever explained. We know that we can swap from Cartesian to polar, therefore x = x(r, θ) and therefore dx = ∂x/∂rdr + ∂x/∂θdθ is the total derivative, and same for y. For the integral, we already directly substitute for x and y, so why don't we also just substitute for dx and dy directly too? ∫f(x, y)dx∧dy= ∫g(r, θ)(∂x/∂rdr + ∂x/∂θdθ)∧(∂y/∂rdr + ∂y/∂θdθ) Where g(r, θ) = f(x(r, θ), y(r, θ)) is just what you get from plugging in polar coordinates into your original function. This looks like a mess, but it's honestly not that bad. The wedge product distributes like an ordinary product would, so expanding using the distributive law gives something which has the form Adr∧dr + Bdr∧dθ + Cdθ∧dr + Ddθ∧dθ for the stuff from the brackets. Each of the terms A, B, C and D are just products of partial derivatives. Now a straightforward consequence of the wedge product is that dx∧dx = -dx∧dx = 0. This means that the A term and D term disappear from the expansion, and we are left with only B and C. We also note that since dr∧dθ = -dθ∧dr, we are overall left with the term (B-C)dr∧dθ, so the integral becomes ∫g(r, θ) (B-C)dr∧dθ and if we evaluate B we get ∂x/∂r ∂y/∂θ and C we get ∂x/∂θ ∂y/∂r which is exactly what you get from the Jacobian method. This is all perfectly explained extremely directly if you use the proper wedge product in your integrals. The wedge product makes the absent terms disappear and inserts the correct sign in front of C. The last thing missing is the absolute value. This is again needed because of the missing wedge product. Since drdθ has no orientation information, we must have ∂(x, y)/∂(r, θ) = ∂(x, y)/∂(θ, r). But algebraically this clearly isn't the case, so to sidestep this issue we throw out all orientation information and focus solely on the magnitude, which is what the absolute value does. With the wedge product, we instead would need to have ∂(x, y)/∂(r, θ)dr∧dθ = ∂(x, y)/∂(θ, r)dθ∧dr, which is automatically true because both minus signs on the RHS cancel out. So there's no need for an arbitrary absolute value. For the life of me I don't know why this isn't taught like this. It's not like any of this is any more difficult than standard multivariable calculus coursework.
@Bryte H I think the signed aspect isn't crucial to understand the Jacobian determinant. Neither really is the formalism of the wedge product (yes it becomes handy in dimensions greater than two, but the fundamental intuition can be garnered by just considering examples in R^2) I think a simple explanation for calc 4 students just involves them understanding a bit of linear algebra. The central intuition is that we are transforming a vector of variables (x,y) to another vector of variables (r, theta). This is a non-linear transformation, but since it is differentiable, locally, within a small range (x,y) to (x+dx,y+dy), the transformation is linear. A linear transformation can be described as a matrix, that matrix is the Jacobian. This can also be seen by introducing to students the multivariable Taylor expansion, showing that the Jacobian is the first term, analogous to a simple derivative, but for more general transformations from R^n to R^m Now that we know the Jacobian is the transformation that takes tangent basis vectors x,y to r,theta, a natural question is what happens to the area under such a transformation. Here is where you introduce to students the idea of a determinant as a signed scale factor for areas under linear transformations. The signed aspect can be understood by giving students a couple simple examples (does the transformation flip the orientation of the basis vectors?) In many ways it is just a more general, multi-dimensional, form of "u-substitution" students are already intimately familiar with.
