Calculus 3 Lecture 13.5: The Chain Rule for Multivariable Functions
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- เผยแพร่เมื่อ 24 ก.ย. 2024
- Calculus 3 Lecture 13.5: The Chain Rule for Multivariable Functions: How to find derivatives of Multivariable Functions involving Parametrics and/or Compositions. Focus will be on deriving the Chain Rule and practice of examples.
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Universities forget how important mathematics really are. They cut down the hours of lessons and professors just give you the main ideas and formulas, no proofs no nothing. However proofs and deep understanding are what get you beyond just passing the semester's exams. They develop your cause-result imagination and allow you to really cultivate original thinking. Thank you Professor Leonard for giving us a chance to choose to differ and study deeply and pointfully.
EXACTLY ! MY SCHOOL EXPLAINS THE BASIC RULES IN LECTURES, SOLVE 1 PROBLEM AND ONTO THE NEXT TOPIC LOL. THEN I HAVE TO TEACH MYSELF EVERYTHING. i am only paying for the degree smh
@@fenerbahceqwe3756 what degree are you studying?
@@CHROMIUMHEROmusic CS
Depends on how the proofs are presented, though.
Some math books present proofs in as confusing ways as possible and use a lot of overly complicated terminology and symbols even when the students for that course aren't expected to be familiar with that stuff.
It's almost as if the authors had written those texts as summaries for _other math professors_ or something, even when they were intended for students, and I really hate that.
Proofs should only be as formal as actually necessary, and keep things in Layman's terms if this is possible without hurting the overall quality of the proofs.
The author David Morin already managed to do that in his physics book "Introduction To Classical Mechanics", which seriously feels _almost_ like a children's book despite bringing up topics like "Principle Of Stationary Action" and space mechanics - it has the same type of charming enthusiasm as a children's science book, and it is good enough to actually be used as course material in one of my physics classes - and yet does an excellent job at explaining all the concepts.
Laurelindo well said
can u give some lectures on body building
Having a good body depends on your genetics, diet, sleep, mental state of mind and training.
I don't think he ever will.
JustAdreamer Forgood genetics aren’t a very big factor in obtaining a “good body” it’s mostly consistent training, proper nutrition, and sleep.
hahahahaha
@@justadreamerforgood69 Nah a body builder level body required good genetic, a good body as in athletic just requires constant effort and progressive overload as well as everything else you mentioned.
00:00 - Intro to Chain Rule with respect to t
44:00 Examples of Chain Rule w/ t
(Organization Tree Method at 47:00 w/ Ex)
56:20 More Intermediate Variables
1:06:00 More Independent Variables
1:10:00 Examples of "More Independent Variables"
1:51:00 Implicit Differentiation of f(x,y) where y=g(x)
god bless you
thank you so much
thank you professor leonard for making these videos. My college teacher is useless and by watching your videos I got 97 on my final. You are a man who needs to be appreciated more.
wow congrats bro
I have a Swedish physics teacher at my university who is about as good as Professor Leonard - his name is Erik Olsson and he works at "Luleå tekniska universitet".
He is cool, in fact he actually has a speech uploaded here on TH-cam where he talks about laser light and waves.
@@Peter_1986
Where is the video ?
@@justadreamerforgood69
The video is called "Laserljus och vågegenskaper; en orienterares perspektiv - Erik Olsson, Luleå tekniska universitet".
@@justadreamerforgood69
Here is the link:
th-cam.com/video/BDwddSrdqaQ/w-d-xo.html
In case you didn't know, professor, English is not even my native language, yet I'm watching your videos instead of my professor's, because you are so smooth and clear with math. It always make me feel like I can follow without sacrificing any comprehension. If I don't, you make it clear what section of your previous videos I should revise.
I'm here in 2020, quarantined with my calc 3 test tomorrow. bless you.
how'd it go ?
Really well actually! Thanks!!
Facing the same thing now :))))))) help
Hannah X how’d it go
Erich Fernandez actually it went well :D thanks for checking 💙
Hearing the laughter at the end of the video made me truly appreciate how valuable of a teacher you are. The manner in which you combine humor and education keeps me engaged and makes my learning experience phenomenal. Because of you, I understand what a good teacher is and how special it is to have one. Thank you so much for uploading your videos. They are incredibly helpful.
This is the kind of a person that should be a millionaire, the best teacher EVER!
