I don't even watch the video anymore cause I've graduated and no longer do math related things, but I come here to like and support because it helped me so much especially in my final year of high school. Thanks again Professor Leonard
@@fuccckckkkkckkck He's probably talking about calculus 1 at high school. As far as I know, you can't take calculus 3 during your four years in high school. The highest I can go right now is finished calculus 2 by senior year, which is by going to local community college. Calculus 3 may be still be possible to finish by high school but damm that's way to fast.
0:00 Proof of Directional Derivatives 37:47 Example 1 56:43 Gradient Vectors 1:09:09 Properties of Gradient Vectors 1:19:57 Inspirational speech 1:34:10 Example 2 1:40:46 Example 3 1:45:12 Example 4 2:00:49 Example 5 2:09:00 Example 6 2:19:31 Example 7 "realistic example"
you have to write a book, about how to study math, and how to solve problems; your lectures as well as your problem solving strategies are very well organized. i think if you write a book, it will be best seller.
Prof Leonard, I just wanted to thank you dearly for all your videos. Been using them since first year and they are highly helpful. We really appreciate it. By the time we cover the sections in class, I am leaps and bounds ahead, all thanks to you. Thank you Sir. God Bless You. Much Love and appreciation from South Africa, Johannesburg.
the next generations will listen to his legendary stories about how he saved the world from dc villains and his crazy math skills from us in the cold winter nights in front of the stove
If Nobel prizes were given in Teaching, I'm sure you would be the first person deserves it. Actually, you would be the only one who could take it but.... I can't describe my grace or how awesome you are in any language. All I can do is a big THANK YOU.
I graduated with a science degree, but never understood Calculus as it was poorly taught. It spins my propeller to go over these lectures and finally understand the Mathematics. Rock on Professor Leonard! Thank you too.
Can we just appreciate this man's passion to teach math unlike 99% of professors in any universities. I don't usually comment on videos but I have the need to after watching your videos. Very helpful when it comes to understanding the topics and solving related problems.
@@ProfessorLeonard Professor Leonard, please don't feel discouraged by the silence from your students in your lecture videos, they are just shy. Lots of students are like that.
I'd just like to say that your videos are the most informative and best math videos I've seen on youtube. After watching it, I completely appreciate the length of time your taking to solidify the understanding of what's happening mathematically instead of just going through the numbers. I'm an Electrical Engineering student taking Electromagetic Fields and was looking for a review on multivariable calculus and your videos are absolutely perfect. Thank you so much for what you are doing here Sir! These lessons are incredible!
For the linear algebra class, no one can be comparable to Gilbert Strang. He likes to play around with all the concepts without having planned about anything before hand, and also lets his students play with him.
@Susuya Juuzou I think there is no stupid person asking if others are stupid or not, and this is the thing about personal preference and opinion of each person, not about the intelligence. At least, I respect your thought of not preferring Gilbert, and I respect enough to not say you are stupid just because your preference is different from mine.
@Susuya JuuzouI don't know what your point is, but no matter what you say, I like Gilbert Strang for Linear Algebra, and Leonard for Calculus. That's it.
He's so good. You can tell he puts a ton of time into lesson planning and thinking through the various levels of understanding a student will have at each stage. I think the final example on this video really brings everything home.
When you explained how you could simplify the denominator to h at 33:49 and then said "what!" I genuinely laughed out loud with amazement and said the same thing as my mind was blown. I've never been that into a lecture before. Your a great teacher and a greater presenter professor Leonard. Thank you for making these videos!
You are an excellent teacher. You make the subject material interesting and informative. I wish there were more college math teachers who have your ability and talent
I want to start a crowdfunding page to get you to produce Linear Algebra and Differential Equations lecture playlists. Not only do I understand this math now, I love it. You've gotten me more excited about math than any other teacher has ever. I feel honored to even watch you on UA-cam.
Every time I zone out and get distracted by something else while watching these videos, Professor Leonard is instantly like "hey, look at the board please, are you listening?".
Professor Leonard, thank you for another well explained and defined video/lecture on Finding Directional Derivatives and Gradients in Calculus Three. The solid explanation of the theory that develops Directional Derivatives and Gradients are very important for all students to understand in Multivariable Calculus. Once students understand the theory, the computational part of this topic is easy to follow. The examples and pictures also helps with understanding the material.
