Lec 21: Gradient fields and potential functions | MIT 18.02 Multivariable Calculus, Fall 2007

[@santiagoalpuy]

Im an student of the engineering faculty of Uruguay, and this helped me a lot. I have always thought that MIT or Harvard were harder than the rest of the universities, but it looks like they are easier because of the amazing teachers they have. Greetings from almost the end of the world!

[@diveintoengineering6089]

Me being an engineering student from Chile thinks the same: these lectures are amazing.

[@ashutoshacharya8]

And, I am from Nepal(THE END OF THE WORLD). Cheers, I think the same.

[@oibekbabayev7739]

And now a student from Uzbekistan, 10 years after the comment has been left, replies in the same way - those lectures are fantastic!

[@GoogleUser-ee8ro]

Just finished watching lecture 20 but cant stop there because of "Tuesday" class he already sold to us. this lecture must be one of the fundamentals before learning Maxwell equations or electrodynamics. It's more comforting to learn some physics from math class than the other way around. btw his french accent is quite cute

[@mattthelearner2797]

Agreed

[@vkv392]

same here...

[@alexhudson502]

Lecture 1: Dot Product Lecture 2: Determinants Lecture 3: Matrices Lecture 4: Square Systems Lecture 5: Parametric Equations Lecture 6: Kepler's Second Law Lecture 7: Exam Review (goes over practice exam 1a at 24 min 40 seconds) Lecture 8: Partial Derivatives Lecture 9: Max-Min and Least Squares Lecture 10: Second Derivative Test Lecture 11: Chain Rule Lecture 12: Gradient Lecture 13: Lagrange Multipliers Lecture 14: Non-Independent Variables Lecture 15: Partial Differential Equations Lecture 16: Double Integrals Lecture 17: Polar Coordinates Lecture 18: Change of Variables Lecture 19: Vector Fields Lecture 20: Path Independence Lecture 21: Gradient Fields and Curl of Vector Fields Lecture 22: Green's Theorem Lecture 23: Flux Lecture 24: Simply Connected Regions Lecture 25: Triple Integrals Lecture 26: Spherical Coordinates Lecture 27: Vector Fields in 3D Lecture 28: Divergence Theorem Lecture 29: Divergence Theorem (cont.) Lecture 30: Line Integrals Lecture 31: Stokes' Theorem Lecture 32: Stokes' Theorem (cont.) Lecture 33: Maxwell's Equations Lecture 34: Final Review Lecture 35: Final Review (cont.)

[@kavoos1000]

right now i am writing my exam, in 3 weeks and and these vids are amazing ..thank you so very much mit and youtube for making this possible ... i ll be always thankfull

[@topilinkala1594]

About weather and curl: The hairy ball theorem says that there must be a calm spot on Earth's weather system. But if there is no curl in Earth's weather system there cannot be a calm spot. But if you think about prevailing winds all over the Earth for example from east to west you'd have cyclones eg. rotating winds (curl 0) on both poles and there would be a calm spot on the eye of the cyclone. You can not smooth a hairy ball.

[@rarulis]

f(x1,y1) = Bless you

[@nefereous9082]

These lectures save me a lot of time. My professor teaches by example which is a pain because he covers this material in 3 lectures. I'd rather come here and get the concepts then sweat through the examples on my own.

[@kylesegal642]

This is all the lectures and their subjects (taken from another guy's comment) Lecture 1: Dot Product Lecture 2: Determinants Lecture 3: Matrices Lecture 4: Square Systems Lecture 5: Parametric Equations Lecture 6: Kepler's Second Law Lecture 7: Exam Review (goes over practice exam 1a at 24 min 40 seconds) Lecture 8: Partial Derivatives Lecture 9: Max-Min and Least Squares Lecture 10: Second Derivative Test Lecture 11: Chain Rule Lecture 12: Gradient Lecture 13: Lagrange Multipliers Lecture 14: Non-Independent Variables Lecture 15: Partial Differential Equations Lecture 16: Double Integrals Lecture 17: Polar Coordinates Lecture 18: Change of Variables Lecture 19: Vector Fields Lecture 20: Path Independence Lecture 21: Gradient Fields Lecture 22: Green's Theorem Lecture 23: Flux Lecture 24: Simply Connected Regions Lecture 25: Triple Integrals Lecture 26: Spherical Coordinates Lecture 27: Vector Fields in 3D Lecture 28: Divergence Theorem Lecture 29: Divergence Theorem (cont.) Lecture 30: Line Integrals Lecture 31: Stokes' Theorem Lecture 32: Stokes' Theorem (cont.) Lecture 33: Maxwell's Equations Lecture 34: Final Review Lecture 35: Final Review (cont.)

[@wadewen1008]

This professor save my grade ! Love you so much and appreciate you !

[@gambitt101]

McGill needs lecturers like him!!..

[@Thegema101]

he is the Ultimate Maths Prof !!!

[@musahlungwani9167]

Bongani Ngema true

[@carlosalbertocuadros5469]

It´s an excellent class Professor. tahnk youuuuu!!!

[@danielmedeiros537]

thank you ,teacher!greatings from Brazil!

[@yermomLeslie]

As a physics enthusiast I found lec 19-21 so related to my interest lol

[@mrkakotube]

I would have wanted to learn more about the intuition behind curl, why the formula has anything to do with rotation and why curl gives twice the angular velocity. Anyway, these are great MIT lectures and Demis is a good professor.

