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

Jacobian matrix and determinant are very important in multivariable calculus, but to understand them, we first need to rethink what derivatives and integrals mean. We can't think of derivatives as slopes if you want to generalise - there are four dimensions to graph the function! This video hopes to explain what the Jacobian matrix and determinant really mean, and essentially why they are actually very natural for changing variables; and also explaining something that might be glossed over when you use them - for example, we require absolute value, and the changing variables function is injective.

In the video, we have only talked about 2D transformations, but the Jacobian can be easily generalised to any number of dimensions you like - you just need to introduce linear maps in higher dimensions! Think about what that means in 3 dimensions for a start!

This video simply aims to introduce the intuition of the Jacobian, and so a lot of things said in the video is not going to be very rigorous - for example, what does approximate mean? It has a specific meaning in mathematics, but we are not getting there; and also not all functions have this nice property of looking like a linear map near a point. These belong to the realm of real analysis, which is well beyond the scope of this video. So please don't shout Fubini's theorem when you see flipping the order of integration at about 17:09.

Video chapter feature:

00:00 Introduction
01:20 Chapter 1: Linear maps
06:01 Chapter 2: Derivatives in 1D
08:08 Chapter 3: Derivatives in 2D
13:01 Chapter 4: What is integration?
17:26 Chapter 5: Changing variables in integration (1D)
19:25 Chapter 6: Changing variables in integration (2D)
22:59 Chapter 7: Cartesian to polar

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[mathemaniac]

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 YouTube 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!

[adamdapatsfan]

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

[NonTwinBrothers]

I love how Youtube is now exploding with math channels

[NuclearSpinach]

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.

[brianhowell]

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

[robertlinder]

After 40 years of college, finally a good explanation .

[leobieker]

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.

[ASASID]

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

Thanks a lot

[gonzalezm]

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 🙏

[RedStinger_]

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!

[Boringpenguin]

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

[ryanj]

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

[abramcz]

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.

[martinjohnsons]

vsauce narrative + 3blue1brown animation

[Aio-Project]

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

[NextazxzNoExuse]

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

[jeremylevitt]

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.

[derrylmartinez]

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.

[arf]

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!

[masonp]

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