@@afroohar Yes, the Jacobian can be explained in a consistent way through what you just described. I just don't see how this is a *better or more intuitive* explanation than simply including orientation in 2D+ integration (something we know it's supposed to have anyway) and doing basic algebra to get the answer we expect. Everything falls out for free, and orientation doesn't really make things more difficult. Doing mental gymnastics and bookkeeping to avoid orientation is significantly more confusing. At least in my personal experience, even though I learned the Jacobian explanation you described in 2nd year uni, because I could never just derive it directly it never stuck with me. If I was expected to substitute x and y and I could come up with expressions for dx and dy, then why couldn't I just directly substitute dx and dy?! "Am I doing something wrong?" "Do differentials not obey algebra?" I had vague notions of 2D Infinitesimal area elements and sometimes remembered there was an absolute value in the formula for some reason, but the fact that none of it seemed like a proper, direct generalisation from the 1D case (where does the absolute value come from...?) meant it was hopeless for me to properly and intuitively grasp it. As a result, I also failed to grasp all subsequent ideas like Green's theorem and Divergence theorem because of the shaky fundamentals. They all seemed like disconnected ideas and I am heavily against rote learning. Maths is beautiful and connected, and if I couldn't see the connection between ideas then I must not be looking hard enough, or I must be misunderstanding something. I blamed myself for not being smart enough, but in reality that disconnect was just an artefact of the poor framework I had been taught. Under a more suitable framework they're all just various expressions of Stoke's theorem, but that's completely non-obvious without the wedge product. Now, there are obviously applications where having an orientation agnostic definition of 2D integration is useful, but it should be understood that those are a specific case and that's not the most general rule. In the same way that ∫f(x)|dx| is sometimes a useful quantity to calculate for certain applications (it's useful for measure theory, for example, where you are largely interested in non-negative quantities), it is *not* the most general 1D integral and shouldn't be treated as such. Many students carry an implicit and incorrect assumption that 2D+ integrals don't inherently include orientation, and it's not until much later (or never, in the case of many engineering and non pure math streams) in a course on differential geometry where we correct the misunderstanding. Why wait that long? Why not just show the proper, most general framework from the beginning? Especially if that framework is a direct generalisation of the 1D case, it allows the results to be obtained easily through basic algebra *and* it fundamentally links all of the other topics (Stokes, Divergence, Green's theorems) covered in multivariable calculus course in a unified framework? Honestly, the explanation I suspect is status quo and legacy. We teach multivariable calculus and differential geometry as two artificially disconnected subjects because we've always done it that way and existing textbooks are written assuming it will be taught that way. Imo that's a very poor reason.
@@Zxv975 You gave a really good explanation. I guess it's kind of to each their own, I can see why to some who really enjoys abstract algebra, ideas like orientation and formalism like wedge products are much more natural. But for someone like me who is a huge fan of geometry, I find such ideas unfamiliar, and prefer to think about manifolds and tangent spaces :D
Love the animation, but just needed to say that it took me a while to understand what the transformation actually does in variable terms, that it takes r and theta from the r-theta domain and transforms/teleports the x-y domain *while* remaining r and theta
I can't pint-point a particular reason but I absolutely love your teaching method. By "teaching method", I'm not really referring to using animations, or the particular example you used. I'm referring to something more abstract and even I am not sure what it is that I am referring to. I just wanted to let you know that it was a brilliant video although I wouldn't really be able to give a solid reason to back my remark, if asked. I hope you continue creating videos and create playlists on different subjects in mathematics. I wish you all the best!
Great video! very nice explanations but I think you could've made the video even better by including a 5 minute segment on what the total derivative and the jacobian matrix are and what they represent and from that one can easily understand why the jacobian determinant tells one how much an infinitesimal area gets scaled when you do a coordinate transformation. From that you can easily talk about general curvilinear coordinate systems.
For a lark I put the double integral in rectangular coordinates into wolframalpha and it produced 18π or ~= 56.549 I computed the double integral with polar coordinates on my hp-prime and I get the same answer that wolframalpha gave above: 18π
The extremely simpler method of calculating the volume is worth noting: integrate the circular area along the z-axis. Integral [0 -> 9] (pi * (9-z) dz)
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wonderful, i finally understood it, you are a very good teacher it is just crazy
since you're taking the absolute value of the determinant, the order of the rows and columns doesn't matter. All that matters is that the old coordinates are in the "numerators" and the new coordinates are in the "denominators", and that the variables line up.
in summary: •det(A^T)=det(A) •det(A but with 2 rows or columns swapped)=-det(A) but since we're taking the absolute value it's fine •det([a/c a/d;b/d b/c]) and det([b/c a/d;a/c b/d]) are unrelated to det([a/c a/d;b/c b/d]) so make sure they line up
this reflects the fact that the order of the differentials doesn't matter, as long as the order of the integrals matches them correctly. dxdy=dydx rdrdØ=rdØdr
This is a truly fantastic video, with great explanations and animations to match. The only part I don't understand is as follows: Why do you keep referring to this as a "volume double integral" at say 10:37, when you are actually simply computing the area of a circle, hence the double (not triple) integral?