Absolutely fantastic. If your want to understand Neural networks in Machine learning, YOU NEED TO See this lecture before you touch neural network algorythm. No one has explained it better than prof Leonard. Thank you sir
My Engineering Analysis 2 class is so focused on calculations we miss a lot of the underlying meaning I am looking for. So thank you for making this and being a resource I can use to better understand concepts that are important to me.
youre an amazing teacher and i will never forget you and how contagious your enthusiasm for math is! really changed my perspective. i like watching these videos as much as watching tv hehe thanks and youre doing a great job!
hello professor , I am one of the students in University Of Toronto St.George Campus. Your videos are freaking awesome, they are much more better than our professor. Hope there will be more fantastic videos in the future.
I managed to understand one topic per day through your vids. It is absolutely amazing, much better than my math1014 prof. I think I may be able to skip all of the math2011 classes with your vids, minimizing the time outside home due to covid. Thanks so much! Math2011, I'm coming to conquer you!
Prof Leonard, you are truly the best, you are probably the best math teacher in the world!! I'm kinda sad i"m almost finished with these videos. We learn and we have fun in your classes
bruh i dont know calc 1 and 2 and am jumping straight to calc cuz online exams helped me pass the previours ones and now i have actualt exams in a month. Plz pray.
I attended lectures from MIT Calculus III, progress made ok sofar, but, I don't understand that, how could you make this thing be so understandable, the way that others can't. I truly appreciate your effort and talents... 🙏
Thank you very much for your classes professor. You explain things so clearly and thoroughly. I love how you emphasize where things come from so we understand what we are calculating. : )
These students really should get use to seeing, in this written form. (which reminds me of inner product spaces, defined in a banach space) I have studied Advanced Integration Methods, Advanced Real Analysis, Complex Analysis, Locally Convex Spaces, Real Analysis, Banach Space Theory, Dimension Theory, Functional Analysis, Operator Theory, Operator Spaces, Operator Algebras, Advanced Convex Analysis. So if they take Calculus 3, they have to be comfortable with a lot of Notation. Thats just a small amout of mathemathematics i have studied, So Mathematical Notation is something these students need to get acquainted with.
Great and depth teaching with clear understanding sir.... Great maths teacher.. Thank you sir..I am as a student have got the clarity on the concepts by ur teaching..tq sir.. Helpful videos sir.. Mainly For the students..
The book "University Physics With Modern Physics" by Young & Freedman is great for physics - preferably the 13th or 14th edition. I don't think you even need to have any prior experience with physics to be able to grasp it - it only seems to expect a familiarity with Calculus 1. It starts with vectors, 2D motion and Newton's laws, and everything is explained very well, and with a relaxed sense of humour.
i was blowned away when he didnt squared the denominator then my breath came back at 1:36:53 ..... even i searched the derivative of 1/x^2+y^2 in google and i even confused google , because that much i trust my professor leonard.....
YOUR SHOULD DEFINATELY DO DIFFERENTIAL EQUATIONS!!! YOU HELPED ME PASS CALC 2 WITH A 99. THIS SEMESTER I HAVE %100 I MY CALC 3 CLASS THANKS TO YOU. PLEASE DO DIFF. EQ.
Really amazing and helpful video. At least not other youtube videos tell me the direction is on xy plane, although it is reasonable and easy to tell. But it's an important fact for me when I go deeper and deeper into the directional derivatives and gradient. Thanks!
I have noticed that a lot of my Engineering Physics courses often refer specifically to this course - it's usually something about vector projections on surfaces (like when they talk about pressure on spherical underwater surfaces in Fluid Statics, for example), gradients in Partial Differential Equations, or the Divergence Theorem.
I believe at 1:36:00 in the partial derivative of x and y, the denominators should be squared (x^2 + y^2)^2. Actually, he corrects this after 1:37:10. All good!
Excellent and graphic explaination of the Directional Derivative. But, Professor, in this derivation (at about 36 mins of the video), we are finding the SLOPE of the tangent line directed ALONG the Unit vector (u^) or to be precise, ALONG a direction PARALLEL to the UNIT VECTOR (u^). If this is so, the GRADIENT of f(x,y) vector will be ALWAYS along u^. Since the SLOPE of f(x,y) ALONG unit vector u^ denoted by D (sub u^) = GRAD f(x,y) . u^, by the above argument that you have so well explained ( and used in deriving the SLOPE of the surface i.e f(x,y) ALONG specific direction u^), it should mean that this "dot" product MUST be = GRAD f(x,y) ALWAYS, isn't it? May be my understanding is inadquate. Can you please explain?
