Differential Geometry - Claudio Arezzo - Lecture 01

[@joshuaceleste6485]

Just a note to anyone confused about the first application of the Mean Value Theorem (I was at first, since its well-known a multivariable MVT doesn't hold). The Mean Value Theorem here is simply being applied component-wise to the curve alpha (hence the different values beta, gamma, delta). You can then combine the resulting equations into a vector valued equation and take the absolute value of both sides, the |t_i-t_{i-1}| factor pulls out, and the remaining factor, as per the definition of f is easily seen to be f(beta,gamma,delta).

[@saddamsultanakbar4342]

Very nice professor. His way of delivering a lecture is very nice.

[@pyrole]

Thank you Sir for such insightful lecture. I really like your informal comments in between, which brings the Mathematics alive...."Its up to us to define the dignity of theorems, while discussing propositions"...I liked this one. Constant references to what is it being a Geometer was nice way of starting to build intuition of the subject. Very detailed lecture with deep insights...I am really glad I found this playlist

[@kanna7789]

The best introductory differential geometry course I have ever seen.

[@guilhermegoncalves110]

1:17 - They're using the textbook from Professor Manfredo, one of the most celebrated Brazilian mathematicians! The moment that he said his name was very satisfying, because it's a really good book, and why not, some of that national pride was felt also :)

[@yerawizardaiden]

Dr. Manfredo's book is a classic text in differential geometry at this point! I hardly know of any university that doesn't use his book as a reference text for any introductory course into differential geometry. Definitely an academic gem to come out of your shores mate!

[@rogeriojunior9459]

I also felt the national pride heheh, especially because i'm also from Maceió, the city where Professor Manfredo was born

[@seefruit750]

1:14:34 - I do not believe this is correct, if I'm understanding the claim being made here correctly. It seems that he is saying that the length of the curve will be the same whether we traverse the segment uniformly, i.e. in a standard (t, f(t)) way, or if it acts as a pendulum. This is not true. Consider alpha = (t + 2 sin (t), 0), and let t be in [0, 2pi]. This traces the line segment from 0 to 2pi on the x-axis. We can also parameterize this trace(!) as (t, 0), and let t come from the same interval. The length of the trace, i.e. the geometric segment itself is obviously 2pi, and our second parametrization gives this. However, the first parametrization does not. You can check this by just using our definition of length and computing the integral. I think what he means is that the length of a curve is invariant under reparametrization, which is true. However, his example is actually of two _different_ curves which have the same trace. The point here being, there is no reparametrization from the "uniform" curve to the "pendulum" curve.

[@edoardofiorani7247]

This lecture is fruit of a tidy mind.

[@randalllionelkharkrang4047]

Slowly, I am witnessing how Differential geometry can be instrumental in Machine Learning. ML is heading for renaissance like how physics had decades ago. excited for this lecture series.

[@LaureanoLuna]

17:29 While f(x) = x^2/3 is a function, the graph proposed for it is not that of a function; note that two different values of y correspond to each positive value of x. If we are to take y as the independent variable, then it should be f(y)=y^2/3. It would have been better to write alpha(I)= (t^3, t^2) instead of (t^2, t^3) and then f(x)=x^2/3 would be OK. It's a bit messy as it stands.

[@NisseOhlsen]

Laureano Luna agree, as it stands it’s wrong.

[@antonioleal3447]

As a matter of fact, that function is on page 3 of the Do Carmo's book as an example. It is alpha(t) = (t^3,t^2). It was a mistake in the order on the video.

[@mou_aissa]

Yes , you're absolutely right cuz it must be for a function that every x has only one image !

[@flav6350]

Thank you for this video! The professor is explaining very clearly and it is enjoyable to listen to. I think there is a small mistake at 36:33: the parameter can go to infinity and the length still be finite, as previously illustrated for the Folium of Descartes at 21:55.

[@Wooflays]

he's talking about [a,b]x[a,b]x[a,b] - compact domain so the graph is closed & bounded. that case doesn't happen.

[@evariste6116]

This first lecture may seem a little abstract, despite Claudio's skill in making each step clear and the interesting examples, but it's very worthwhile following it through as it very efficiently sets up the fundamental concepts of curvature and torsion in the 2nd lecture, which are the basis of the geometry of curves in N-space

[@cisp360]

This is much much better than a differential geometry class I took at UCLA.

