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!
@@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 ??
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.
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.
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.
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!
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 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 🙏
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!
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.
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!
@@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.
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 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
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.
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!
13:30 is important. Del f is into the mountain which is still normal to the curve and it is logical as del f components are in i and j directions. We move in x and y and the result is change in z which the magnitude of the gradient tells us.
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 😊
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.
2:36 I don't get what f(x,y) is wrt r(t). Like, do you mean to say you slice a mountain, take its cross-section, then the f(x,y) is built on it which is at a height C from r(t) on the z-axis such that f(x,y) is also a level curve?
I've read conflicting definitions of the gradient vector. Sometimes it's a normal vector perpendicular to the tangent plane at point P on a surface. In other definitions the gradient is a vector pointing "downhill" and lies *within* the tangent plane.
The issue is whether we are talking about the gradient of f, where z=f(x,y), or the gradient of F where F(x,y,z)=f(x,y)-z. The former gives a 2D vector in the x,y-plane (note: not the tangent plane), while the latter gives the normal to the surface
@@DrTrefor @Dr. Trefor Bazett Thank you, this is a helpful clarification! The method of Lagrange multipliers matches the grad f directions for two functions, correct? And I think statistical or machine learning methods that use gradient descent to find solutions for data modeling also utilize grad f, not grad F.
@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.
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
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?
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!
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.
dude I love ya. that cleared everything about gradients in my mind. Thanks a lot bud.
Glad it helped!
Man, you really teach what's important to understand the concepts, and you explain yourself perfectly! Amazing! You've gained a new subscriber 😁
I have never seen such a beautiful explanation ever of Gradients love you
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!
Great derivation and application! The derivation of the gradient-vector formula and its justification were both quite easy to follow!
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! :)
Beautifully presented! It's such a cool topic, and using mountains as an analogy makes everything so intuitive.
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!
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
The great combination of theory and a real example of a mountain in Vancouver. I enjoy the lesson series of Calculus so much.
Best math channel on TH-cam!
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 🙏
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!
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.
after visualizing these concepts it became easier for me to perform the mathematical formulas thank you so much sir for the valuable information
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!
The example at the end really helped a lot in getting the concept , such a great explanation , I can't thank u enough.
Thank God I found this channel 🙌
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.
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!
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!
this video is one of the greatest one's that you can find on this topic
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!
you'll make us love calculus and maths!!
thanks for including practical example of Vancouver island
haha I had fun with that part!
Your demonstration is just amazing Sir....the best explanation of gradient vector on TH-cam....
Thanks a ton!
my favorite math teacher on youtube
Thank you!
Sir, you taught the topic deeply and with real life application...I think now I become your fan❤
Thank you sir.
I am doing all possible steps to take this channel to a bigger audience
i talked about him to an audience of about 150 people!!
This video is amazing. First time I'm seeing these concepts clearly since I started taking this course.
Fantastic explanation. Now I understand the gradient for our purposes lies in the xy plane and that it points into the mountain.
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
rewatched it several times, started losing hope but then it clicked and I was like 'wait that makes sense!'
Example of a mountain was superb to explain gradient..thanks bro
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.
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!
I wish there was the option of giving more than one like. Superb explanation!
This channel is truly underrated
13:30 is important.
Del f is into the mountain which is still normal to the curve and it is logical as del f components are in i and j directions.
We move in x and y and the result is change in z which the magnitude of the gradient tells us.
Excellent explanation
Amazing explanation!!!! thank you so much, you make great influence in the world..
understood the beauty of multivariable calculus and gradient operator. Thanks a lot sir :)))
Happy to help!
Thank you professor for the Golden Hinde application.
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 😊
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 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.
Perfectly explained. Thank you sir
Fantastic way of teaching!!! I recommend the classes here in Brazil!
Great video that gives a brilliant straight explanation for the gradient vector. Hope to have your class in UVic.
This video is really helpful. Thank you so much,👏
Best math channel. Massive respect. Thank you sir..❤️
amazing and so interesting! keep it up
Such a clarity in your explanation....thank you so much sir, you cleared my hardest doubt...😊😊❤❤😊❤❤❤
A legend is living among us!!!!!!!!!
Love your teaching sir...""LOVE""
The Geometric element is fascinating. But the algebraic dot product provide a solid conclusion.
excellent video as always, nice example :)
As a rock climber I like when he uses analogies involving climbing and mathematical concepts. The light bulb goes on in my head!
great example, thank you for your video
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
Thank you sir ❤! You are clearing my such deep buried doubts .
Absolutely fantastic explainer! Way to go!
You’re literally perfect
Such a brilliant video, it truly heps
I cannot thank you enough kind sir.
Thanks a lot sir 🔥🔥🔥
Great video, made it easy to understand. Thanks, professor Trefor!
Absolute trivial explanation.
thank you so much for clearing the doubt. The video was very helpful.
At 8:39, shouldn’t you say 0 slope instead of minimum slope? I think you get the minimum slope when theta is -π
Yess....Rate of change of function is minimum in the direction opposite to gradient vector that is at angle of 180°
Thank you ,for the video. It was a great explanation.
after 99999years of searching, I finally found a video that could make me understood
2:36 I don't get what f(x,y) is wrt r(t). Like, do you mean to say you slice a mountain, take its cross-section, then the f(x,y) is built on it which is at a height C from r(t) on the z-axis such that f(x,y) is also a level curve?
Super thanks Dr.
May you be blessed
Brilliant explanation again :). It would be very good if you linked the previous explanations, i.e., the tension vector.
So clever haha, loved the explanation
Oh god it helped me so much thanks sensai
Great illustration, thank you!
I've read conflicting definitions of the gradient vector. Sometimes it's a normal vector perpendicular to the tangent plane at point P on a surface. In other definitions the gradient is a vector pointing "downhill" and lies *within* the tangent plane.
The issue is whether we are talking about the gradient of f, where z=f(x,y), or the gradient of F where F(x,y,z)=f(x,y)-z. The former gives a 2D vector in the x,y-plane (note: not the tangent plane), while the latter gives the normal to the surface
@@DrTrefor @Dr. Trefor Bazett Thank you, this is a helpful clarification!
The method of Lagrange multipliers matches the grad f directions for two functions, correct? And I think statistical or machine learning methods that use gradient descent to find solutions for data modeling also utilize grad f, not grad F.
Very well-explained.
Glad you think so!
thankyou so much! this one's great!!🌟
Great explanation! Thanks alot
Simply Brilliant
Thank you!
@9:00 Why wouldn't the smallest magnitude be at 180 degrees where
COS -theta would be minus- 1?
I love so much u are a genius teacher
Think you sir,,,,,,,Respect from Bangladesh 🇧🇩🇧🇩🇧🇩🇧🇩🇧🇩
thanks a lot for the great explanation
Awesome video!
Does any know in 5:32 why the tangent goes along the curve?
Great Video. Simple and understanding.
I love you, thank you!
In starting, too many concepts to grasp. But the example in end was very interesting.
very informative
4:00 anyone knows how he moved from the "sum of two things multiplied" to the dot product of two sums?
@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.
Great video!!! Is this one part of a series? How do we know the order in which to watch these great videos?
Yup, check out the links in teh description, have a whole multivariable playlist:)
Just amazing
amazingly clear. thank you