I feel like there's a much simpler and more intuitive way to do this: The directional derivative of f should be the rate of change of f as you move in a certain direction, so we'd like to parameterize a path through a point in some fixed direction, and then we will calculate the rate of change of our function as we move along that path. Clearly this means we should start by trying to parameterize a path. We need a point and a direction Take the path (x(s), y(s)) = (x0 + a*s, y0 + b*s). We can write this as (x0, y0) + s*(a, b), and written this way it's clear that this is a straight path through the point (x0, y0) in the direction of the vector (a, b). We'll think of (x0, y0) as our starting point. We are at (x0, y0) when s=0, and as we vary s to be non-zero we move away from the starting point. We'd also like for (a, b) to be a unit vector. Why? Because then when s = 7, say, then we've moved a distance 7 away from the starting point, since (a, b) has length 1 so 7(a,b) has length 7. Therefore, s doesn't just control how far away we are from the starting point; it is literally the distance* we are from the starting point. Now we are ready to calculate the directional derivative at (x0, y0). The ordinary derivative is the rate of change of f with respect to distance traveled along the the axis (usually considered the x-axis). The directional derivative is no different; it's the rate of change with respect to distance traveled, but this time as we move along this new "axis" given by the straight path we've parameterized. Along this path, f is given by f(x(s), y(s)) = f(x0 + a*s, y0 + b*s)), and since s literally is the distance traveled, the rate of change at (x0, y0), what we call the "directional derivative" at (x0, y0), is just df/ds evaluated at s=0. Notice that it was important that (a, b) is a unit vector because only then is s the distance traveled. Now, using the multivariable chain rule, df/ds = df/dx * a + df/dy * b, where by df/dx and df/dy I really mean the partial derivatives (can't write that symbol on youtube) evaluated at (x0, y0). Therefore the directional derivative is (df/dx, df/dy) dot (a,b). The first thing in the dot product is the gradient of f evaluated at (x0, y0) and (a, b) is the unit vector indicating the direction.
32:58 I love how he just pulls out another pen from his pocket instead of picking up the one he dropped. Ultimate prepared math teacher with dry-erase markers to spare in his utility belt. Is he Batman or Superman?
Professor Leonard, it took me a while to understand how you obtained the partial derivative of fx and fy at 1:36:00. I believe you forgot to take the denominator to the power of 2. that is, -2x/(x^2 +y^2 )^2, and -2y/(x^2 +y^2 )^2 respectively. nvm I spent 1 hour trying to figure out just to find out someone in the class pointed it out a few minutes later... lol
35:00 i don't undrstand why the directional derivative defined as limit evaluates nicely as dot product of gradient and the unit vector along that direction. now i am looking for that proof. great explaination btw, it cant be explained better than this.
Was studying for an exam, paused the video, procrastinated for 30 minutes, then came back to this tab at 19:31 and found a disappointed professor Leonard starring at me
I don't even watch the video anymore cause I've graduated and no longer do math related things, but I come here to like and support because it helped me so much especially in my final year of high school. Thanks again Professor Leonard
lol high school...
lol middle school grammar at most
What kind of high school did you go to if you already learned how to do this lol
@@fuccckckkkkckkck He's probably talking about calculus 1 at high school. As far as I know, you can't take calculus 3 during your four years in high school. The highest I can go right now is finished calculus 2 by senior year, which is by going to local community college. Calculus 3 may be still be possible to finish by high school but damm that's way to fast.
Go Ham not all people live in an English-speaking country, you need to remember that.
0:00 Proof of Directional Derivatives
37:47 Example 1
56:43 Gradient Vectors
1:09:09 Properties of Gradient Vectors
1:19:57 Inspirational speech
1:34:10 Example 2
1:40:46 Example 3
1:45:12 Example 4
2:00:49 Example 5
2:09:00 Example 6
2:19:31 Example 7 "realistic example"
I believe there is an error on example #2. The partial of x should be -2x/(x^2+y^2)^2 or -2x/(x^4+y^4+(xy)^2)
Yeah, and he already fixed it!
