Geometric Meaning of the Gradient Vector

Рет қаралды 169,807

Күн бұрын

What direction should you travel to increase your height on a mountain as fast as possible? What direction should you travel to keep your height constant (i.e. travel on a contour aka a level curve)? In this video we discuss the math of this problem, assuming we had some nice function describing the height of the mountain. The gradiant vector, whose components are the respective partial derivatives gives us the answer to the direction of maximal increase. Indeed, we saw previously how directional derivatives could be written in terms of the gradient vector. At the end we look at all of this with an actual topographical map full of contours.
0:00 The Mountain Problem
2:24 Deriving the Gradient Formula
6:07 Directional Derivatives
10:54 Topographical Maps
****************************************************
COURSE PLAYLISTS:
►CALCULUS I: • Calculus I (Limits, De...
► CALCULUS II: • Calculus II (Integrati...
►MULTIVARIABLE CALCULUS (Calc III): • Calculus III: Multivar...
►DIFFERENTIAL EQUATIONS (Calc IV): • How to solve ODEs with...
►DISCRETE MATH: • Discrete Math (Full Co...
►LINEAR ALGEBRA: • Linear Algebra (Full C...
***************************************************
► Want to learn math effectively? Check out my "Learning Math" Series:
• 5 Tips To Make Math Pr...
►Want some cool math? Check out my "Cool Math" Series:
• Cool Math Series
****************************************************
►Follow me on Twitter: / treforbazett
*****************************************************
This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.
BECOME A MEMBER:
►Join: / @drtrefor
MATH BOOKS & MERCH I LOVE:
► My Amazon Affiliate Shop: www.amazon.com/shop/treforbazett

Пікірлер: 253

@RyanMcCoppin 2 жыл бұрын

I was very confused when people said the gradient was "normal" to the curve. I thought they meant the function itself, not the "level curve". Now it makes complete sense! Thanks!

@juanbautistasoria757 3 ай бұрын

Same!

@sumittete2804 3 ай бұрын

Is rate of change of function minimum in the direction of tangent vector or in the direction opposite to gradient vector ?

@Suyogya77 2 ай бұрын

@@sumittete2804 rate of change of the function is minimum in the direction of the tangent vector i.e. in you move perpendicular to the gradient vector

@sumittete2804 2 ай бұрын

@@Suyogya77 But if i move opposite to gradient vector i.e 180° I'm getting rate of change of function as negative which is less than 0. Moving along tangent vector the rate of change of function is zero. So how ??

@Suyogya77 2 ай бұрын

@@sumittete2804 do you use telegram or something?

@9888565407 3 жыл бұрын

dude I love ya. that cleared everything about gradients in my mind. Thanks a lot bud.

@DrTrefor 3 жыл бұрын

Glad it helped!

@destreme9189 Ай бұрын

My professor rambled on for 2 hours and I didn't understand anything. Here you are explaining it perfectly in 15 minutes. THANK YOU SO MUCH

@zwhizzo4185 2 жыл бұрын

This is just brilliant. Clear voice, sound logic and great example.

@johnnytoobad7785 3 жыл бұрын

Man if my college calculus profs. were as articulate as Dr. Bazett..I would have gotten better grades in those classes. I'm now a retired software "geek" and really love watching these presentations. Very few folks who understand advanced math (and EE-Comp Sci) are good at teaching it to "undergrads". Of course I love the animations also.

@codstary1015 Жыл бұрын

Couldn't agree more, lot of lectures I have i can barely understand what they are trying to present. It's pretty funny that a you tuber can present ideas in a much more clear and straight forward matter.

@focusmaestro4013 10 ай бұрын

@@codstary1015 LOL, Its pretty naive of you to assume that Dr.Trefor is just another youtuber!

@dustincondon5557 3 жыл бұрын

Beautifully presented! It's such a cool topic, and using mountains as an analogy makes everything so intuitive.

@alkankondo89 3 жыл бұрын

Great derivation and application! The derivation of the gradient-vector formula and its justification were both quite easy to follow!