Does the Jacobian (scaling factor) always exist within the integral expression when we're changing the coordinate system? For example if I am "changing" the coordinates from x-y to y-x, the Jacobian should just be 1, since there isn't going to be any shearing or change in shape of the plane.
Why the determinant? IIRC because (in 3D) it corresponds to the triple product and thus the volume of the differential element relative to the volume of the untransformed differential element.
My first approach was to calculate dx then dy and express the product in terms of the new variables r an theta, and their differentials dr and dtheta -- just like doing a standard change of variables. The result I find almost looks like the correct Jacobian sandwiched in between the desired differential. But I find an extra term which is a function of theta -- and that's not coherent because, by symmetry, the Jacobian has to be invariant by rotation. The details of my calculation (theta is noted as t for simplicity): dx = dr.cos(t) - r.dt.sin(t) dy = dr.sin(t) + r.dt.cos(t) dx.dy = r.dr.dt.[cos²(t) - sin²(t)] + [a bunch of higher order differentials] dx.dy = r.dr.dt.cos(2t) When integrating, the cos(2t) term just gives 0 -- another hint of incoherence. So where's the blunder?
The base problem is to add up the volume of each micro wedge piece. When you set up a math problem you also have to ensure that your math is describing the reality.
Wait so are r and theta represent a nonlinear transformation and the Jacobian is a local linear approximation for our differentials so they are in agreement with your identity cartesian coordinate system? And determinates determine how much areas are scaled so..
took calc 3 this year and this was far better explaination of change of variables and why dx dy = r dr d0. how did i not know the change was due to the jacobian lol
I can't say I like determinants, they seem like a frig that works. This video is great at explaining the purpose of the Jacobian but having to recourse to old, dry and mysterious determinants is opaque. Is there a modern, more intuitive way of expressing the way the Jacobian is calculated?
The teaching techniques used in this video are vastly superior to classical math education. The animations are a massive improvement. And the open why and how approach is refreshing.
This is the fantastic visualization of how Jacobian works. It may not be a very secrets concept for those who attended the higher level math course. But for those student new to advance calculus, this is definitely a precious gem --- Quickly guide them to the Multivariable mapping world.
If only math professors learn you like this. 3D sketch are fundamental
@@gagadaddy8713 The Jacobian is the key to the total derivative, which is one of the most important concepts in diff geo and advanced calc.
@@krishyket Can't agree more!
I keep telling people that local educators are almost never going to be able to teach as well as the best RUclips video.
I remember my cal prof at the time saying "this is the formula for the jacobian, no need to know what it is just compute it" and i never even realized the areas of the shapes don't match when going from cartesian to polar ... you just blew my mind. Thank you
He himself didn't know. What do you expect 😂😂😂😂He is just glad, he has a well paid job that pay his bills🙏
I don't expect to learn everything from university. The teachers don't give me the rigor of detail that makes me satisfied. I learn more from self-study and these external sources.
I just think of university as being there to give me an outline of what to learn and to hand me my diploma. To be fair, the laboratory facilities and libraries also facilitate my learning, but never the professors.
@@frieden6298I’m sorry you and many others have had this experience. This is a common/standard undergraduate experience. During my degree obtained at my university, I was very fortunate to dive into the “why”, and that motivated me to pursue graduate studies.
My standard undergraduate multivariable calculus course went over the conversions from Cartesian to polar-and spherical-and discussed the Jacobian and its importance at length
Serbian? I’m in 53 rn
Easily the best explanation I've seen on the Jacobian determinant. You're an absolute legend
A great refresher on stuff I've forgotten over 20 years. After you've been out of school for a while, you just robotically insert "r dr dθ" for "dx dy" without even considering why you are doing it. Jacobians are just things you vaguely remember when trying to code up some algorithm you found in a paper and cursing how much linear algebra you have forgotten.