Thank you So Much
you have to write a book, about how to study math, and how to solve problems; your lectures as well as your problem solving strategies are very well organized. i think if you write a book, it will be best seller.
which book,,,,,,tell me pls
jawad he didnt write a book zack said he wish he did. anyways go get james stewart book.
Maybe the world would be better off if we weren’t spoon fed
The quality of your teaching is top notch; everything is explained well and in depth. You're doing such a great deed professor, thank you!!
Been watching this man since I was in yr.11, now I am in my 2nd year of Mechanical Engineering degree.
Hearing this 1:20:08 meant a lot. :)
Prof Leonard, I just wanted to thank you dearly for all your videos. Been using them since first year and they are highly helpful. We really appreciate it. By the time we cover the sections in class, I am leaps and bounds ahead, all thanks to you. Thank you Sir. God Bless You. Much Love and appreciation from South Africa, Johannesburg.
0:00 - 37:45 = Directional Derivative "Proof"
56:00 - 1:33:30 = Gradient "Proof"
Dude, thanks so much.
A Daily Planet salary don't mean much these days...Nice to see Clark Kent still going strong.
I pay my employees well
@@ParabolicMind no you dont you just give them minimum wage
the next generations will listen to his legendary stories about how he saved the world from dc villains and his crazy math skills from us in the cold winter nights in front of the stove
this guy is proof that math is way more than just random numbers on paper. You are a truly amazing math teacher...
If Nobel prizes were given in Teaching, I'm sure you would be the first person deserves it. Actually, you would be the only one who could take it but.... I can't describe my grace or how awesome you are in any language. All I can do is a big THANK YOU.
ua-cam.com/video/l6YOfcOf_3c/v-deo.html
This professor will always hold a special place in my heart for being the absolute best
I graduated with a science degree, but never understood Calculus as it was poorly taught. It spins my propeller to go over these lectures and finally understand the Mathematics. Rock on Professor Leonard! Thank you too.
Can we just appreciate this man's passion to teach math unlike 99% of professors in any universities. I don't usually comment on videos but I have the need to after watching your videos. Very helpful when it comes to understanding the topics and solving related problems.
Thank you! I really do appreciate it and I'm glad you have found the videos useful
@@ProfessorLeonard
Professor Leonard, please don't feel discouraged by the silence from your students in your lecture videos, they are just shy.
Lots of students are like that.
I'd just like to say that your videos are the most informative and best math videos I've seen on youtube. After watching it, I completely appreciate the length of time your taking to solidify the understanding of what's happening mathematically instead of just going through the numbers. I'm an Electrical Engineering student taking Electromagetic Fields and was looking for a review on multivariable calculus and your videos are absolutely perfect. Thank you so much for what you are doing here Sir! These lessons are incredible!
plz professor... do some videos about linear algebra!
gilbert strang is your friend.
@Susuya Juuzou yeah gilbert strang would only be good after already taking a course in linear algebra imo
For the linear algebra class, no one can be comparable to Gilbert Strang. He likes to play around with all the concepts without having planned about anything before hand, and also lets his students play with him.
@Susuya Juuzou I think there is no stupid person asking if others are stupid or not, and this is the thing about personal preference and opinion of each person, not about the intelligence. At least, I respect your thought of not preferring Gilbert, and I respect enough to not say you are stupid just because your preference is different from mine.
@Susuya JuuzouI don't know what your point is, but no matter what you say, I like Gilbert Strang for Linear Algebra, and Leonard for Calculus. That's it.
He's so good. You can tell he puts a ton of time into lesson planning and thinking through the various levels of understanding a student will have at each stage. I think the final example on this video really brings everything home.
Professor Leonard is the best teacher! Thanks to him I pass calculus 1, now i will pass calculus 2!!!!
When you explained how you could simplify the denominator to h at 33:49 and then said "what!" I genuinely laughed out loud with amazement and said the same thing as my mind was blown. I've never been that into a lecture before. Your a great teacher and a greater presenter professor Leonard. Thank you for making these videos!