Professor, I am currently taking Calculus 3. I am an engineering student at the University of Alaska. I sat through the lecture on this topic today. I have never been so confused. Usually my professor does a pretty good job but today it just didn't click. I was pretty worried about my understanding. I watched this video and now I understand. Thank you for this Professor. :-)
He has got to be the best math professor of all time, I just finished all my math classes for engineering and I can say without question, he is the best math teacher I have ever seen
I started loving calculus after watching your videos, Professor Leonard. I've never thought it would be this interesting
Thank you very much for the fun and interesting videos
"I watched Leonard on one half speed..." Cracked me up. Understanding this chain rule material is incredibly useful for pretty much any kind of higher physics and diff eqs. Fantastic work.
MY WHOLE LIFE i never got excited about math till this DAY that changed. The way he shows proofs are so interesting and i seriously wanna keep listening. Your students are so lucky, my professor for calc3 literally reads word per word off of a power point.
I LOVE how much he cares if people understand.
Most teachers: If that didn't make sense, you're broken.
Prof L: Did everyone understand that?? Everyone??
You are one talented superhero. I just froging wish all teachers were like yourself. You deserve more attention! Much love from sweden sir.
Could you PLEASE do Differential Equations for us? (Please)
please
Pretty please with a cherry on top...
Seriously Professor, need some Diff Eq
Please!!!!!
I topped the calculus 3 class by just learning the concept through your videos!! Thank you so much, Sir!! We all need a prof like you. Now, could you please add "differential equations" to your video lectures, please? 🙏
Learning this is a MUST for knowing how Artificial Neural networks work. Thank you Prof Leonard.
Professor LEonard, I came to this video about multi variable chain rule from MIT Open course Ware Denis Aoroux lecture. Some of the lectures that Denis aoroux gave, particularly on second derivative test and the gradient vector were not clear to me. I really enjoyed your explanation about how the chain rule actually works, around 25:00. The best explanation ever, because it is very simply explained, instead of using fancy mathematical words and symbols. This demonstrates that you sir absolutely dominate your field, and you have a talent for a very clean explanation. Thank you very much.
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XD
+C Park
Yeah, he is freaking buff as hell, holy crap.
I love this guy, really explains the math in a manner that is digestible by the student. Wish i had a prof like him.
Watching your videos helped me a lot. I didn't understand what was going on in class. Since the topics are somewhat related to one another, I thought I was doomed. Watching your videos helped me a lot. I got an A- for Calculus. THANK YOU.
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I would buy it even if just a simple eBook, for a course I'm not even taking. These videos are invaluable and he should be compensated in some form, other than the measly TH-cam check he gets.
almost 6 figure salary from his college and if you watch his videos without ads hes making a decent amount... maybe his patreon
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Greatest math' teacher , bar none. These students should relish this education, for they will find none better, anywhere.
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These videos are timeless. Masterpiece of teaching
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I am reviewing this playlist in conjunction with my current course in Electromagnetic Field Theory;
that whole course is essentially one huge application of Calculus 3.
Thank you so much prof. You make me love maths again. After all this engineering entrance and all, i lost my passion in mathematics.
Teachers like you are blessing.
Professor Leonard ,thank you for a powerful analysis of the Chain Rule and Implicit Differentiation for Multivariable Functions in Multivariable Calculus. Although I encountered this material in Calculus one, I fully understand the Chain Rule and implicit Differentiation that is presented in Calculus Three.
2:02:29 Yah Our Applied Maths II professor is just like, and thanks to you I only go to class when I have exam, otherwise nope.
Love from Ethiopia
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same
You’re awesome man!!!! 2020 and I am preparing for my masters in economics and my bachelors didn’t teach me shit about calculus 3. You’re really saving me. Thanks
I like to watch these vids to feel fancy even though I have no idea what's happening.
+TheMrEM4N Keep it up and you might figure out what's going on
I like to watch these videos cause not failing is my passion *Cries in Silence*...
If you start with his Cal 1 videos, like I did last summer (as well as taking the course), you will have no reason not to understand. Professor Leonard has a gift to make complex things very easy to grasp. I'm now in Cal 3 and these videos are genuinely the only thing keeping me afloat....(with an A)
@@KLCII88 i couldn't agree more !
@@KLCII88 The gift is patience more than anything. It takes a great deal of patience to teach in the way that he does: to take everything at a snails pace by default, to painstakingly tie every new concept to what the student has already learned, etc. He teaches for the benefit of the stupid and inattentive without demanding too little of them, which is actually quite impressive.
