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