Lec 20: Path independence and conservative fields | MIT 18.02 Multivariable Calculus, Fall 2007

Lecture 20: Path independence and conservative fields.
View the complete course at: ocw.mit.edu/18-...
License: Creative Commons BY-NC-SA
More information at ocw.mit.edu/terms
More courses at ocw.mit.edu

[@LzJPage]

his english is absolutely flawless. just has a thick accent which makes him seem even more intelligent, almost as if he mastered english overnight

[@ch.ajaysingh]

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.)

[@saubaral]

ocw.mit.edu/courses/mathematics/18-02-multivariable-calculus-fall-2007/video-lectures/

[@maazadnan117]

Thats what i was looking for.👍🏻

[@priyanshubansal6776]

why you do this in every lecture bro

[@MsSujoy]

I am highly obliged for your time and labour.... 🙏🙏

[@ishadev01]

thanks

[@jimmydu444]

That is one of my most elegant proof of the fundamental theorem of line integrals I've ever seen.

[@lucastorres12]

Thank you Denis, you're a brilliant lecturer.

[@veloster9616]

lol, really? I guess u dont see really brilliant lecturer.....

[@isaacmandell-seaver7223]

@@veloster9616 what

[@maliksalman5884]

For those confused about Auroux's loose definition of "potential," especially with regard to electrical potential, it comes down to an extra charge in the numerator that he somehow neglects. The integral of (gradient of f dot dr) where f is the electric force (denoted by F) is equal to the Work, and also equal to the change in U, the potential energy. The integral of (gradient of f dot dr) where f is the electric field (denoted by E) is equal to the change in V, the electrical potential, which varies from U in that it has one charge in the numerator, rather than two. In other words: F --> U by line integral ; E --> V by line integral. F does not --> V by line integral.

[@diveintoengineering6089]

Thank you Prof. Auroux. Best wishes from Chile.

[@ycz6]

@brilliantdiamonds 18.02 is a required class for all MIT graduates, so the professor doesn't always go into the details of the analysis and whatnot. 18.024 is a more rigorous class for mathy people. On the other hand, I do have to say that this class is actually more rigorous than the calculus courses I took in high school; certainly more so than AP Calc, which was mostly formula dumping.

[@osvaldotapia6967]

Buena exposición, con una clara forma de comunicar la idea de un campo conservativo y las interpretaciones físicas del mismo.

[@ashwinvishwakarma2531]

the way dis nigga talks is so nice

[@user-fb4zo8wd5n]

Physicists insert a minus sign to get a nice looking energy conservation law: K + V = constant. Without the minus sign, conservation of energy would look like this: K - V = constant. Here K is kinetic energy and V is the potential energy.

[@vishwaajithn.k3266]

well,that gives us another quantity,the lagrangian

[@kingmoses8866]

his chalk work is beautiful, smooth

[@mrkakotube]

It's clear that he likes physics. That's good.

[@Ensign_Cthulhu]

He's some form of applied mathematician is my guess.

[@alexc.r2793]

@@Ensign_Cthulhu yeah. the concept of vector field was created by physicist.

[@TheVisualEdison]

@@Ensign_Cthulhu he actually did a BSc Physics equivalent in france, but his graduate research in math is extremely theoretical

[@edutnavarro1992]

Great class, helping me a lot in Calculus III. Greetings from a studant from Polytechnique School of the University of São Paulo, Brazil!!

[@crzyaboutstuff]

@brco2003 because the point on the unit circle is given by (cos [angle] , sin [angle] ). Here, your angle is pi/4 . Which you probably know is square root of 2 over 2. But this is just simplified from 1 over the square root of 2, which is the point he has.

[@hidetsuguhiraki3307]

I prefer to first divide the integration according to the path, then choose the appropriate coordinate system for each path. Thank you.

[@mariomaruf]

@schwarzegarde This is the second semester in math mit has. Their first semester is Calculus 1 and 2 put together and their second shown here is calculus 3.

[@alinapol]

the lecture is brilliant!

[@HotPepperLala]

43:25 I think he meant electrical potential difference is voltage, electrical potential itself is not the same as voltage.

[@imegatrone]

I Really Like The Video From Your Path independence and conservative fields.