@rafidahmed2796 Жыл бұрын

I usually don't comment on videos but that's the best explanation i've ever watched to understand... i had this confusing for a long time and this lecture cleared that up! you deserve more subs!

@Anonymous-nz8wd 3 жыл бұрын

I have never seen such a beautiful explanation ever of Gradients love you

@user-wu8yq1rb9t 2 жыл бұрын

Of course I enjoyed it. For better understanding of the Gradient, I searched this subject, fortunately, I saw you and I just clicked! Thank you so much

@liuqing1995 3 жыл бұрын

The great combination of theory and a real example of a mountain in Vancouver. I enjoy the lesson series of Calculus so much.

@Party21304 Жыл бұрын

This video is spot on! Very nice. You just clarified gradient, level curves and the directional derivative in an intuitive way. I know understand the meaning behind the math. Thank you so much!

@chernihivka 3 жыл бұрын

great videos, Trefor, I have been looking for the explanations with geometrical insights vs just algebra on the board. This really helps to "see" the math. thanks!

@sadpotato5111 2 жыл бұрын

It is a wonderful thing to see your passion about mathematics, I'm assure you it is contagious and I love you because of it. I wish best for you with my all heart. Please do continue to make videos like that.

@zukofire6424 Жыл бұрын

had seen the videos pop up in the search results and never found the time to have a look. Now just did : I'm a fan! Thanks Prof. Bazett! :)

@wildertapiasaenz2333 18 күн бұрын

I found myself really “down the rabbit hole” with this concept because it just doesn’t mean anything until you visualise it. Your videos really helped me, thank you 🙏

@nicogovindsamy9022 3 жыл бұрын

This video is amazing. First time I'm seeing these concepts clearly since I started taking this course.

@HPLovecraftsCat9 Жыл бұрын

Thank you for this video. You are a very clear example of the fact that you don't need 3Blue1Brown levels of visual editing in order to explain something intuitively and clearly.

@milesborowsky8140 3 жыл бұрын

This is awesome. You're uploading right as I'm learning this material in class, and it's super helpful. I'm a math and education major too - this is such a great conceptual explanation. Thank you!

@PantheraOncaTV Ай бұрын

Man, you really teach what's important to understand the concepts, and you explain yourself perfectly! Amazing! You've gained a new subscriber 😁

@minnngb Жыл бұрын

i have been looking for a video to help me understand directional derivatives and the gradient for a week and this one was the most helpful!! the visual examples are amazing :)) thank you

@hope-wq9jd 2 жыл бұрын

after visualizing these concepts it became easier for me to perform the mathematical formulas thank you so much sir for the valuable information

@lakshitha8720 2 жыл бұрын

Best math channel. Massive respect. Thank you sir..❤️

@aln4075 3 ай бұрын

this video is one of the greatest one's that you can find on this topic

@Conorstew 3 жыл бұрын

My new favourite video of yours, the mountain example was great :) You taught me calc1 at Uvic last year and now you are teaching me calc 3. A true godsend, thanks Trefor!

@DrTrefor 3 жыл бұрын

Thanks Conor, really appreciate that. Good luck with math 200!

@youssefdirani 2 жыл бұрын

Super thanks Dr. May you be blessed

@engrfawadiqbalkhan786 Жыл бұрын

I am a civil engineer , now a days I am pursuing master's in structural engineering ,in structural engineering we use these concepts to find maximum stresses/strains , Before this video I tried a lot but couldn't get into the depth of concept but after watching this video ,I got the concept of it ,animations are very helpful .thankyou and keep up the good work.

@ishraqhossain1235 3 жыл бұрын

your bio says you try to do "evidence based pedagogical practices". This is just beautiful man. Hope you come up with more intuitive calculus videos

@viniciusdeobber8239 2 жыл бұрын

Great video, made it easy to understand. Thanks, professor Trefor!

@walac 3 жыл бұрын

Best math channel on KZfaq!

@RSS18 2 жыл бұрын

Perfectly explained. Thank you sir

@julioreyram 3 жыл бұрын

Brilliant explanation again :). It would be very good if you linked the previous explanations, i.e., the tension vector.