[@ianbrown482]

Here's some intuition. In the plane we can have either clockwise or counterclockwise rotation, so let's define counterclockwise as the positive direction for rotation and curl. So, if I have a point with positive curl, then to the right of the point, that is, for x-values greater than that of the point, we will have upward motion. Similarly, to the left of the point we will have downward motion. So for lower x-values the vertical motion around the point is downward or negative, and for greater x values the vertical motion is upward or positive. So, around the point, as x increases, the vertical motion, which is the partial derivative with respect to y, increases. In other words, the rate of change of the partial derivative with respect to y, with respect to x, f sub yx or N sub x, is greater than 0 (in the case of positive curl). For the horizontal motion, which takes place above and below the point, the reasoning is similar. Above the point the motion is horizontal and to the left, while below it's to the right. Above the point the partial derivative with respect to x is negative, while below it's positive. Since above the point the y-values are greater, we see that as y increases, the horizontal motion component decreases by becoming more negative. So the rate of change of the partial derivative with respect to x, with respect to y, f sub xy or M sub y, is less than 0 (again, in the case of positive curl). Since M sub y will be negative, but we want a positive result to indicate positive curl, curl is defined as N sub x - M sub y, because N sub x > 0 and -(M sub y) > 0. Curl is just the sum of the parts of motion indicating clockwise or counterclockwise rotation around a point, necessarily described by the second partial derivatives.

[@maxim.aleksa]

You know a teacher cares about you where you're in a class of 200 people and he says "Bless you!" when you sneeze.

[@stefanoromagnoli9891]

He is such a good teacher!

[@kartik6110]

Always love those blackboard moments. XD

[@oolongtea0922]

thanks MIT

[@danieljulian4676]

Well, that sure puts a nice spin on vector fields. It's an old spin, but nice. An oldie but a goodie. Definitely in the top 40.

[@TonyG-n1m]

Bro what

[@rinkaghosh7961]

Thank you So much Sir 🙏 .. thanks a lot !!!

[@abbasbookwala]

to me a direct derivation of force field giving torque as a curl would have been more intuitive rather than merely showing an analogous equation at the end of the lecture. For velocity field, I could very well and easily see why the equation of curl represents rotation but I am failing to see vide the same equation how force field would give torque as curl.

[@Amvalson]

Thank you very much for the video. Hopefully it helps everybody.

[@imicca]

this guy helped me a lot !

[@not_amanullah]

Thanks ❤️🤍

[@not_amanullah]

This is helpful ❤️🤍

[@briandaugherty3589]

One strange thing here - why are they still using chalk? I can't remember when I last saw that here - it has definite health problems.

[@khunshamehmood8399]

This is from 13 years ago

[@athenanguyen9455]

I wish this professor came back to MIT.

[@expertnoobFTW]

Really wish the current professor verbally explained stuff more in his lectures. Meanwhile, all we have now is the shadow region.

[@هذاأنا-ذ3ث]

He went to Berkeley for about a decade and now he is at Harvard, not far from MIT.

[@NSBeverything]

at the end of lecture, it was aid that curl of force field gives torque....is it give exact value of torque or it gives twice value?

[@randomgirl7000]

When he said that we have seen that f(xy)=f(yx) at 3:50, I didn’t get that this to derivateves are equal.

[@hangjiang858]

Damn, he is good

[@yonatanable]

The curl measure how intense the rotational measure at that point...

[@the_eternal_student]

How do we know that a constant depends on y?

[@1995a1995z]

can someone please tell me why they always cheer when he wipes the board??

[@Mumfin]

That means they don't have to.

[@KCIsMe]

1995a1995z I'm pretty sure they cheer because he is able to wipe the board completely before the board on top of it comes down.

[@muntoonxt]

1995a1995z It's a 'trick' the professor is well known for. From what I've read, it sounds as if it's almost like a meme at MIT.

[@isaacmandell-seaver7223]

He’s an absolute pro at wiping the chalk into submission. They’re cheering for his godly blackboard powers

[@akrishna1729]

he's legendary for his speed-erasing technique

[@times2508]

I think it should modules of curl f =nx-my

[@hennet08]

Hello, can you please tell me where i can find the exercises for that week. I'll appreciate it.

[@battlewing221]

i would donate to MIT OCW if I had my own money .

[@proghostbusters1627]

Yeah me too. Hope you get money of your own someday to donate to MIT ocw :D

[@battlewing221]

@@proghostbusters1627 lol thanks

[@wcsah]

18:22 - lol

[@athenanguyen9455]

I am so confused all the time.

[@captainmeowmeow2]

18:00 best lol

[@SphereofTime]

1:04

[@basharalmashni645]

18:20 Future scientists 😂😂😂😂

[@vdoslayer]

the MIT kids are a harassing bunch! and dr auraux is so benign never even scolds!

[@TrangNguyen-fr8yl]

They love him. There may be some cultural differences in how students and teachers relate here. They enjoy his class and love him. If you let that be a possibility, you might feel the beauty of what's happening in this class.

[@funnyidiot3752]

What is teacher name?

[@mitocw]

Denis Auroux. See the Scholar version of the course on MIT OpenCourseWare for more information and materials at ocw.mit.edu/18-02SCF10.