I don't want to take calculus 3 but professor has made me fall in love with the calculus 3
"I'm giving you tools here folks, tools."
Professor Leonard 1:05:15
Spoken like a hollywood movie star
25 professor "dislike" that are can't reach the level of perfection that is these lectures
Hahaha, wonderful. My EC problem on my Midterm required the use of the derivative tree. Thanks Professor Leonard!
22:05 dt -> dx & dy -> dw
~42:00 dw as partials of x y
Leonard, your vids are so amazing. You will definitely save my math2011.
I managed to understand half of the content before the semester starts. That's just amazing.
Just took my final exam for Calculus II (which has many Calc III concepts like directional derivatives and triple integrals) and I felt so good and confident walking out of there thanks to your videos. I went from a 47% on the first exam, to 89% on the second, ending with a 92% on the last exam before the final. I used to try and read the textbook and pay attention in class but was not really absorbing any of the content. I stumbled through the homework and was lost on the smaller quizzes in the beginning. Your videos and explanations helped me actually understand what was going on rather than copy a formula down with no idea where it came from. Thank you so much! Legit the only reason I passed Calc *all the way from a D to a B+ going into the final
I agree with the others -- I hope you can do a Differential Equations course at some point. I am learning it from my textbook and teacher, but I feel like I'm lacking on theory because of how formulaic it can get. I would love to get intuition and really understand DE, and I feel like your teaching would make it much better.
This cannot get any better. BEST EXPLANATION EVER. Thank you super man :)
came here to say that I, like many, will soon owe my degree and success to Leonard. THANK YOU!
You are a lifesaver Professor Leonard. I am an online student and my online professors explain nothing. I wish I could take your class in person. I love the jokes too.
@1:28:20 It should be dx/du, since there's only one independent variable that relates to x. Similarly, dy/dv. These shouldn't be partials.
you sure? I was wondering about it too.
1:52:02
Noticed a lot of people pointing this out. Does anyone know why he took a partial derivative there? I'm still confused.
you are right dude!
Prof. Lenard the idea of using the tree really help me, thank you
Can't describe how thankful i am Professor Leonard , Big thnx from Sweden
Prof. Leonard, are you going to make Differential Equations videos? Please consider doing so :( You're the best !!!!
+Karam Alnadi
If you need material for differential equations, then check out the site Paul's Online Math Notes.
Great site with very clear explanations and lots of example problems.
Dear professor Leonard
You are awesome
I'm watching ur videos about intermediate algebra
I'm too excited to reach here
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All best wishes for you sir
I wish my professors were like that. Thank you for sharing this lecture
i watch these lectures on 2x speed and it makes him look like he's REALLY good at writing on a board
AHAHAH that pun @ 1:37:10. Man these videos help me so much! The tree trick is paramount so that one does not get lost when dealing with more than 2 intermediate variables and middlemen equations that contain one of the 2 ind. variable. Cheers professor leonard for another well explained lecture.
Hey, I see a lot of comments thanking Prof. Leonard, but y'all really should go LIKE his videos. It'd help him out! thanks all.
thank you Prof!
video so good, I HAD to leave a comment. Thank you Professor. You are my superman.
at first I was like: ahh that tree method sucks, then about at 1:30:00 I realized how good it is xD
I cant be more thankful! He goes to such a basic! Thanks!!!
Waiting for another video Mr Professor. It is really helpful. God bless you.
I love your passion for teaching.
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The God of Mathematics, ladies and gentlemen!
man we need more professors like you... professor that actually give a shit about ensuring that everyone understands...
I am from Bangladesh. Outstanding class
I wouldn't be able to make any sense out of this topic without your lectures, the textbook SUCKS. thank you!
Professor Leonard is awesome! Funny too! Thanks for the videos they’re super helpful!
this gets recommended to me as i just had a lesson on this🎉
dependent or independent that won't change the fact that you are a great professor
Hi professor, your videos are very much appreciated and inspirational. Even when you don't the right feedback from your pupils you don't despair and are willing to go over the topic again until it becomes clear to them, patience personified. May I just ask you a question?. At approx. 1:14:57 on this video the partial w with respect to u or v on the left hand side of the equation is set equal to the expressions on the right hand side, however shouldn't the partial Dx/Dx and Dy/Dy [ I use " D " here to mean partial ] cancel out giving the right hand side of the equations twice the value of the left hand sides of the expression. This I don't understand......any clarification would be gratefully appreciated. Thanks , and kindest regards .Barry.