You are an excellent teacher. You make the subject material interesting and informative. I wish there were more college math teachers who have your ability and talent
I want to start a crowdfunding page to get you to produce Linear Algebra and Differential Equations lecture playlists. Not only do I understand this math now, I love it. You've gotten me more excited about math than any other teacher has ever. I feel honored to even watch you on UA-cam.
Great delivery! The approach makes internalization effortless!
I'm so glad your doing calc 3. I started to panic when I didn't see 13.6! I'm glad you keep making these videos. Thanks a million.
I was so immersed I almost raised my hand haha! great video
Every time I zone out and get distracted by something else while watching these videos, Professor Leonard is instantly like "hey, look at the board please, are you listening?".
I must be ULTRA IMMERSED, I raise my hand and respond 😂
Same :D
I always raise mine.
your stupid for raising your hand. your not in his class
Professor Leonard, thank you for another well explained and defined video/lecture on Finding Directional Derivatives and Gradients in Calculus Three. The solid explanation of the theory that develops Directional Derivatives and Gradients are very important for all students to understand in Multivariable Calculus. Once students understand the theory, the computational part of this topic is easy to follow. The examples and pictures also helps with understanding the material.
you are best professor i had ever know
Hats off for this math wizard
Absolutely fantastic. If your want to understand Neural networks in Machine learning, YOU NEED TO See this lecture before you touch neural network algorythm. No one has explained it better than prof Leonard. Thank you sir
I didn't know what I was gonna do without your videos ... I have been watching your videos for 3 years ... very very helpful ... God bless you
My Engineering Analysis 2 class is so focused on calculations we miss a lot of the underlying meaning I am looking for. So thank you for making this and being a resource I can use to better understand concepts that are important to me.
10:30 valid more than ever in 2020
Agreeeeeee
So true
you beat me to the joke, I applaud you sir.
youre an amazing teacher and i will never forget you and how contagious your enthusiasm for math is! really changed my perspective. i like watching these videos as much as watching tv hehe thanks and youre doing a great job!
hello professor , I am one of the students in University Of Toronto St.George Campus. Your videos are freaking awesome, they are much more better than our professor. Hope there will be more fantastic videos in the future.
What an absolutely fantastic professor. I am so impressed by his teaching ability.
I managed to understand one topic per day through your vids. It is absolutely amazing, much better than my math1014 prof. I think I may be able to skip all of the math2011 classes with your vids, minimizing the time outside home due to covid. Thanks so much! Math2011, I'm coming to conquer you!
Thank you very much ! Best youtube channel for calc.
Almost raised hand and responded to your questions : )
Again you are the absolute best !!
Excellent teaching.
I am re-learning calculus, especially multi-variable calculus.
As a math major and math tutor, this guy is freakin' GOOD.
professor Arnold is my immortal hero
Prof Leonard, you are truly the best, you are probably the best math teacher in the world!!
I'm kinda sad i"m almost finished with these videos. We learn and we have fun in your classes
quite possibly the best teacher on youtube
fell behind in class and am using this to catch up appreciate it so much!
bruh i dont know calc 1 and 2 and am jumping straight to calc cuz online exams helped me pass the previours ones and now i have actualt exams in a month. Plz pray.
Thank you professor! Your teaching way is much easier to understand than any teachers.
YOU ARE THE BEST I'VE EVER MET
This lecture is a masterpiece!
The man whom I will like to call my math teacher is not my UNI instructor, rather he is that guy....love the way he emphasis important points...
Truly magnificent stuff, mixing clear teaching with a great sense of humour. You really have the whole shebang.
1:36:00 deadass gave me an early mid-life crisis. I thought I couldn't do derivatives right and was having a meltdown.
This just happened to me lmao, i went backed and watched a patrickJMT video to make sure
Bro just happened to me i was going crazy thanks for this comment haha xD
what is a dead ass
DR1FTER omg same
same. especially when he didn't correct the squared part above
You are a god damn legend
15:53
Laying down the law!
Great lectures for visual learners.
I attended lectures from MIT Calculus III, progress made ok sofar, but, I don't understand that, how could you make this thing be so understandable, the way that others can't. I truly appreciate your effort and talents... 🙏
Thank you very much for your classes professor. You explain things so clearly and thoroughly. I love how you emphasize where things come from so we understand what we are calculating. : )
I felt your excitement when you dropped the marker. That was honestly a smooth way of proving it
Thank you so much for all of the videos that you do, I am very grateful for them all.
