Geometric Meaning of the Gradient Vector

[@RyanMcCoppin]

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]

Same!

[@sumittete2804]

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

[@Suyogya77]

​@@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]

@@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]

@@sumittete2804 do you use telegram or something?

[@destreme9189]

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

[@9888565407]

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

[@DrTrefor]

Glad it helped!

[@peterfriedman4912]

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]

Glad it was helpful!

[@johnnytoobad7785]

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]

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]

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

[@Anonymous-nz8wd]

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

[@dustincondon5557]

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

[@PantheraOncaTV]

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

[@wildertapiasaenz2333]

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 🙏

[@hope-wq9jd]

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

[@ManojKumar-cj7oj]

Thank God I found this channel 🙌

[@KaviPriyan-qt6vc]

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

[@aln4075]

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

[@HPLovecraftsCat9]

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.

[@rafidahmed2796]

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!

[@alkankondo89]

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

[@engrfawadiqbalkhan786]

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.

[@at6727]

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

[@DrTrefor]

haha I had fun with that part!

[@sadpotato5111]

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.

[@ikarus1060]

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]

Glad it helped!

[@walac]

Best math channel on KZbin!

[@tanayasharma7776]

my favorite math teacher on youtube

[@DrTrefor]

Thank you!

[@user-wu8yq1rb9t]

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

[@AS-ix3qd]

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]

glad it helped!

[@zukofire6424]

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! :)

[@liuqing1995]

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

[@Conorstew]

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]

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

[@olehborys1462]

great example, thank you for your video

[@chernihivka]

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!

[@habilismayilov838]

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

[@maximlavrenko1164]

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

[@Party21304]

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!

[@electricalabc3411]

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

[@nicogovindsamy9022]

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

[@maani315]

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

[@DrTrefor]

Thanks a ton!

[@Indik47]

Excellent explanation

[@chrismarklowitz1001]

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.

[@rizalpurnawan3796]

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

[@DrTrefor]

You are welcome!

[@sumittete2804]

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

[@carultch]

@@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.

[@hrkalita159]

Man, you are Richard Feynman of our time

[@DrTrefor]

That is high praise!

[@Abdullahezzat1893]

​@@DrTrefor more than him!

[@sanujitacharya794]

Literally the truth🫡☺️

[@ABHISHEKSHARMA-gj6xp]

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

[@muhammedabdullvek7123]

​@@Abdullahezzat1893of course not. What a stupid reply🤦🏽‍♂️

[@rupaupadhaya8888]

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

[@rickgrimes47]

I cannot thank you enough kind sir.

[@RSS18]

Perfectly explained. Thank you sir

[@joliettraveler]

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

[@xiliu3526]

Such a brilliant video, it truly heps

[@gibs40]

This channel is truly underrated

[@joaomattos9271]

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

[@shivarapolu3180]

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

[@DrTrefor]

Happy to help!

[@scratchyourhead2383]

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

[@sergiolucas38]

excellent video as always, nice example :)

[@milesborowsky8140]

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!

[@minnngb]

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

[@LinLin-rs2bv]

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

[@rivkamutka2833]

Amazing explanation. Thank you!

[@lakshitha99]

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

[@shashankjhaa6221]

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

[@itsnadchaos]

amazing and so interesting! keep it up

[@bimankalita2960]

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

[@TheEmad619]

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

[@ishraqhossain1235]

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

[@quynhngadau5482]

Great illustration, thank you!

[@optimistprime6741]

You’re literally perfect

[@subhajitbarman4672]

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

[@TheWildStatistician]

Absolutely fantastic explainer! Way to go!

[@wildertapiasaenz2333]

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 😊

[@muktasane6046]

Thanks so much it helped me understand the gradient concept.

[@somanathdash3153]

Oh god it helped me so much thanks sensai

[@continnum_radhe-radhe]

Thanks a lot sir 🔥🔥🔥

[@shubhamvadagave8961]

Great explanation! Thanks alot

[@anandita166]

thankyou so much! this one's great!!🌟

[@hemrajkumawat801]

Just amazing

[@viniciusdeobber8239]

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

[@shreyapandey7154]

very informative

[@pawanadhikari110]

Absolute trivial explanation.

[@youssefdirani]

Super thanks Dr. May you be blessed

[@BoZhaoengineering]

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

[@AutiEShap]

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

[@noonesland2471]

I love you, thank you!

[@seblaredo1058]

when i pass calc 3 and eventually become an electrical engineer itll be thanks to you thank you 😭😭

[@nickknight5373]

Very well-explained.

[@DrTrefor]

Glad you think so!

[@ahmadwael8541]

thanks a lot for the great explanation

[@AK-sh8df]

Wonderful video

[@santhoshkumar-lv4ph]

I love so much u are a genius teacher

[@Silver-bq6td]

great job.

[@sujithkumar7158]

Simply Brilliant

[@DrTrefor]

Thank you!

[@julioreyram]

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

[@alejrandom6592]

Awesome video!

[@thankyouthankyou1172]

after 99999years of searching, I finally found a video that could make me understood

[@nalinkumarsharma6075]

In starting, too many concepts to grasp. But the example in end was very interesting.

[@Niglnws]

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.

[@RobertWF42]

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.

[@DrTrefor]

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

[@RobertWF42]

@@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.

[@edgbaston149]

Thank you so much🙏 You're amazing. 😊😎

[@Ramu12987]

I think that the direction you have shown for Gradient Vector is the path that a climber would have to take while the actual direction of Gradient at any point would be paralel to XY plane.

[@naikyasir2871]

Really enjoyed

[@thisukainol]

wow. great explanation

[@punditgi]

Excellent!

[@wandrew307]

Amazing, Thank you!

[@satyamtripathi2234]

thank you

[@peterpetrov4867]

Great video!!! Is this one part of a series? How do we know the order in which to watch these great videos?

[@DrTrefor]

Yup, check out the links in teh description, have a whole multivariable playlist:)

[@101_huzefajohar7]

Hii Im from India, and I really love how you explain such difficult concepts in such a simple and brilliant way, I never could have understood these concepts had I been dependent only on my college professor’s class, as he himself does’nt clearly know these concepts like you! However, I could still not connect with the idea of the directional derivative as a dot product of the gradient and the vector component, intuitively. Can you explain how is it actually helping in finding the slope of the function in any particular direction?

[@Gealamusic]

i think that it relates to what he said about cos(theta). If the dot product is equal to 0, the two components are orthogonal. And in this case you have gradient of f going up and the tangent going across (when you look at the mountain) this means that there is a 90° angle between them and cos(90°) = 0.

[@richardslesinski4137]

amazingly clear. thank you