Lec 22: Green's theorem | MIT 18.02 Multivariable Calculus, Fall 2007

[@mj_friedman]

6:11 Prof Auroux meets an alien... "Wow another life form! How fascinating! You must tell me... Do you use the same sign convention for curl??"

[@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.)

[@yash1152]

thanks

[@JustinGreen22]

Wow, MIT is awesome for putting these up on KZbin.

[@matron9936]

I love this professor. Amazing explanation!

[@raulhdez8592]

Soy una

[@raulhdez8592]

Te ñ

[@raulhdez8592]

Peres 0ppñ

[@ZopteY]

Thank you MIT, I now understand green's theorem!

[@Brewsto]

Anyone who tries to attempt the double integeral by hand at 13:23. Remember when switching to polar coordinates dA is r drdθ and x=2+cosθ because r=1 and center of circle is at (2,0). Aso it's really weird at first since you try to calculate the area of a circle on shifted x and you think that the area of the circle remains the same since the radius is the same. You would think Α=πr^2 no matter what x is but no. The area of the circle depends on the center (x,y) because of the force field.

[@eecakir]

i didnt get the part where he picked x to be equal 2 + costheta, ty for pointing out, cheers.

[@eecakir]

This is funny how i replied an explanation from 3 years before

[@liodony7195]

it's because the center of the circIe in this case isn't locate in the origin, if you set theta=0 you'll get x=cos(0)=1 which isn't true for this circle, that's why he added 2 to account for the shift of the center from the origin to the point (2,0)

[@PhysicsMath]

The v(r)=k/r. At center r-->0 . So the area inside the contour

[@dark_all_day9311]

I set up the integral as x r dr dθ with 0

[@zhaoxingdeng5264]

Beautiful lecture!

[@hershyfishman2929]

41:58 You don't get M back from ∂M/∂y in cases where M is the sum of a function of x only and another function since you lost all information of how M depends on x only (for example M = x+y, ∂M/∂y = 1, integrating 1dy will never give M back). However, you do get back M(x, f2) -M(x, f1) because what depends only on x cancels out.

[@rishavdhariwal4782]

yeah thanks was wondering about this

[@kaushaltimilsina7727]

For more intuition about the Green's Theorem, can it be thought of as- the potential energy stored in the rotational motion stored in the line integral of the closed curve is sum of the potential at the area inside curve, sort of like if you start twisting a rubber sheet, you get the work done on the closed curve when the rotation propagates radially outwards-building up from the work done on the closed curves of increasing radius?

[@marcuspoon4625]

that is a very cool analogy, thanks very much

[@gibsonrulezz]

very helpful for self-study also.

[@elamvaluthis7268]

Mathematics is comprehensive like these professors.

[@yonatanable]

The Green's Theorem is only to work if the curve is closed...

[@vickyc859]

I am loving mathematics now... which makes life easier at NIT

[@seungchullee221]

Q. How much it cost? A. About thousand dollars, that's why I didn't bring it here. Professor really answers every questions even in Microeconomics.. :)

[@not_amanullah]

This is helpful ❤️🤍

[@shakesbeer00]

hope to see one planimeter someday, which seems cool.

[@footballCartoon91]

@14:39, how come he got the area of the circle = 2*pi ? Area of a circle = pi*r*r if r=sqrt(2) so area=2*pi.. Maybe because radius=sqrt(2) I guess.

[@not_amanullah]

Thanks ❤

[@봄여율-u9f]

25:00 Proof of GREEN's THM

[@hangjiang858]

how many times more will they applaud for him erasing the blackboard ?

[@DeathJustice]

Denis rules. Top class.

[@TrevorKafka]

@yonatanable Yep! That's what the ∮ symbol indicates... an integral about a closed curve.

[@aashsyed1277]

Amazing!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

[@tropicalverktaki]

He is french, btw

[@TTFMjock]

How did they engineer the chalk outline? That has to be some effect.

[@muntoonxt]

Life of Brian Probably heavy use of sharpening on edges.

[@sifiso5055]

Hagoromo Chalk

[@proexcel123]

I thought in Lecture 20, Prof Denis said that all closed loop integral of F.ds = 0? Then where does Green's theorem comes from?

[@mlst3rg]

if the line integral is not independent of path, then the line integral is not necessarily zero over a closed loop! but for conservative vector fields F, we have curl F = 0.. so the double integral and line integral over a loop are both 0.

[@proexcel123]

Oh... Thanks!

[@KevZyro]

26:44 haha thats awesome

[@drizzlenumba1]

Le board.... It moves!!! =D

[@patrickyu8470]

44:04 oh

[@todsaponster]

Vary good

[@priyanshubansal6776]

13.00 that idea never came to me

[@yash1152]

13:00

[@paulwittekind5071]

Got any Americans who know calculus? Paul Revered

[@FariborzGhavamian]

great lecture. why on earth do they clap every single time he erases something !

[@fadyb4031]

Because they like how he erases the board while the board above is lowering without stopping. P.S. are you still alive?

[@basharalmashni645]

18:50

[@yash1152]

18:31 26:42 oh lololol :D

[@ankushankush4665]

in every lecture of this teacher, why the students are laughing and coughing very much. that's kinda very disrespectful. the teacher seems very nice and sweet but the students are just making fun. they should respect him, atleast for mit when their lectures are on internet.

[@theblinkingbrownie4654]

They worked hard af to get into mit, if they didn't respect the teachers they wouldn't have worked so hard. Leave them alone, they had to do a lot of work and entertainment is scarce so they enjoy the simple things. Also how you gonna tell someone to not cough? They don't choose to.

[@ankushankush4665]

@@theblinkingbrownie4654 I don't have any problem due to their coughing but they are pretending sooooo much, seems like they don't care about the teacher. I am also a student and I really know how coughing sounds like when it is pretended. I am not making fun of their coughing...

[@Bignic2008]

18:42 Why did they start clapping? Lol

[@echo-channel77]

Because he erases the board quickly and beats the declining boards. I kid you not.

[@Originalimoc]

See previous videos and you'll know why😂

[@lestercrest5965]

There's history there

[@yash1152]

wow, really? which one?

[@prazors]

...because they're goobers, probably a freshman class.

[@GHOST3R]

THAT IS SEXY

[@aashsyed1277]

?

[@MarikoIchigo]

"ZERRO"...funny accent.