What do you do integrals for
As a disclaimer I am a graduate of A&M college of engineering (graduated 20 years ago and never looked back). I came across this video skeptically, and to my surprise, I'm very impressed.
There is a simpler way to think of this concept without having to understand why the jacobian is neccessary. dy and dx are graphically the deriviatives of lengths. The corresponding derivatives of lengths in polar coordinates is dS and dr, not d(theta) and dr, where S is arc length. dS = rd(theta). dA then = dy dx = dr dS = r dr d(theta). It works better with visuals, but I always found this a helpful way to understand this concept.
A simple, clear and concise explanation.
Thanks. That was very helpful.
Understanding the Jacobian helps to generalize whatever transformation you'd like to do, regardless how much you can "visualize" the new integral.
However, the geometric example using dS, as r dθ, helps to visualize "what's going on" when beginning in integral change of variables.
Jacobian is general
thank you very much Joseph, this is waht I am looking for.I can't tell you what you did for me
If your confused A is area and switch the “then” and the “dA”
If Mathematics were taught in schools and universities more like this...imagine the world that would be possible. Thank you!
The best of all videos explaining the Jacobian I´ve seen.
For someone who just started learning integral techniques, this video was surprisingly well explained
Brazilian book of calculus from the mathematician Hamilton Guidorizzi is the best book of calculus! He explained every detail of the Jacobian with high class!
The animation did the trick, I didn't understand anything before with just formulas but now after watching the video everything is clear ☺
Best explanation ever on the Jacobian. Amazing. I graduated almost a decade ago, but this was fun to revisit.
This was the most fabulous explanation on change of variable, which I have ever come across!
If I had to rate the best videos on change of variable integration that I've ever seen, this is 1 through 7. maybe even 1 through 8.
This is the best explanation I found on YT even much more popular channels just come up with the transformation matrix, but fail to come up with a good example.
Magnificent way to visualize the Jacobian. Congratulations!!
You helped me understand jacobian in one small lecture ... God , engineering was easy, if I had teachers like you ... gosh .. But thank you so so much
I knew I needed to just sit down and find somebody who can explain this stuff conceptually but it's still surprising when I'm doing it
Would love seeing a vid about why using the jacobian determinant works. Now this just feels like magic
Excellently explained. If all math concept is explained with a visual, and concise comment, all math learners will love math. 100% guranteed.
Wowwww! I could never understand the Jacobian when I was studying in college. This is by far the best video I have come acrossed
By far the most concise and clear explanation on change of variable I have watched.
ive been trying to understand this for the longer part of an hour, your visuals helped me understand it within minutes
Superb explanation of the Jacobian as a scale factor. And fantastic animation. Makes the concept of the Jacobian so clear. Great video.
I always learn well with animation.. thank you sir for this perfect teaching
What I do not think is explained is why the determinant of this is precisely what you need to make the conversion. Maybe it is based on top of a prior video where this meaning is explained. Anyway, good video and well explained. I also like the comment about how this can be understood without Jacobians. I always have used that way of thinking, but thinking about the same topic in multiple ways is helpful
This is too good!! Why didn't I come across this video back when I was taking Calc 2 class in early 2022?!?! Thank you sooo much for this videoo! I might need it for a future encounter with the Jacobian
U did stuff like this in calc 2? I’m only familiar with it being a calc 3 concept
@@pianodan1608
I think every math department classifies it differently. With the current curriculum, my uni has four calculus classes:
- Calc 1 (single variable calculus): limits, derivatives, integrals, integration techniques, FToC, IVT, MVT, trigonometric and hyperbolic functions, surfaces of revolution
- Calc 2 (multivariable calculus): partial derivatives, iterated/double/triple integrals, polar/cylindrical/spherical coordinates
- Calc 3 (sequences and series): improper integrals, Taylor series, sequence/series convergence (Dirichlet's test etc), orthogonality conditions, Fourier series
- Vector calculus: divergence, curl, Gauss' divergence theorem, Stokes' theorem
@@bismajoyosumarto1237 I see!
