Geometric Meaning of the Gradient Vector
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- Опубліковано 29 тра 2024
- 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
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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!
Same!
Is rate of change of function minimum in the direction of tangent vector or in the direction opposite to gradient vector ?
@@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
@@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 ??
@@sumittete2804 do you use telegram or something?
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
dude I love ya. that cleared everything about gradients in my mind. Thanks a lot bud.
Glad it helped!
This is just brilliant. Clear voice, sound logic and great example.
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.
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.
@@codstary1015 LOL, Its pretty naive of you to assume that Dr.Trefor is just another youtuber!
Man, you really teach what's important to understand the concepts, and you explain yourself perfectly! Amazing! You've gained a new subscriber 😁
Beautifully presented! It's such a cool topic, and using mountains as an analogy makes everything so intuitive.
Great derivation and application! The derivation of the gradient-vector formula and its justification were both quite easy to follow!
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!
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!
I have never seen such a beautiful explanation ever of Gradients love you
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!
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.
Glad it was helpful!
The great combination of theory and a real example of a mountain in Vancouver. I enjoy the lesson series of Calculus so much.
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!
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.
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.
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 🙏
This video is amazing. First time I'm seeing these concepts clearly since I started taking this course.
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
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
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!
Thanks Conor, really appreciate that. Good luck with math 200!
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.
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! :)
after visualizing these concepts it became easier for me to perform the mathematical formulas thank you so much sir for the valuable information
Perfectly explained. Thank you sir
this video is one of the greatest one's that you can find on this topic
Best math channel. Massive respect. Thank you sir..❤️
Great video, made it easy to understand. Thanks, professor Trefor!
Fantastic way of teaching!!! I recommend the classes here in Brazil!
Absolutely fantastic explainer! Way to go!
Amazing explanation!!!! thank you so much, you make great influence in the world..
Thank God I found this channel 🙌
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
Glad it helped!
Such a brilliant video, it truly heps
Example of a mountain was superb to explain gradient..thanks bro
Super thanks Dr.
May you be blessed
you'll make us love calculus and maths!!
thanks for including practical example of Vancouver island
haha I had fun with that part!
Amazing explanation. Thank you!
Best math channel on UA-cam!
I am doing all possible steps to take this channel to a bigger audience
This video is really helpful. Thank you so much,👏
Fantastic explanation. Now I understand the gradient for our purposes lies in the xy plane and that it points into the mountain.
Thank you sir ❤! You are clearing my such deep buried doubts .
Such a clarity in your explanation....thank you so much sir, you cleared my hardest doubt...😊😊❤❤😊❤❤❤
Great video that gives a brilliant straight explanation for the gradient vector. Hope to have your class in UVic.
thank you so much for clearing the doubt. The video was very helpful.
Love your teaching sir...""LOVE""
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.
glad it helped!
Excellent explanation
great example, thank you for your video
amazing and so interesting! keep it up
Your demonstration is just amazing Sir....the best explanation of gradient vector on UA-cam....
Thanks a ton!
excellent video as always, nice example :)
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.
Thanks so much it helped me understand the gradient concept.
Great explanation! Thanks alot
Brilliant explanation again :). It would be very good if you linked the previous explanations, i.e., the tension vector.
rewatched it several times, started losing hope but then it clicked and I was like 'wait that makes sense!'
thanks a lot for the great explanation
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
Amazing lecture!
It essentially proves the notion that the gradient is orthogonal to the level set.
Thanks a lot Sir Trefor.
You are welcome!
Is rate of change of function minimum in the direction of tangent vector or in the direction opposite to gradient vector ?
@@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.
I wish there was the option of giving more than one like. Superb explanation!
amazingly clear. thank you
wow. great explanation
Amazing explanation! Thank you so much!!
Glad you enjoyed it!
awesome stuff! i love this subject
thankyou so much! this one's great!!🌟
Good works! Thank you.
It's awesome interpretation💖
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 😊
very informative
This channel is truly underrated
Thanks a lot sir 🔥🔥🔥
I love so much u are a genius teacher
Really enjoyed
understood the beauty of multivariable calculus and gradient operator. Thanks a lot sir :)))
Happy to help!
Wonderful video
Absolute trivial explanation.
Thank you so much🙏
You're amazing. 😊😎
Very well-explained.
Glad you think so!
I love it Great sir
Oh god it helped me so much thanks sensai
Awesome video!
Just amazing
great job.
Thanks a lot sir...
You’re literally perfect
Excellent!
The Geometric element is fascinating. But the algebraic dot product provide a solid conclusion.
Thank you very much
bruh, u nailed it🙏🙏
A legend is living among us!!!!!!!!!
Simply Brilliant
Thank you!
thank you
Man, you are Richard Feynman of our time
That is high praise!
@@DrTrefor more than him!
Literally the truth🫡☺️
Not high praise , the amount of hard work you have done to develop these is phenomenal ❤
@@Abdullahezzat1893of course not. What a stupid reply🤦🏽♂️
I love you, thank you!
Well i know what's ur reply
Glad to know it was helpful
this was excellent thank youuu
Glad it helped!
@Trefor Bazett - another great video! And another question from me regarding the gradient pointing perpendicular to level curves/surfaces. I can't see 'intuitively' why the gradient should be perpendicular to a level surface. To use your mountain analogy. Let's say that I am standing at a point on a contour on a mountain, and I'm 'designing' the mountain, can't I just pick a direction that's not perpendicular to the contour as the direction of steepest ascent?
And second question, is there a way of 'intuitively' seeing why the grad vector points in the direction of the maximum rate of change? To me it is quite a surprising fact - I can't actually see intuitively why this should be the case without going through the algebra. Am I meant to be surprised by this or is there a way that makes this intuitively obvious?
hey I think you can but this is where you have to use lagrange multiplier to find the max. which you can determine in witch you go is the steepest. I mean I maybe wrong, but this is what I think.