@ManojKumar-cj7oj 3 жыл бұрын

Thank God I found this channel 🙌

@TheWildStatistician 6 ай бұрын

Absolutely fantastic explainer! Way to go!

@scratchyourhead2383 2 жыл бұрын

Thank you sir ❤! You are clearing my such deep buried doubts .

@peterfriedman4912 3 жыл бұрын

Thank you so much. I read my textbook and understood about half of this material and watched this video a couple of times and now understand the gradient vector much better. You really helped me.

@DrTrefor 3 жыл бұрын

Glad it was helpful!

@electricalabc3411 3 жыл бұрын

Example of a mountain was superb to explain gradient..thanks bro

@LinLin-rs2bv 2 жыл бұрын

Great video that gives a brilliant straight explanation for the gradient vector. Hope to have your class in UVic.

@KaviPriyan-qt6vc 3 жыл бұрын

I am doing all possible steps to take this channel to a bigger audience

@TheEmad619 3 жыл бұрын

thank you so much for clearing the doubt. The video was very helpful.

@rivkamutka2833 Жыл бұрын

Amazing explanation. Thank you!

@joaomattos9271 Жыл бұрын

Fantastic way of teaching!!! I recommend the classes here in Brazil!

@nadenglish3811 3 жыл бұрын

amazing and so interesting! keep it up

@xiliu3526 3 ай бұрын

Such a brilliant video, it truly heps

@sergiolucas38 2 жыл бұрын

excellent video as always, nice example :)

@rupaupadhaya8888 2 жыл бұрын

This video is really helpful. Thank you so much,👏

@joliettraveler 3 жыл бұрын

Fantastic explanation. Now I understand the gradient for our purposes lies in the xy plane and that it points into the mountain.

@gibs40 Жыл бұрын

This channel is truly underrated

@hrkalita159 3 жыл бұрын

Man, you are Richard Feynman of our time

@DrTrefor 3 жыл бұрын

That is high praise!

@tarekezzat1893 Жыл бұрын

@@DrTrefor more than him!

@sanujitacharya794 Жыл бұрын

Literally the truth🫡☺️

@ABHISHEKSHARMA-gj6xp Жыл бұрын

Not high praise , the amount of hard work you have done to develop these is phenomenal ❤

@muhammedabdullvek7123 9 ай бұрын

@@tarekezzat1893of course not. What a stupid reply🤦🏽‍♂️

@olehborys1462 Жыл бұрын

great example, thank you for your video

@subhajitbarman4672 3 жыл бұрын

Such a clarity in your explanation....thank you so much sir, you cleared my hardest doubt...😊😊❤❤😊❤❤❤

@Indik47 6 ай бұрын

Excellent explanation

@at6727 3 жыл бұрын

you'll make us love calculus and maths!! thanks for including practical example of Vancouver island

@DrTrefor 3 жыл бұрын

haha I had fun with that part!

@sashamuller9743 3 жыл бұрын

awesome stuff! i love this subject

@anandita166 3 жыл бұрын

thankyou so much! this one's great!!🌟

@shashankjhaa6221 Жыл бұрын

Love your teaching sir...""LOVE""

@habilismayilov838 Жыл бұрын

Amazing explanation!!!! thank you so much, you make great influence in the world..

@ahmadwael8541 Ай бұрын

thanks a lot for the great explanation

@muktasane6046 Жыл бұрын

Thanks so much it helped me understand the gradient concept.

@maani315 3 жыл бұрын

Your demonstration is just amazing Sir....the best explanation of gradient vector on KZfaq....

@DrTrefor 3 жыл бұрын

Thanks a ton!

@ikarus1060 2 жыл бұрын

Ahhh, finally I fully got it, thx man:) The map help a lot. I knew what gradient is, but i strugled to get the geometric meaning

@DrTrefor 2 жыл бұрын

Glad it helped!

@chuz1818 3 жыл бұрын

Amazing explanation! Thank you so much!!

@DrTrefor 3 жыл бұрын

Glad you enjoyed it!