Thank you for an exceptional video/lecture on the Change of Variables and the Jacobian in Calculus Three.
This is a great video! Your explanation is very calm and clear, which makes it easy enough to follow what is happening in what can sometimes be a little overwhelmingly confusing math. The video really helped me understand the topic better, thank you!
The best explanation for the Jacobian I found out there! Thanks!
Excellent. This is good explanation for canonical coordinate transformation from one set of coordinate to another. I am happy to see this.
Great, easy to understand reminder. Allowing me to pass it on to my son.
These illustrations are literally so nice
Oh my god this video is amazing, such an intuitive explanation of it! It clicked to me right as you said the areas change and the jacobian is the determinant of the matrix (which represents the area of the thing, and since things are in derivatives, I’m guessing it translates to the change in area). Definitely subscribed
Best explanation of jacobian i have ever heard thank you so much
but you didnt explain why the determinant of the jacobian scales it correctly
Worth noting that, because of how determinants work, you can swap the rows/cols and transpose the matrix as much as you want and the |J| won't change
True, the only way you could actually mess up the det is if you put the partials of x and y on the same row
Amazing video! Very well explained. Its unusual how a channel of such quality doesnt have at least a couple of thousand subscribers.
This video got me excited for calc 3 next semester, this was so fun
Excellent explanation of the Jacobian! If I'd been able to watch this 40 years ago when I first learned this, I could have learned in 15 minutes what took me a month or so to learn!
Best explanation of Jacobian on RUclips
Excellent video and explanation of the Jacobian - thank you for creating !!!
The most satisfying video on change of variables.
Thanks❤
Nice application of a shell balance to show the differential element in the theta direction is r*d(theta).
this is the best video i've seen on Jacobian explanation !!
Thank you!!!
This makes me finally understand a concept I was confused about for quite a while now.
The visual interpretations in this video are really helpful!
Thank you. Visualization is the key.
Nicely done; I am proficient at using the Jacobian. But that explained WHY it works really well
Super great and interesting video. You put so much effort in it. The outcome is impressive. Thank you so much.
This doesn't explain why the Jacobian (as a determinant of partial derivatives) is the scale factor for the differential area of the transformation.
Good question. The tl;dr is that the typical 2D integral is wrong and is missing a wedge product. If you include the wedge product, then deriving the Jacobian is simple algebra and nothing mystical at all.
A (very long) but more complete explanation is as follows. Honestly you could make an entire video on properly explaining this this idea...
First, note that the double integral we're familiar with is missing something crucial. We know from 1D integration that integrals are signed; they carry information about orientation. This is still present in 2D with swapping bounds, but it isn't present when swapping the order of integration. For example, calculate the volume of a cube, given by f(x, y) = 1 from 0 to 1
∫₀¹∫₀¹ f(x, y)dxdy = ∫₀¹∫₀¹dxdy= 1
This is positively oriented: you're tracing out the 2D square in the x-y plane in the anti-clockwise direction (we arbitrarily call this "positive")
However, if we swap our order of integration we should get something negatively oriented, but we don't
∫₀¹∫₀¹dydx = 1
This *should* be negative because it traces out the square in the opposite orientation (clockwise), but it's not. What gives?
The answer is that dxdy was the incorrect area form, and should instead be dx∧dy. This wedge operation behaves like the cross product in 2D and 3D (but it also works in arbitrary dimension), so it has the rule dy∧dx = -dx∧dy. This alone fixes the problem and recovers orientation.
You could also arrive at the same conclusion by trying to integrate along axes that aren't perpendicular (i.e. using infinitesimal parallelograms as the fundamental unit instead of infinitesimal squares, as both can tile the plane both should be valid in principle). The standard dxdy has no nice way of handling anything other than perpendicular axes natively, but with the wedge product
dx∧dy = dx×dy = sin(ϕ)dxdy
we can handle it easily. This is the same answer you'd get through the Jacobian, but presupposing that would somewhat be circular logic.
