Geometric Meaning of the Gradient Vector

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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This video was created by Dr. Trefor Bazett. I'm an Assistant Teaching Professor at the University of Victoria.

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[destreme]

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

[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!

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

[PantheraOncaTV]

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

[peterfriedman]

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.

[Anonymous-nzwd]

I have never seen such a beautiful explanation ever of Gradients love you <3

[HPLovecraftsCat]

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.

[wildertapiasaenz]

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 🙏

[dustincondon]

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

[johnnytoobad]

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.

[engrfawadiqbalkhan]

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.

[rafidahmed]

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!

[alkankondo]

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

[wuyqrbt]

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

[zukofire]

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! :)

[sadpotato]

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.

[hope-wqjd]

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

[liuqing]

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

[aln]

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

[AS-ixqd]

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.