Change of Variables and the Jacobian

Changing variables can sometimes make double integrals way easier to compute, but fully converting over from one coordinate system to another can be tricky, because in addition to changing the integrand and the integration bounds, you also have to throw in this additional factor called the Jacobian. But why? And what is the Jacobian doing?

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This video was funded by Texas A&M University as part of the Enhancing Online Courses grant.

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The animations in this video were mostly made with a homemade Python library called "Morpho". You can find the project here:

[JoannaPirieHill]

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.

[mikexbox]

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

[StratosFair]

Easily the best explanation I've seen on the Jacobian determinant. You're an absolute legend

[josephjacobus]

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.

[BangkokBubonaglia]

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.

[necaro]

If Mathematics were taught in schools and universities more like this...imagine the world that would be possible. Thank you!

[dcterr]

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!

[Teo-uwmh]

but you didnt explain why the determinant of the jacobian scales it correctly

[james]

This was the most fabulous explanation on change of variable, which I have ever come across!

[lenoxbaloyi]

The animation did the trick, I didn't understand anything before with just formulas but now after watching the video everything is clear ☺

[horacioguillermobrizuela]

The best of all videos explaining the Jacobian I´ve seen.

[plantnt]

For someone who just started learning integral techniques, this video was surprisingly well explained

[afroohar]

This doesn't explain why the Jacobian (as a determinant of partial derivatives) is the scale factor for the differential area of the transformation.

[JhonatanNeneh]

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!

[moneymo]

Best explanation ever on the Jacobian. Amazing. I graduated almost a decade ago, but this was fun to revisit.

[wilisonilagi]

I always learn well with animation.. thank you sir for this perfect teaching

[comoenclase]

Magnificent way to visualize the Jacobian. Congratulations!!

[rudypieplenbosch]

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.

[phd_anime_cats]

The best explanation for the Jacobian I found out there! Thanks!

[keithkeller]

Excellent video and explanation of the Jacobian - thank you for creating !!!