[@homerthompson416]

Always wondered if 120A was any good. I took 121 (topology) instead for my geometry requirement and the class was awesome since it was taught by Terry Tao. Was funny because he was younger than most of the grad students in the class at the time.

[@umarkhan-hu7yt]

His gesture is also good.

[@femmegeometry]

These lectures are inspiring me so much . One day i want to teach differential geometry :)

[@aplusscience3834]

the moment he referenced Do Carmo book, i knew i'll like this lecture. I freaking love that book, an absolute masterpiece.

[@Tommy_007]

I haven't read that book for about 25 years. I didn't like it. I remember I found it imprecise several places. I have a Ph.D. in mathematics.

[@SebastianRamirez-qw9qv]

Thanks a lot 🙏 this is a very nice explanation, what a great teacher 🍃

[@greg55666]

He's a very good teacher.

[@dariosilva85]

Wow, italian quality even in math. It is like drinking a good wine.

[@High_Priest_Jonko]

36:35 "We can measure lengths only of finite pieces" What about the foil of Descartes thats defined on (-1, infinity) but approaches the origin and appears to have finite length?

[@billguastalla1392]

Great lecture, thank you

[@joseantoniovargasmendoza7141]

Good lecture, but so far I have seen two mistakes. The example of a cusp is y = x^(3/2) and its graph must be on the non-negative x half plane. Also the folium should be of Descartes.

[@joabrosenberg2961]

Definition of Length; Invariance under Isometry; reparameterization 1:20:00

[@deconfinedQPT]

How can we have an access to the given assignments ?

[@imapublicagent]

Someone recommended these lectures to supplement the following book by do Carmo. It's a classic in the field with problems after each section and selected hints and answers in the back: www.amazon.com/Differential-Geometry-Curves-Surfaces-Mathematics/dp/0486806995/ref=sr_1_1?crid=1ZM1WFX1C14FI&keywords=do+carmo+differential+geometry+of+curves+and+surfaces&qid=1580545497&s=books&sprefix=de+carmo+diff%2Cstripbooks%2C220&sr=1-1

[@mohammadrafee3314]

Excellent 💯 teaching.

[@anantbadal6045]

Where can we get some homework and self assessment problems for each lectures?

[@rogeriojunior9459]

Manfredo's book has some really good problems to practice

[@around_mathematics]

Thanks professor for this excellent lecture.

[@timeearning6782]

Are other lectures are uploaded of differential geometry series

[@Rookie1706]

very interesting class that I will take eventually.

[@psychohosi]

I've never seen that version of the mean value theorem, can't find it anywhere on the internet and quite frankly don't know what it is even saying here.

[@许克-s9n]

it is a theorem in calculus I.

[@psychohosi]

@@许克-s9n Yeah but that theorem doesn't apply to curves which is what he does here (you can even find counter examples for this on wikipedia). Taking the norm doesn't fix either because he would have to take the norm *before* applying the theorem - which is not what is done here.

[@psychohosi]

@@许克-s9n nvm after watching it again just now I get what he does here.

[@gamzeonder8763]

i thought that might be a mistake also. Really confused here.

[@YUYANGHONG]

At 17:00 it should be y=x^(3/2), right?

[@JansthcirlU]

I think the prof meant to write alpha(t) = (t^3, t^2) because that's the curve he drew

[@lucasmcguire1554]

He corrects it by putting x along the usual y direction

[@Sana_Ullah99]

From where can I get the contents which will be covered in this course

[@RSciOfficial]

Differential Geometry of Curves and Surfaces by Manfredo do Carmo

[@Tommy_007]

I would get another book than do Carmo. It is not mathemacally precise.

[@kashafrajput8231]

Can somebody tell me how to check diffeomorphism of a curve and how can we check that two or more curves belongs to same equivalence class

[@leonardodavinci1329]

I know nothing about this subject but since no one has helped you in a year, I will personally learn it just to help you. Give me some time.

[@carlkpsplucky5554]

Imagine a coordinate plane, to calculate diffeomorphism, we must remember y=mx+b. To calculate the differential slope we must..

[@strength6206]

Why is there a bra above the board?