[@rarulis]

My class uses stewart's multivariable calculus and I don't like that book. These videos and ostebee's multivariable calculus make me one of the best in class.

[@mariomariovitiviti]

question: a f(x,y) with gradient y,x shouldn't be xy + c? (nothing changes in practice)

[@not_amanullah]

Thanks ❤🤍

[@nosms6581]

hello.I have a question; the lecturer said that if a force field is conservative then you can't have a particle moving for ever in this field. but we know that gravitational field is conservative so why the moon moves around the earth or planet around the sun? thanks in advance.

[@user-gm3pk5cp3z]

your question is highly physics related , I mean that there is a physics principal missing out of the picture , I think understanding why the earth doesn't fall on (to) the sun will solve the problem , check that on KZbin .

[@quagmire444]

physicist potentials are the opposite of mathematicians potentials looool.

[@elamvaluthis7268]

Very very great.

[@pinkman9620]

This lecture is basically a Physics lecture

[@SwissJustMe]

omg i watched some of the MIT lectures they all are so simple and dont go deep into stuff. its like school...extremeley easy related to our universities oO

[@codingWorld709]

Thanks sir

[@gaymo69]

@PolskaUrban Well isn't math reality? Like, vector fields are useful in describing electromagnetism and forces etc., which are real.

[@dw-ud8bg]

@Λ well, what constitutes reality? Mathematics is indeed incorporeal in that it does not manifest itself in a physical, tangible form, but one can argue that it is real because its existence undeniably impacts the physical world. In a way, our mathematical capacity directly enables this very comment section. For that, I opine that Mathematics is real. And needless to say, physics is real, too. Finally, I cannot fully fathom what it means to be real either; this which I present is merely my personal point of view and I respect your expressed opinion.

[@dw-ud8bg]

@Λ, a reasonable perspective, however, I'd still argue that the quality of being real is not fully contingent on whether an object is physical or not. Instead, an object can be considered real if it influences the physical world in a tangible fashion, which Mathematics certainly does. Additionally, as to the proposed physical nonexistence of mathematical constructs, it is not a clear cut either; for example, the notion of number -- and thus, the study of it -- emerges very intrinsically the physical world as the natural quantifier, independent of human practices. Another relevant instance in relation to this is how there exist many physical objects which document mathematics, are enabled by it, or both. Furthermore, like physical phenomena, mathematical entities can also be consistently perceived, gauged, and actioned upon in a physical capacity. In fact, physical phenomena cannot be contemplated effectively in a practical manner without the consideration of mathematical quantities. Any discussion of physical phenomena devoid of actual, observed quantities is often theoretical, abstract, and therefore, incorporeal, divorced from the physical space. In synopsis, my argument is that directly physical existence is merely a subcategory of reality, that an object should be considered real if it indirectly influences the physical world in a tangible, consequential, and indirectly physical manner, and that to an extent, mathematical constructs do manifest physically, albeit indirectly.

[@anonym498]

I have a question: is exact differential the same as total differential?

[@yashagarwal3999]

what did the professor smoke at 19:05

[@indiablackwell]

good pick up

[@djordjedo96]

42:00 Are the things he's saying correct or he was confusing the words "potential" and "potential energy" during the whole lecture? Shouldn't force be a gradient of potential energy and electric field intensity a gradient of electric potential?

[@ruben-en4jz]

maybe it's to late, but the "potential" of the force is the "potencial energy" ( F=-grad(U) ) and the "potential" of the field (the electric field for example) is the "potential" ( E=-grad(V) ) i used the physics convention

[@user-fb4zo8wd5n]

Yeah, technically potential and potential energy are not the same thing. However, they are close enough since they differ by a constant factor like charge or mass, depending on the context.

[@user-zk3st7qd1i]

It is good to watch video or to read book ,,, please tell me anyone 😢😢😢😢

[@battlewing221]

here is the link to the whole playlist kzbin.info/www/bejne/hqmmqZ-pocSMrbs

[@remsenking]

@SwissJustMe honestly i noticed that too....i think the professors are just easier..

[@josephwheelerton]

if someone could explain what he is saying at 34:35, that would be great. I don't really understand it. I wish I could be more specific in explaining my confusion!