@thisukainol Ай бұрын

wow. great explanation

@rizalpurnawan3796 3 жыл бұрын

Amazing lecture! It essentially proves the notion that the gradient is orthogonal to the level set. Thanks a lot Sir Trefor.

@DrTrefor 3 жыл бұрын

You are welcome!

@sumittete2804 3 ай бұрын

Is rate of change of function minimum in the direction of tangent vector or in the direction opposite to gradient vector ?

@carultch 2 ай бұрын

@@sumittete2804 The direction of the gradient tells you the direction of steepest ascent, and the magnitude tells you the slope of that ascent. Opposite the gradient vector, is the direction of steepest descent. The rate of change of the function is minimized, if the input point travels perpendicular to the gradient vector. The contour lines are perpendicular to the gradient vector. The principle behind Lagrange multipliers comes from this idea. Given that the point in question follows a constrained path, the candidates for the local extreme value of the function's output will occur when the path and the contour line, share a common direction.

@AS-ix3qd 3 жыл бұрын

my mind is blown, finally I understand how the tangent unit vector gives a direction along which f(x,y) is constant, Thx alot Dr.

@DrTrefor 3 жыл бұрын

glad it helped!

@shubhamvadagave8961 2 жыл бұрын

Great explanation! Thanks alot

@bimankalita2960 8 ай бұрын

I wish there was the option of giving more than one like. Superb explanation!

@continnum6540 2 жыл бұрын

Thanks a lot sir 🔥🔥🔥

@richardslesinski4137 2 жыл бұрын

amazingly clear. thank you

@somanathdash3153 2 жыл бұрын

Oh god it helped me so much thanks sensai

@maximlavrenko1164 Жыл бұрын

rewatched it several times, started losing hope but then it clicked and I was like 'wait that makes sense!'

@MamToCos 7 ай бұрын

Good works! Thank you.

@user-do1kj6ov6z 11 ай бұрын

It's awesome interpretation💖

@mrnicesmell2022 6 ай бұрын

A legend is living among us!!!!!!!!!

@naikyasir2871 Жыл бұрын

Really enjoyed

@BoZhaoengineering 3 жыл бұрын

The Geometric element is fascinating. But the algebraic dot product provide a solid conclusion.

@vaibhavgaur261 2 жыл бұрын

bruh, u nailed it🙏🙏

@edgbaston149 2 жыл бұрын

Thank you so much🙏 You're amazing. 😊😎

@pawanadhikari110 3 жыл бұрын

Absolute trivial explanation.

@optimistprime6741 2 жыл бұрын

You’re literally perfect

@shreyapandey7154 10 ай бұрын

very informative

@alejrandom6592 3 жыл бұрын

Awesome video!

@chrismarklowitz1001 Жыл бұрын

I like to think of the normal vector to the contour plot as the shortest distance between two points is a straight line. And when the distance between them next contour is infinitesimal. It is a parallel line. And anything other than normal is going to be more than just going normal.

@hemrajkumawat801 3 жыл бұрын

Just amazing

@punditgi 3 жыл бұрын

Excellent!

@shivarapolu3180 3 жыл бұрын

understood the beauty of multivariable calculus and gradient operator. Thanks a lot sir :)))

@DrTrefor 3 жыл бұрын

Happy to help!

@AK-sh8df 2 жыл бұрын

Wonderful video

@santhoshkumar-lv4ph 3 жыл бұрын

I love so much u are a genius teacher

@nickknight5373 3 жыл бұрын

Very well-explained.

@DrTrefor 3 жыл бұрын

Glad you think so!

@noonesland2471 2 жыл бұрын

I love you, thank you!

@BscPhysicsRGLectures 3 жыл бұрын

Love you sir!

@dodo-js5gw 3 жыл бұрын

Bonjour ! And thank you for yours amazing videos. I would line to know if you have an intuitive way to explain the gradient of a vector valued function that gives a matrix. Do we have also the notion of direction and highest variation for matrices ? Thanks !

@wildertapiasaenz2333 18 күн бұрын

Also really important how you pointed out grad f as in the x-y plane as that also can be very confusing initially thinking about it as the gradient itself but of course that’s why we need 😊