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.
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!
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.
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!
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
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.
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!
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!
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 🙏
@@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.
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 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 so much, I struggled with understanding why is the gradient vector pointing in the direction of the fastest increase, but now I think I get it:))
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 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?
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 am not sure if I understand about the direction of gradient. How can a gradient be living in the x,y plane and yet point to the direction of maximum ascent or descent which is in z direction? The gradient is going inward or outward but is in x,y plane still. If this is because of flattening of z on to x,y plane on a 2-d contour map then once we expand it back to 3-d map, shouldn't the direction of gradient also change to align with z as we go back from 2-d to 3-d?
Why is the magnitude of the gradient vector said to be the RATE of maximum ascent? When I see "rate", I think slope. Why isn't the rate of ascent simply the partial of y divided by the partial of x.? Isn't this the slope of the gradient....i.e. change in y over the change in x? What am I missing? thanks
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.
Good afternoon Fellow mathematicians, could you please help? There is a function of two variables. I need to find its minimum. I start from a certain point. Next, what is the minimum search algorithm? For example: an increment of one variable is taken, a function is considered, we look - less or more than the previous step. Further we take an increment by the same change or another? Or it is necessary to take an increment at once of two variables?
@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.
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.
what if the normal vector of dr/dt was pointing in the other direction?... I mean, the vector that makes 180 degres with the gradient vector is still normal to the tangent, but it does not point in the direction of maximum slop, what does that mean?
Does the gradient point to the top of the mountain? In your diagram your gradient pointed to a 'local max' but you would have needed another gradient to actually reach the top?
The dot product of the gradient with the tangent line is 0, meaning that the angle between the two is 90°, and since the tangent goes along the curve, the gradient points away at a 90° angle, or in other words is orthogonal to the contour of the curve
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?
Man, you really teach what's important to understand the concepts, and you explain yourself perfectly! Amazing! You've gained a new subscriber 😁
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!
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!
I have never seen such a beautiful explanation ever of Gradients love you
Great derivation and application! The derivation of the gradient-vector formula and its justification were both quite easy to follow!
Beautifully presented! It's such a cool topic, and using mountains as an analogy makes everything so intuitive.
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! :)
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.
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!
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 UA-cam!
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 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!
after visualizing these concepts it became easier for me to perform the mathematical formulas thank you so much sir for the valuable information
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!
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!
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 🙏
The example at the end really helped a lot in getting the concept , such a great explanation , I can't thank u enough.
this video is one of the greatest one's that you can find on this topic
Your demonstration is just amazing Sir....the best explanation of gradient vector on UA-cam....
Thanks a ton!
Thank God I found this channel 🙌
you'll make us love calculus and maths!!
thanks for including practical example of Vancouver island
haha I had fun with that part!
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.
Sir, you taught the topic deeply and with real life application...I think now I become your fan❤
Thank you sir.
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 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!
my favorite math teacher on youtube
Thank you!
This video is amazing. First time I'm seeing these concepts clearly since I started taking this course.
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 so much, I struggled with understanding why is the gradient vector pointing in the direction of the fastest increase, but now I think I get it:))
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.
Amazing explanation!!!! thank you so much, you make great influence in the world..
Example of a mountain was superb to explain gradient..thanks bro
I wish there was the option of giving more than one like. Superb explanation!
Fantastic explanation. Now I understand the gradient for our purposes lies in the xy plane and that it points into the mountain.
This channel is truly underrated
Excellent explanation
understood the beauty of multivariable calculus and gradient operator. Thanks a lot sir :)))
Happy to help!
Fantastic way of teaching!!! I recommend the classes here in Brazil!
Thank you professor for the Golden Hinde application.
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!!
Great video that gives a brilliant straight explanation for the gradient vector. Hope to have your class in UVic.
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
This video is really helpful. Thank you so much,👏
excellent video as always, nice example :)
Perfectly explained. Thank you sir
Best math channel. Massive respect. Thank you sir..❤️
great example, thank you for your video
amazing and so interesting! keep it up
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.
Absolutely fantastic explainer! Way to go!
Such a brilliant video, it truly heps
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°
Such a clarity in your explanation....thank you so much sir, you cleared my hardest doubt...😊😊❤❤😊❤❤❤
Great video, made it easy to understand. Thanks, professor Trefor!
Love your teaching sir...""LOVE""
Thank you ,for the video. It was a great explanation.
rewatched it several times, started losing hope but then it clicked and I was like 'wait that makes sense!'
Brilliant explanation again :). It would be very good if you linked the previous explanations, i.e., the tension vector.
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?
A legend is living among us!!!!!!!!!
So clever haha, loved the explanation
thank you so much for clearing the doubt. The video was very helpful.
Thank you sir ❤! You are clearing my such deep buried doubts .
4:00 anyone knows how he moved from the "sum of two things multiplied" to the dot product of two sums?
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 😊
Great illustration, thank you!
You’re literally perfect
10:15 how do you determine which directions are positive for the two vectors?
@9:00 Why wouldn't the smallest magnitude be at 180 degrees where
COS -theta would be minus- 1?
I cannot thank you enough kind sir.
I am not sure if I understand about the direction of gradient. How can a gradient be living in the x,y plane and yet point to the direction of maximum ascent or descent which is in z direction?
The gradient is going inward or outward but is in x,y plane still. If this is because of flattening of z on to x,y plane on a 2-d contour map then once we expand it back to 3-d map, shouldn't the direction of gradient also change to align with z as we go back from 2-d to 3-d?
The Geometric element is fascinating. But the algebraic dot product provide a solid conclusion.
Great explanation! Thanks alot
Absolute trivial explanation.
Why is the magnitude of the gradient vector said to be the RATE of maximum ascent? When I see "rate", I think slope. Why isn't the rate of ascent simply the partial of y divided by the partial of x.? Isn't this the slope of the gradient....i.e. change in y over the change in x? What am I missing? thanks
Awesome video!
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:)
Sir ,please recommend me a mathematics book for engineering, and for gate and IIT jam exams.
Very well-explained.
Glad you think so!
Oh god it helped me so much thanks sensai
Super thanks Dr.
May you be blessed
Thanks a lot sir 🔥🔥🔥
thanks a lot for the great explanation
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.
Thanks so much it helped me understand the gradient concept.
Good afternoon Fellow mathematicians, could you please help? There is a function of two variables. I need to find its minimum. I start from a certain point. Next, what is the minimum search algorithm? For example: an increment of one variable is taken, a function is considered, we look - less or more than the previous step. Further we take an increment by the same change or another? Or it is necessary to take an increment at once of two variables?
@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.
thankyou so much! this one's great!!🌟
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.
Does any know in 5:32 why the tangent goes along the curve?
Great Video. Simple and understanding.
what if the normal vector of dr/dt was pointing in the other direction?... I mean, the vector that makes 180 degres with the gradient vector is still normal to the tangent, but it does not point in the direction of maximum slop, what does that mean?
Does the gradient point to the top of the mountain? In your diagram your gradient pointed to a 'local max' but you would have needed another gradient to actually reach the top?
The gradient lives in two dimensions, i.e. it is a direction on a map. It doens't have an up/down component.
so with a with the gradient we can determine the most optimal way to climb up the mountain? For example the rate of climb?
Please can anyone explain me why the gradient is orthogonal to the contour of the curve? Thank you for such a nice video
The dot product of the gradient with the tangent line is 0, meaning that the angle between the two is 90°, and since the tangent goes along the curve, the gradient points away at a 90° angle, or in other words is orthogonal to the contour of the curve