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 KZfaq video.

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”

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

What do you do integrals for

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!

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.

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