Multivariable Chain Rule Proof
HTML-код
- Опубликовано: 1 апр 2025
- In this video, I provide a really intuitive proof of the chain rule in several variables, and show how the derivative of fg is just the matrix multiplication of the derivative of f times the derivative of g. Enjoy linear algebra and multivariable calculus in its pure awesomeness!
You misspelled Chen lu in the title
Hahahaha
blackpenredpen ...it's the southern spelling.
For about 2 years I haven't had any intuition behind the chain rule, nor any intuition about its proof. And in under 30 minutes, now I do. Thank you so much!
Hello there!!!! From 7th grade to 10th grade I was taught to memorize math formulas and memorize the ways to answer math questions the same goes for physics they made memorize definitions too that of course led to me to hate these subjects, in 11th grade in 11th grade my dad introduced me to an amazing (math - physics)sudanese teacher who eventually to taught me to understand the intuiton behind everything rather than memorizing , when I was taught to memorize definitions I was to forget them pretty fast but then when I understood the intuition and visualized the Math and Physics concepts I never forgot the information and it was more entertaining and I realized something too that every math problems that I had hard time trying to answer was not due to the way the book does it was due to linkage of concepts so I brushed up my Math basics by revising every topic with intuition and the results were amazing . Now I am in 12th grade and finished the math curriculum before my class mates (errr to not brag still have two lessons in statistics but they are easy). After I finish 12th grade I aim to study advanced Mathematics for preparing for univeristy and also understanding your amazing concepts. Sorry that the text is too long 😂😂😂 . By the way I will study math and physics from brilliant.org it got amazing problems.
The trick was in the definition of e(w)... Nice proof Dr. Peyam :)
Amazing as always Doctor!
Would you like to make a video about Taylor Series in Rn? It would be a nice follow up from this last video about the Chen Lu. (Don’t wanna make BPRP angry)
love your enthusiasm
Ok.. I just thought that I have to take the time to tell you that you are amazing! I love calculus and hence I love this channel. One day I'll surpass you!!
🥰🥰🥰 Thank you!!!
@@drpeyam But mind you! I will surpass you!
You are a fantastic teacher!
🥰
I love how you write the differential of a map at a point in the same order as I do (most teachers I know write $dg_p$, $dg(p)$ or even $T_pg$, but never $d_pg$) BUT you use uppercase D and I use lowercase d. So close yet so far.
EDIT: also anecdote time when our calc teacher proved the HD chen lu he proved the matrix thing using induction (because he proved it for \mathbb{R}^n \longrightarrow \mathbb{R}^m). If the teacher wasn't so bad-tempered, I would have suggested using a lemma from linear algebra that literally states the matrix of the composition is the product of matrices. Like, it's literally a result you see in algebra I...
If you would use indicies and summation more, you would know the chain rule is just matrix multiplication from the first moment you see it. :D
Who is Chen Lu and why is the chain rule named after him?
I don't think it's actually named after anyone, in particular, but rather that it's more of a mere linguistic coincidence.
How do you get to write the gradient of f as one 2x2-matrix?
As you have two different functions f, should not each of those have its own 1x2- or 2x1-matrix?
Actually it's not that surprising that matrix multiplication appears, since a derivative is nothing but an approximation with a linear function. So if you approximate g ° f with a linear function, it will just be the >composition< of the two derivatives of f and g . And now composition of linear maps is, of course, realized by matrix multiplication.
You just have to view the derivative >at a single point< as a function, as opposed to viewing the function f' which maps every point to the derivative at that point. I think the confusion arises, when you only look at 1-dimensional functions, and the derivative "function" is just a single number (the slope of the tangent line), so it doesnt look like a function at all. (But compositions of two linear R->R functions is also nothing else than multiplication of the slopes)
When you said slope I got the "It's calculus! (Looking at slopes)" song from Vi Hart in my head
The most general chen lu!!!
MG Chen Lu!!
@@drpeyam now do chen lu on manifolds dr peyam :D
Omg 😳
You said in a video that you don't like the indicies and summations, or something similar. Do I remember correctly? If you did say something like that, I would like to convince you otherwise. :D The multivariable chen lu written in that format is just so elegant in my opinion. It's same as your matrix notation in the video, of course.
Good point, but I still hate the summation convention 😂
@@drpeyam make a video about it, so I can convince you otherwise. Or don't just give me your reasons. Your life will be so much easier if I succeed.
@@drpeyam Why? It's the best thing to happen to physics lol
@@HilbertXVI agreed. There are also some proofs in differential geometry, where you need to choose a chart (at least I never heared of a proof where you don't). The summation convention is almost necessary there. Plus the ... can be ambiguous sometimes.
@@zoltankurti I took a differential geometry course in my degree and it was hard to follow thanks to the summation convention. Sorry but I stand with Dr Peyan on this one
Proud to be like no 22!!
Greetings from Turkey.
Tesekürler :)
do you have a video in which you explain the total derivative?
super!