Cheers!! Ive my first Midterm on 9th of April and this is the last topic which is included.... Thank you, Mr professor you helped me allot.
Studying this before I take my machine learning class is extremely helpful!
These students really should get use to seeing, in this written form. (which reminds me of inner product spaces, defined in a banach space) I have studied Advanced Integration Methods, Advanced Real Analysis, Complex Analysis, Locally Convex Spaces, Real Analysis, Banach Space Theory, Dimension Theory, Functional Analysis, Operator Theory, Operator Spaces, Operator Algebras, Advanced Convex Analysis. So if they take Calculus 3, they have to be comfortable with a lot of Notation. Thats just a small amout of mathemathematics i have studied, So Mathematical Notation is something these students need to get acquainted with.
The best video on Du. I'm glad I found you Prof. Leonard.
Great and depth teaching with clear understanding sir.... Great maths teacher.. Thank you sir..I am as a student have got the clarity on the concepts by ur teaching..tq sir.. Helpful videos sir.. Mainly For the students..
Wish you did physics
The book "University Physics With Modern Physics" by Young & Freedman is great for physics - preferably the 13th or 14th edition.
I don't think you even need to have any prior experience with physics to be able to grasp it - it only seems to expect a familiarity with Calculus 1.
It starts with vectors, 2D motion and Newton's laws, and everything is explained very well, and with a relaxed sense of humour.
Best Math Professor Ever👍
i was blowned away when he didnt squared the denominator then my breath came back at 1:36:53 ..... even i searched the derivative of 1/x^2+y^2 in google and i even confused google , because that much i trust my professor leonard.....
1:39:01 is the Fx & Fy equation realy correct?😥
Twelve minutes in and I already have a better concept of what’s going on.
YOUR SHOULD DEFINATELY DO DIFFERENTIAL EQUATIONS!!! YOU HELPED ME PASS CALC 2 WITH A 99. THIS SEMESTER I HAVE %100 I MY CALC 3 CLASS THANKS TO YOU. PLEASE DO DIFF. EQ.
Yes definitely what I am looking for. Your explanation is resolved my question about transport equation sir! Thank you so much!
Really amazing and helpful video. At least not other youtube videos tell me the direction is on xy plane, although it is reasonable and easy to tell. But it's an important fact for me when I go deeper and deeper into the directional derivatives and gradient. Thanks!
I have noticed that a lot of my Engineering Physics courses often refer specifically to this course - it's usually something about vector projections on surfaces (like when they talk about pressure on spherical underwater surfaces in Fluid Statics, for example), gradients in Partial Differential Equations, or the Divergence Theorem.
For those who just wanna go directly to the juicy stuff without actually understanding the mechanism, go to 37:43! There, saved you half an hour!
I believe at 1:36:00 in the partial derivative of x and y, the denominators should be squared (x^2 + y^2)^2. Actually, he corrects this after 1:37:10. All good!
Thanks for a awesome lesson! I wouldn't have understood what was going on in my cal 3 class right now if it wasnt for this video lecture
You are hilarious and very entertaining, plus your explanations are really good. Thanks! :)
Im sorry, but at 1 hour an 35 min when he takes the partial derivatives inst it suppose to be -2x/(x^2+y^2)^2 for fx and similarly for fy?
Nice a student caught it! Fixed
yup, you are right!
His student corrected him at 1:36:57
you are the best professor leonard
gradient - 56:45
Bookmarks for future me:
1:15:00 - properties of del f
Thank you ,sir. Huge respect from Nepal.
1:08:55 properties of gradient
its helpful lecture thanks
Professor Leonard
"don't do math for the sake of maths, that sucks" is such a good quote. What a guy
0:05
You know a professor is brilliant when he introduces a new idea, the student says “yesss”
Excellent and graphic explaination of the Directional Derivative. But, Professor, in this derivation (at about 36 mins of the video), we are finding the SLOPE of the tangent line directed ALONG the Unit vector (u^) or to be precise, ALONG a direction PARALLEL to the UNIT VECTOR (u^). If this is so, the GRADIENT of f(x,y) vector will be ALWAYS along u^. Since the SLOPE of f(x,y) ALONG unit vector u^ denoted by D (sub u^) = GRAD f(x,y) . u^, by the above argument that you have so well explained ( and used in deriving the SLOPE of the surface i.e f(x,y) ALONG specific direction u^), it should mean that this "dot" product MUST be = GRAD f(x,y) ALWAYS, isn't it? May be my understanding is inadquate. Can you please explain?