[@alute5532]

Book M De Carmo geometry of curves &surfaces -What's geometry is about? You've seen geometry in Euclidean Space & analysis in it Study functions defined on (open sets on In) Or infinite dimensional spaces But some sense `vector spaces` All space you know till now male Translations Geometric term:flat spaces - or characterize spaces in terms of Curvature Able to study functions defined on opens sets on Rn Even infinite dimensional -studies how curvature Our ability to study differential equations .main problem what's curvature? Given sense of What's sth curved to sth flat Almost half of problem! Start w lowest Dimension ,then surfaces 2D Have all features make thing complicated.first topic Curves(4lecs): 4:00 Most common definition of curve is a map imGe of map & map itself Differentiable Csup infininty I -> R3 in R Map subset R in subset of R3 Vector A'=x'(t), y'(t) ,z'(t) Tangent vector Takeaway differential curve is planar curve 2d P in R3 s.t. The image of map A is contained in Smooth x y z smooth functions s.t Curve lies in plane P Takeaway Curve can have self intersections 2. Graph A isa (A has no corners) -If our theory doesn't cover lines & circles then our theory is to be thrown away Center c radius R Finding norm sqrt (x^2 +y^2 +z^2) From it find its continuous as norm can be >= 0 May not be differentiable.( can be 0) Uniformly continuous (on compact subsets) Transition is linear map (if linear can add (metric) matrix using standard basis) whose associated matrix A satisfies A At=I Aka orthogonal transformation Not 1.composition 2. rotation 3.translation 4.reflection Out this equation, cannot decide determinant of A=1 (+1 or -1?) 1:07:00 +1 direct isometry -1 inverse isometry Doesn't matter these are just.. Names Conclude tower definition of length (of curve) is in variant By isometry length curves is preserved Translations if move the length doesnt change If act on a segment by an orthogonal matrix it's length doesn't change That's limit of lengths of segments Length is geometric quantity | A(b)-A(a)| =L(a,b)(A) is segment This will force you to think One of problems you thing when you understand the Question, the solution is immediate, but understanding the question is not, because there's an ambiguity here What's a segment? 1 geometric object 2 parametrized object? I. E. A line is form A(t)= t*v+ v0 with t in some "fixed interval" one way But aren't stmt won't be true Out of the length, you're not able to reconstruct the way the particle is moving on the curve (true) Ormstein uhlenbeck You cannot decide if point is going from The motion of particle is uncertain A(a) to A(b) will it go in Uniform? motion way. Or going like a pendulum ? Length of curve is same. (motion won't be so) Here adjest: is segment up to paramerization -parametrized well(enough) - Geometric meaningful question We wanna do geometry We're using analytic tools to So some geometry Parametrizing object is v. Convoneint, but introduces a problem You can parametrized same geometric thing In am infinte number of ways Q. So how do i go back to geometric properties of the objectwhich quantities independent of way particles move Using smooth functions is easy till now Price to Pay: Many geometric things can be described in completely different Analytical ways Since I'm interested in geometry of this object, Need in someway to kill this freedom (of possible paramerizations) A is made of smooth functions -image A can be smooth without corners -idea behind smooth Not true A could have corners despite that A=t 2, t3 A(I) is graph of function f x 2/³ 15:49... Diffeomorphism: I &j are open intervals Nope between 2 open intervals 1.Smooth 2.Invertible 3. smooth inverse Given 1.parametrized curve A From I-> R3 We can construct a new curve Beta Simply composing with map phi j -> I beta go from J -> R3 A compass of Phi. so A take it to R3 As beta A0*phi As geometries yes same curve image is the same Going around this object in a different way But Object is same Image of Beta & image of Alpha are the same Beta: reparametrization of alpha Phillip from J to I diffeo morphism A is smooth curve a b closed sub interval (of J) Phone maps closed sub interval into another closed interval call [c d] Then length between a b of A composed Phi = length L( c d)(A) With property c inf smooth with inverable C If not true raise all start from scratch Geometric property of image These 2 objects have the same image If these not true we are on wrong track Solution compare norm tangent vector of that with norm tangent vector or lhs by chain rule: Norm (A comp phi)' (t)= |A' phi(t)|*|phi'(t) Phi is map from r to r 1st norm(vector) 2nd absolute value of a number Phi is diffeomorphism Phi' never Be ZERO either positive or negative everywhere S=phi(t) - int(derivative A comp phi')(t) *dt (polynomial " Comparing integral a b of derivative of this wrt to A'phi(t)*abs(phi'(t)*dt The only delicacy here If phi'always positive , absolute value = phi' itself (no error) If Phi'>0 By change vars integral: Int(c,d) |d/ds L(A(s)|*ds (laplacian?) If Phi'negative formula is true too Switch interval integration (c, d) assumes symmetry object (c

[@alvaropaz3817]

Great, great but need to prove the rigid rotation considering the definition of rigid body

[@BArdekani]

38:40 what's to guarantee that \alpha(t_i) line up like that? That is, as t goes from a to b \alpha(t) is moving in one direction. The particle could be reverse direction for a while and go back.