Now, with that established, deriving the Jacobian is a straightforward exercise in algebra. The entire reason that the Jacobian feels like a leap is because this entire backstory I presented here is swept under the rug and only implicit in the Jacobian. This means that you are indeed taking a leap as none of this is ever explained.
We know that we can swap from Cartesian to polar, therefore x = x(r, θ) and therefore dx = ∂x/∂rdr + ∂x/∂θdθ is the total derivative, and same for y. For the integral, we already directly substitute for x and y, so why don't we also just substitute for dx and dy directly too?
∫f(x, y)dx∧dy= ∫g(r, θ)(∂x/∂rdr + ∂x/∂θdθ)∧(∂y/∂rdr + ∂y/∂θdθ)
Where g(r, θ) = f(x(r, θ), y(r, θ)) is just what you get from plugging in polar coordinates into your original function. This looks like a mess, but it's honestly not that bad. The wedge product distributes like an ordinary product would, so expanding using the distributive law gives something which has the form
Adr∧dr + Bdr∧dθ + Cdθ∧dr + Ddθ∧dθ
for the stuff from the brackets. Each of the terms A, B, C and D are just products of partial derivatives. Now a straightforward consequence of the wedge product is that dx∧dx = -dx∧dx = 0. This means that the A term and D term disappear from the expansion, and we are left with only B and C. We also note that since dr∧dθ = -dθ∧dr, we are overall left with the term (B-C)dr∧dθ, so the integral becomes
∫g(r, θ) (B-C)dr∧dθ
and if we evaluate B we get ∂x/∂r ∂y/∂θ and C we get ∂x/∂θ ∂y/∂r which is exactly what you get from the Jacobian method.
This is all perfectly explained extremely directly if you use the proper wedge product in your integrals. The wedge product makes the absent terms disappear and inserts the correct sign in front of C.
The last thing missing is the absolute value. This is again needed because of the missing wedge product. Since drdθ has no orientation information, we must have ∂(x, y)/∂(r, θ) = ∂(x, y)/∂(θ, r). But algebraically this clearly isn't the case, so to sidestep this issue we throw out all orientation information and focus solely on the magnitude, which is what the absolute value does. With the wedge product, we instead would need to have ∂(x, y)/∂(r, θ)dr∧dθ = ∂(x, y)/∂(θ, r)dθ∧dr, which is automatically true because both minus signs on the RHS cancel out. So there's no need for an arbitrary absolute value.
For the life of me I don't know why this isn't taught like this. It's not like any of this is any more difficult than standard multivariable calculus coursework.
@Bryte H
I think the signed aspect isn't crucial to understand the Jacobian determinant. Neither really is the formalism of the wedge product (yes it becomes handy in dimensions greater than two, but the fundamental intuition can be garnered by just considering examples in R^2)
I think a simple explanation for calc 4 students just involves them understanding a bit of linear algebra. The central intuition is that we are transforming a vector of variables (x,y) to another vector of variables (r, theta). This is a non-linear transformation, but since it is differentiable, locally, within a small range (x,y) to (x+dx,y+dy), the transformation is linear. A linear transformation can be described as a matrix, that matrix is the Jacobian. This can also be seen by introducing to students the multivariable Taylor expansion, showing that the Jacobian is the first term, analogous to a simple derivative, but for more general transformations from R^n to R^m
Now that we know the Jacobian is the transformation that takes tangent basis vectors x,y to r,theta, a natural question is what happens to the area under such a transformation. Here is where you introduce to students the idea of a determinant as a signed scale factor for areas under linear transformations. The signed aspect can be understood by giving students a couple simple examples (does the transformation flip the orientation of the basis vectors?)
In many ways it is just a more general, multi-dimensional, form of "u-substitution" students are already intimately familiar with.
@@afroohar Yes, the Jacobian can be explained in a consistent way through what you just described. I just don't see how this is a *better or more intuitive* explanation than simply including orientation in 2D+ integration (something we know it's supposed to have anyway) and doing basic algebra to get the answer we expect. Everything falls out for free, and orientation doesn't really make things more difficult. Doing mental gymnastics and bookkeeping to avoid orientation is significantly more confusing.