56:43
1:40:46
done
I feel like there's a much simpler and more intuitive way to do this:
The directional derivative of f should be the rate of change of f as you move in a certain direction, so we'd like to parameterize a path through a point in some fixed direction, and then we will calculate the rate of change of our function as we move along that path.
Clearly this means we should start by trying to parameterize a path. We need a point and a direction Take the path (x(s), y(s)) = (x0 + a*s, y0 + b*s). We can write this as (x0, y0) + s*(a, b), and written this way it's clear that this is a straight path through the point (x0, y0) in the direction of the vector (a, b). We'll think of (x0, y0) as our starting point. We are at (x0, y0) when s=0, and as we vary s to be non-zero we move away from the starting point. We'd also like for (a, b) to be a unit vector. Why? Because then when s = 7, say, then we've moved a distance 7 away from the starting point, since (a, b) has length 1 so 7(a,b) has length 7. Therefore, s doesn't just control how far away we are from the starting point; it is literally the distance* we are from the starting point.
Now we are ready to calculate the directional derivative at (x0, y0). The ordinary derivative is the rate of change of f with respect to distance traveled along the the axis (usually considered the x-axis). The directional derivative is no different; it's the rate of change with respect to distance traveled, but this time as we move along this new "axis" given by the straight path we've parameterized. Along this path, f is given by f(x(s), y(s)) = f(x0 + a*s, y0 + b*s)), and since s literally is the distance traveled, the rate of change at (x0, y0), what we call the "directional derivative" at (x0, y0), is just df/ds evaluated at s=0. Notice that it was important that (a, b) is a unit vector because only then is s the distance traveled. Now, using the multivariable chain rule, df/ds = df/dx * a + df/dy * b, where by df/dx and df/dy I really mean the partial derivatives (can't write that symbol on youtube) evaluated at (x0, y0).
Therefore the directional derivative is (df/dx, df/dy) dot (a,b). The first thing in the dot product is the gradient of f evaluated at (x0, y0) and (a, b) is the unit vector indicating the direction.
You are very good. I am glad I found you
32:58 I love how he just pulls out another pen from his pocket instead of picking up the one he dropped. Ultimate prepared math teacher with dry-erase markers to spare in his utility belt. Is he Batman or Superman?
I appreciate your lectures.
Yes, I needed help with the Dorito!(gradient) Thank You Professor Lenorad!!
Professor Leonard, it took me a while to understand how you obtained the partial derivative of fx and fy at 1:36:00. I believe you forgot to take the denominator to the power of 2. that is, -2x/(x^2 +y^2 )^2, and -2y/(x^2 +y^2 )^2 respectively.
nvm I spent 1 hour trying to figure out just to find out someone in the class pointed it out a few minutes later... lol
Lol, I was looking for this comment. I was so confused as well.
Calc III was started to seem pretty dry, but this right here, this is the good stuff.
I'm here to learn, but also to continue the "bless you" chain at 1:55...BLESS YOU
I shall also continue the chain at 1:55, bless you!
Thanks for the upload...you are carrying me through this semester. I may as well be taking an online course! Get accredited! :)
Paused at 1:36:21 to read the comments and verify if the partial derivatives were correct...
Note to self:
First find del f at point
Write in vector comp then plug in
Second find unit vector
Third times del f with unit v
35:00 i don't undrstand why the directional derivative defined as limit evaluates nicely as dot product of gradient and the unit vector along that direction. now i am looking for that proof. great explaination btw, it cant be explained better than this.
I bet his students REALLY have no questions when he asks "Do you have any questions"
you are amazing man ! you helped me quite much to understand all that stuff ! Keep going ! Greetings from Germany
here i am back at this video 8 years later since i went back to school for my masters lol
Was studying for an exam, paused the video, procrastinated for 30 minutes, then came back to this tab at 19:31 and found a disappointed professor Leonard starring at me
Damn you're buff as hell and 10x better than my professor...thank you for this free education
Directional der 51:50