[@Simon-xi8tb]

This lecture is very zmooth

[@costyakurlaev9473]

Nonlinear into the linear space

[@vaibhavnayel]

this is the most italian math dude I've ever seen

[@tavindersingh3098]

At 10:13 can someone clear that is that Z Stands for 3rd dimension's coordinates or just for something else. Idk here in India Z stands for 3D but in diagram they confuse in Z and Y. Because where he showed Z Is actually we show as Y And Y is where he showed Z and X is where he showed Y. it's confusing so please clear it someone.

[@noriotakemoto1462]

MIKEY SINGH yes Z is just a coordinate of the 3 dimensional space. The diagram shows the XYZ axes as the right handed Cartesian coordinate system, which is commonly used.

[@TheAzwxecrv]

I have a doubt, Professor Arezzo. Curve is defined as map from I to R3. That is, there is no mention of continuity. Then, what if I map every point in I to completely disconnected points in R3? How can we call it a curve? (Pls note that we can't claim that because t, which is from interval I, is continuous, alpha (t) also continuous - obviously, it is the basic Calculus course. x can be continuous, but f(x) need not be continuous.)

[@gamzeonder8763]

What kind of a Mean Value Theorem is that? Im really confused. isnt it supposed to be f'(...)*(t_i-t_i-1)=(alpha(t_i)-alpha(t_i-1)) ???

[@JM-ty6uq]

Im trying to figure out the same thing right now.

[@JM-ty6uq]

So you take the regular mean value property (see wikipedia) alpha'(t*_i) = ( alpha(t_i) - alpha(t_{i-1}) ) / (t_i - t_{i-1}), rearrange and take absolute values will give you (t_i - t_{i-1}) |alpha'(t*_i)| = | alpha(t_i) - alpha(t_{i-1})|, but f will equal |alpha'(t*_i)| at some point (beta_i,gamma_i,delta_i) since it is a more general function, which gives the final result.

[@gamzeonder8763]

oh, i see. thanks a lot!

[@JM-ty6uq]

@@gamzeonder8763 No Problem! This lecture series looks really good and I think I am going to continue - it would be great to have someone to bounce thoughts with. If you want to do that, feel free to add me on discord: jm123#0117

[@Kirqos]

alpha is a vector valued function and there is no MVT for vector valued functions. I'm also confused as to what he's doing there, I checked another source and the proof there seems to be quite more involved, by checking the coordinate functions individually (to which you can apply MVT)

[@seanpaul2562]

1:23:55 i don't understand why derivative of smooth function with smooth inverse can't be zero ?

[@Julianorla7]

If your derivative is equal to zero in some point, the function won't have smooth inverse for all its domain, since this would imply that there is a vicinity where the function is not injective.

[@seanpaul2562]

@@Julianorla7 okay now i got it if minima or maxima exists inverse is not smooth thanks

[@Julianorla7]

To be clearer, if your derivative is equal to zero at some point, by the chain rule you can check that the derivative of the inverse function is indetermined at the preimage of that point, meaning that the inverse is not smooth. Moreover, if your derivative changes sign, you can show that your function isn't injective, so the inverse is not well defined.

[@seanpaul2562]

Thanks man

[@pronaybiswas7524]

amazing lecture

[@luisbreva6122]

11:56 e.g is a shortcut to say for example OK? In case you didn't.... Ok lmaooo

[@udaymuhammdi9298]

Thanks very much

[@omermuharremyagcioglu]

handsome and intellectual mathematic proffesor..thank you sir

[@kparag01]

What are prerequisite for this class?

[@excalipr]

Parag Kulkarni probably multi variable calculus, linear algebra, and real analysis?

[@Fightclub1995]

excalipr complex analysis is also handy

[@konev13thebeast]

multivariable calculus, differential equations, variational calculus and topology should give you a sufficient baseline.

[@sverrirjensson9772]

I think calculus ,linear algebra , formal logic and mathematical proofs would be suffice a prerequisite.

[@costyakurlaev9473]

What is your native? Is it italian or romanian?

[@ce6535]

Really diverse class, this is a great thing.