[@maxsash]

Read about Perpetual machines if you haven't already. That will help clear the doubt. Below, I will try to explain but don't bother if you don't understand. There are much better explanations. The point basically is that if you have 'extra' energy after completing a closed loop, then it can be used to perform that loop therefore the loop will get completed without any external input, and this can be repeated to get infinite outputs without inputs.

[@fateplus1]

mmmmmmmmmmmmmmm i love this sooo much

[@starkw9575]

how is the line intergral limit of example 1: c3 curve from 0 to 1/root2 ??? why is that point (1/root2,1/root2)

[@mhamadhantro9818]

It's a unit circle (r=1) and the angle at this point is ø=pi/4 relative to the x_axis, by plugging in those in x=rcosø and y=rsinø you get your point.

[@hershyfishman2929]

Alternatively, since r =1, and y = x (as angel pi/4 is diagonal line y = x), solve for x and y by Pythagorean theorem: 2x^2 = 1

[@brco2003]

Can anyone tell me how he gets the coordinates (1/root2, 1/root2?

[@mhamadhantro9818]

It's a unit circle (r=1) and the angle at this point is ø=pi/4 relative to the x_axis, by plugging in those in x=rcosø and y=rsinø you get your point.

[@mukilraj9855]

In 22:38 isnt the dF/dt a vector? Can anyone explain. Thank you

[@DeadPool-jt1ci]

df/dt is the rate of change of f with respect to t.It's definitely not a vector. That's why if u take the integral of that with respect to t , you get back f itself.Remember , we are starting with vectors , but gradF is , dr which is a vector is , if you take the dot product of those u get fxdx+fydy. Now you can use the trick that dx = dx/dt * dt , and dy = dy/dt *dt , then factor out the * dt part.So you are left with (fx dx/dt + fy dy/dt) * dt. But the parenthesis is just df / dt. Remember from implicit differentiation , if f = f(x,y) , then df = fxdx + fydy , where fx= partial f with respect to x and fy= partial f with respect to y

[@user-rm6rb3tp5u]

24:44

[@PolskaUrban]

Did my A level and had just enough of this math stuff...can some maybe explain for dummies wherefore you would need this math stuff in reallity?

[@navs8603]

In most of the cases, these are what are driving actions behind the scene. We are always told that these are not "applied" in reality, however, every bit of these are used by us everyday -- just that we don't see them.

[@fatihorhan9355]

After proving line integral for gradients is end point - start point, weren't the consequences already obvious?

[@chaos00000]

why is multivariable calculus actually hard i love his french accent but man i understand less and less of what hes saying as we go deeper into the lectures

[@YouTodayKing]

Wow, zero dislikes as of now!!!!

[@chadliampearcy]

Since then only two people disliked it. Well if people don't like the series why would they be this far?

[@vfxenthusiast1244]

@@chadliampearcy +1 as of june 19

[@ahorribleperson3302]

@@vfxenthusiast1244 +3 as of July 21

[@user-ce1eh]

It's May 22 and the dislike count was made invisible sometime back.

[@tomrichter9021]

Doesn't he make a big mistake in the first example of the video (when he calculates the work done from the field on the particle that is doing the closed path)? To me it seems as if he completely ignores the effect of the vector field when he calculates c2 and c3, as no mention is made of the look of the vectors and their direction in relation to the trajectory. Instead, as he never gave any algeabric description of the vector field, he implicitly assumes (and I think by mistake) that the vector field is tangent to the trajectory along the entire trajectory (i.e. F dot dr = |F||dr|).

[@borisbarron1839]

Your wrong, he uses the formula where the components of 'r' were already expanded. In other words (note everything I write is a vector). F= and r = , so dr = . That means that F * dr = * = Fx dx + Fy dy. At this point he paramatized all the variables in terms of the angle. So everything he did was spot on.

[@not_amanullah]

This is helpful 🤍❤️

[@shafiatif]

wow

[@criticalthinking575]

Does anyone realise that he sometimes try to add humour to his lectures?

[@DeltaAccel]

He's a bit slow explaining in my taste, but still I think he's a great professor

[@DeltaAccel]

I find it endearing

[@Ramix09]

+DeltaAccel Just set the youtube speed to 1.25 or 1.5 :P

[@CreativeVery]

Lmao. If you think this guy is slow.

[@dreia2405]

DeltaAccel You wish you had a teacher like him