At least in my personal experience, even though I learned the Jacobian explanation you described in 2nd year uni, because I could never just derive it directly it never stuck with me. If I was expected to substitute x and y and I could come up with expressions for dx and dy, then why couldn't I just directly substitute dx and dy?! "Am I doing something wrong?" "Do differentials not obey algebra?" I had vague notions of 2D Infinitesimal area elements and sometimes remembered there was an absolute value in the formula for some reason, but the fact that none of it seemed like a proper, direct generalisation from the 1D case (where does the absolute value come from...?) meant it was hopeless for me to properly and intuitively grasp it. As a result, I also failed to grasp all subsequent ideas like Green's theorem and Divergence theorem because of the shaky fundamentals. They all seemed like disconnected ideas and I am heavily against rote learning. Maths is beautiful and connected, and if I couldn't see the connection between ideas then I must not be looking hard enough, or I must be misunderstanding something. I blamed myself for not being smart enough, but in reality that disconnect was just an artefact of the poor framework I had been taught. Under a more suitable framework they're all just various expressions of Stoke's theorem, but that's completely non-obvious without the wedge product.
Now, there are obviously applications where having an orientation agnostic definition of 2D integration is useful, but it should be understood that those are a specific case and that's not the most general rule. In the same way that
∫f(x)|dx|
is sometimes a useful quantity to calculate for certain applications (it's useful for measure theory, for example, where you are largely interested in non-negative quantities), it is *not* the most general 1D integral and shouldn't be treated as such. Many students carry an implicit and incorrect assumption that 2D+ integrals don't inherently include orientation, and it's not until much later (or never, in the case of many engineering and non pure math streams) in a course on differential geometry where we correct the misunderstanding. Why wait that long? Why not just show the proper, most general framework from the beginning? Especially if that framework is a direct generalisation of the 1D case, it allows the results to be obtained easily through basic algebra *and* it fundamentally links all of the other topics (Stokes, Divergence, Green's theorems) covered in multivariable calculus course in a unified framework?
Honestly, the explanation I suspect is status quo and legacy. We teach multivariable calculus and differential geometry as two artificially disconnected subjects because we've always done it that way and existing textbooks are written assuming it will be taught that way. Imo that's a very poor reason.
@@Zxv975 You gave a really good explanation. I guess it's kind of to each their own, I can see why to some who really enjoys abstract algebra, ideas like orientation and formalism like wedge products are much more natural.
But for someone like me who is a huge fan of geometry, I find such ideas unfamiliar, and prefer to think about manifolds and tangent spaces :D
@@afroohar your explaination is very clear, thank you.
Love the animation, but just needed to say that it took me a while to understand what the transformation actually does in variable terms, that it takes r and theta from the r-theta domain and transforms/teleports the x-y domain *while* remaining r and theta
I can't pint-point a particular reason but I absolutely love your teaching method. By "teaching method", I'm not really referring to using animations, or the particular example you used. I'm referring to something more abstract and even I am not sure what it is that I am referring to.
I just wanted to let you know that it was a brilliant video although I wouldn't really be able to give a solid reason to back my remark, if asked. I hope you continue creating videos and create playlists on different subjects in mathematics. I wish you all the best!
By far one of the best videos I've seen.
Great video! very nice explanations but I think you could've made the video even better by including a 5 minute segment on what the total derivative and the jacobian matrix are and what they represent and from that one can easily understand why the jacobian determinant tells one how much an infinitesimal area gets scaled when you do a coordinate transformation. From that you can easily talk about general curvilinear coordinate systems.
Best video there is on Jacobians
For a lark I put the double integral in rectangular coordinates into wolframalpha and it produced 18π or ~= 56.549
I computed the double integral with polar coordinates on my hp-prime and I get the same answer that wolframalpha gave above: 18π
Cannot thank you enough for this explanation
Such a great combination of using animations for visual understanding and actually showing how the Maths gets computed! Count me subscribed ;)
0:00 Thanks to Texas Uni for funding. Great!!
The extremely simpler method of calculating the volume is worth noting: integrate the circular area along the z-axis.