[@LaTortuePGM]

Dubious Claims well, i guess he meant everyone could understand some stuff from the course some way and share more diverse thoughts with other students.

[@LaTortuePGM]

oh i didn't think about races lmao

[@LaTortuePGM]

i thought he knew the students

[@StatelessLiberty]

"No white people in the audience, this is a great thing."

[@davidfoley8546]

​@@StatelessLiberty "International center" is literally in the name--if the audience was not diverse, it wouldn't really be accomplishing its mission.

[@MuhammadBilal-gq9el]

Sir I need elementary differential geometry revised 2nd edition (BarrettO'Neill) solution manual chpter#01,2

[@darovan4398]

Wonderful lecture! Thank you so much!

[@fcalin21]

Allot of information for one leson.

[@دکتراقبالی]

Thanks

[@costyakurlaev9473]

3 parameters to measure

[@satyajitsaha29ss]

Wonderful lecture...

[@haimbenavraham1502]

Great teacher.

[@llliiliiiiiililiiiliiiilllllli]

Does this follow a particular book? Would Kreyszig's Differential Geometry be appropriate?

[@llliiliiiiiililiiiliiiilllllli]

Thanks!

[@llliiliiiiiililiiiliiiilllllli]

also RIP do Carmo :(

[@jamshaiddhothar1838]

Sir this lecture is extremely helpful for students.

[@jobbione]

Grazie prof

[@aj-uo3uh]

Great teacher and great stuff. I like the view points of the process and struggle for defining something that we know intuitively and the problems that can arise. But the definition and proof that the limit exists for arc length of a curve I still don't understand. A much easier and simpler but similar (with mean value theorem) way is on wikepedia: en.wikipedia.org/wiki/Arc_length

[@qbtc]

Download a pdf copy of the recommended textbook here, www.math.purdue.edu/~li2285/courses/562f/docarmo.pdf

[@jamshaiddhothar1838]

Sir is it possible to share notes on differential geometry in PDF format ?

[@munozariasjm]

Really good lecture!

[@costyakurlaev9473]

i'm facinated with the 5d spaces

[@rogeriojunior9459]

Can you see them?

[@Ravigupta-vo1yl]

Thank you sir.

[@John-lf3xf]

jair bolsonaro if he was actually smart

[@universum-theuniverseexpla6565]

1:18:33

[@mumtazhamid]

Wonderful : )

[@johnmckeown6573]

23:44 open set of WHAT ? 23:58 ... plus something like this .. some chalk marks on the board and ... WHAT ? Pls explain !

[@chasebender7473]

Open sets of the folium in the subspace topology, meaning that a set U is open in the folium C if and only if there exists an open set V of R^2 (the disk that he drew) such that U=C intersect V. This as opposed to the push forward topology of alpha, meaning a set U is open in C if and only if there is an open set V of R with alpha(V)=U

[@salehhasan9385]

yh

[@bhavna6565]

It's a planned confusion XD

[@gianlucanotarangelo3098]

Geogebra tells that the figure was actually right with the usual axes.

[@harrycraig2077]

Hu A

[@xiaocute1107]

这口音我也是醉了

[@crjon3638]

哈哈哈

[@neolam9208]

xiao cute 這種沒水準的留言才令人醉

[@antoniomantovani3147]

i dont like this lecture, first, you must define parametrizations and equivalene class, then what is a gragh, and then curves

[@anthonyramirez3200]

Give me an example of where this type of math will be useful on my every day life.

[@NotLegato]

if you're a theoretical physicist studying general relativity, for example. there's no way you can have hang ups about "real life" applications at THIS point of your mathematics education.

[@Suav58]

It helps to understand a lot about real life. For instance: We have this naive notion of invertibility. Invertibility works fine on highest levels of abstraction (group, number field) but as soon as one allows for one bat of an eye it stops being obvious that there is only one inverse. That is one can return from B to A in a number of ways. This path integral question was bothering Feynman, but it is the one bothering modellers of economical systems and political systems as well...

[@johnmckeown6573]

@Slawomir P Wojcik that was your real world example ?

[@someperson9052]

Why does it need to be useful for your everyday life? Not everything that is learnt needs to be applied to you walking to work, sometimes it is learnt for a specific job. A lot of people learn engineering but that's not useful for "everyday life" it's learnt by a specific group of people.

[@hongchulnam1630]

I study computer graphics. Computer graphics requires differential geometry for rendering(which creates realistic images for animations for example).

[@crehenge2386]

unreadable writing...