49:38 "tryna be cool, shouldn't have been a math teacher" gave me a sensible chuckle.
ok im gonna start uploading my quiz grades lol cause leonard i am doing phenonenal because of you
I only got through Algebra in school and didn't understand it b/c it wasn't explained worth a crap. I'm actually understanding Calculus from watching this guy!
Great lecture! Thank you Professor Leonard :D
You make studying math interesting thank you so much for your efforts
51:47 the "dependent variable" header is supposed to be the "independent".
I have noticed this as well haha but its ok professor havent seen it
The number of components of the derivative of a specific independent value is equal to the number of intermediate variables that are composed of that independent variable! I think that sums it up!
Thank you professor Leonard!! you are a hero!!
Too bad you don't even get to see these concepts shine when you first learn them.
This is so useful for linear approximations of nonlinear systems of differential eqns.
For all the people asking him to do differential equations and linear algebra. As you may know by now brilliance takes work. You are all well on your way to that if you're concerned
about diff eq. Instead of just mimicking what Professor Leonard does you should've also been learning how to learn difficult material. So apply his method of finding out HOW things became that way and WHY a particular method works. You are all on your way to shaping our future, you cannot expect someone else to do all the work for you. Take pride in the pleasure of finding things out. Do not be so cavalier with the discomfort of not understanding something. It is essential in the discovery process. It is essential for growth. it is essential for you to become the better version of yourself.
I have aspbergers syndrome and don't think like you at all. I need this comprehensive insite and we all mimic something as you say, I do not pick up on inferred info in class nor in conversation.
Thank you Professor Leonard.
THANKYOU SO MUCHFDNFJDSJAAAAA this it didnt make any sense to me before it was just a rule and now i understand where it comes from!!!! this is amazing thank u
At around 55 minutes, t is labeled as a dependent variables. However, since both x, y, and w respond to t, doesn't that make t the independent variable?
Someone who knows please reply to this comment. I was very confused by this part of the video as well. Thank you
I just noticed at 1:52:00 he explains that writing dependent instead of 'independent' was a mistake he made in writing one of the first trees. The mistake, I believe, was at 47:30
26:43 First time I have seen a math teacher give the finger to his students, lmao.
It’s not a rigorous proof, but it makes good intuitive sense!
i think i have a crush on Mr.leonard I can't stop watching his lectures.
@Professor Leonard, just a suggestion. At 1:50:19 you start talking about implicit differentiation for multivariable calculus. I think you should mention that on the the title of the video.
Thank you, prof. You are the best.❤❤❤❤❤❤
Wow...... An amazing lecture!
Why is it that when we do the partial derivatives of a non-parametric multivariable function [f(x,y)] we do NOT add the partial derivatives, but here we CAN add the partials ( multiplied by the full derivatives with respect to time) to get a single derivative with respect to time?
@ 1:27:50 why are we taking the partial derivative of "x" w/ respect to "u" when "x" has "u" as the its only independent variable? why not take a normal derivative "dx/du"? I'm aware that the outcome will be the same, I just want to get my notation right.
1:51:00 why it has to be partial derivative since only w depends on one independent variable which's x?
I really hope other chapters are coming soon/ he is not done posting calc 3 videos. I really really need them...
Are you gonna have videos for linear algebra too ?
+Bibek Adhikari I hope so
I think that Linear Algebra should be presented before any Calculus classes.
You don't need Calculus to deal with Linear Algebra, all you really need is an understanding for ordinary Algebra and some basic Trigonometry and you are good to go, whereas you actually have a lot of use for Linear Algebra in some Calculus courses, like the Wronskian matrix for example.
how about all who benefit from pro Lenard donate and ask for differential equations? I did
thank you so much for the video Professor
I love you professor leonard :)
01:27:53 Earlier you said that when there's only one path to choose (only one dependent variable on which the intermediate depends on), you use an ordinary derivative. So why do you use a partial derivative here? x depends only on u.
Professor, you said at time 1:27:54 that to get from X to U we use a partial derivative. This contradicts your statement towards the beginning of the video about "If there is only one independent variable we are connecting the intermediate variable by a normal d" (normal derivative NOT PARTIAL). So, my question is why do you use a partial in that case?
I was wondering the same thing. Did you end up figuring out why that is?
what a video man
Thank you very much sir, God bless you
1:37:40 We watch you professor! 💚
Great lecture!