Integral [0 -> 9] (pi * (9-z) dz)
wonderful, i finally understood it, you are a very good teacher it is just crazy
Why this page is so underrated
I am surprised why you have so less views. You have extraordinary sense of visualisation of maths like 3b1b
since you're taking the absolute value of the determinant, the order of the rows and columns doesn't matter. All that matters is that the old coordinates are in the "numerators" and the new coordinates are in the "denominators", and that the variables line up.
in summary:
•det(A^T)=det(A)
•det(A but with 2 rows or columns swapped)=-det(A) but since we're taking the absolute value it's fine
•det([a/c a/d;b/d b/c]) and det([b/c a/d;a/c b/d]) are unrelated to det([a/c a/d;b/c b/d]) so make sure they line up
this reflects the fact that the order of the differentials doesn't matter, as long as the order of the integrals matches them correctly. dxdy=dydx
rdrdØ=rdØdr
Absolute banger of a video.
Very good video ! Why is it that the Jacobian is the matrix of these partial derivatives ?
bravo my guy, these visualizations are so good!
You got me in 2:49. I was so easily going with your flow. Haha
This is a wonderful video! Very clearly explained.
This is a truly fantastic video, with great explanations and animations to match. The only part I don't understand is as follows: Why do you keep referring to this as a "volume double integral" at say 10:37, when you are actually simply computing the area of a circle, hence the double (not triple) integral?
This is god damn clear. Amazing!
I love you so much. This genuinely help me a ton.
Excellent video on the topic
nice explanation with good graphics
A true masterpiece
Amazing! Reminds me of Morphocular
EDIT: I KNEW IT
Does the Jacobian (scaling factor) always exist within the integral expression when we're changing the coordinate system?
For example if I am "changing" the coordinates from x-y to y-x, the Jacobian should just be 1, since there isn't going to be any shearing or change in shape of the plane.
Why the determinant?
IIRC because (in 3D) it corresponds to the triple product and thus the volume of the differential element relative to the volume of the untransformed differential element.
الف رحمة على والديك ...😊
Sir, just keep going. Your channel is amazing 🙏❤️
Brilliant lesson. Thanks.
My first approach was to calculate dx then dy and express the product in terms of the new variables r an theta, and their differentials dr and dtheta -- just like doing a standard change of variables.
The result I find almost looks like the correct Jacobian sandwiched in between the desired differential. But I find an extra term which is a function of theta -- and that's not coherent because, by symmetry, the Jacobian has to be invariant by rotation.
The details of my calculation (theta is noted as t for simplicity):
dx = dr.cos(t) - r.dt.sin(t)
dy = dr.sin(t) + r.dt.cos(t)
dx.dy = r.dr.dt.[cos²(t) - sin²(t)] + [a bunch of higher order differentials]
dx.dy = r.dr.dt.cos(2t)
When integrating, the cos(2t) term just gives 0 -- another hint of incoherence.
So where's the blunder?
Best explanation ever ❤❤❤❤
THIS VIDEO IS DIVINE CONTENT
The base problem is to add up the volume of each micro wedge piece. When you set up a math problem you also have to ensure that your math is describing the reality.
Fantastic explanation!
Beautifully done
Wait so are r and theta represent a nonlinear transformation and the Jacobian is a local linear approximation for our differentials so they are in agreement with your identity cartesian coordinate system? And determinates determine how much areas are scaled so..
Amazing, just amazing!
Great use of Manim
simply brilliant
took calc 3 this year and this was far better explaination of change of variables and why dx dy = r dr d0. how did i not know the change was due to the jacobian lol
This is so beautiful, thank you
11:27 can’t you visualize a 3rd diagramm once Jacobi applied? Or in other words, visualizing Jacob matrix depending on r and theta 🤔
Oh thank you! Helped me so much.
you earned a subscription with this one
Awesome explanation :)
Finally i undrestand jacobian 😂
Thank you ❤
amazing explanation, thank you!
I can't say I like determinants, they seem like a frig that works. This video is great at explaining the purpose of the Jacobian but having to recourse to old, dry and mysterious determinants is opaque. Is there a modern, more intuitive way of expressing the way the Jacobian is calculated?