As geometers we ask...Is it all?...loved the lecture...the idea of 0 dim objects in 1 dim universe was awesome...can’t wait how this generalises to infinite dimensions
Idea: instead of regarding arc length as a function of some function, alpha, from I into R3, let's regard arc length as a function of the Image of alpha into R3. This seems more natural, since, from the point of view of arc length, the set of functions, alpha, having the same image forms an equivalence class. We then let each equivalence class be represented by the one curve, alpha*, for which the arc length grows linearly. In effect, we have changed the domain for arc lengths to be the set of curves traces out by points moving at constant speed :-).
The image of alpha is a curve, therefore your function is defined on the curve. The advantage of defining functions on the intervals is that we already built Calculus and therefore we have all the apparatus of calculus to study the properties of such functions. What you are suggesting is to throw away that apparatus; functions defined on the curve are completely different from functions defined on intervals. In a sense, the whole point of differential geometry is to build the study of functions defined on curved spaces (like curves) using what we already know about functions defined on flat spaces (like intervals). So your idea is the aim of the course, not something we should imbue in definitions in the beginning of the course
Nice lecture except in the definition of Frenet (direct) triedron, correctly defined by the cross product, but falsely determined on the picture by use of the LEFT HAND!
This is very important, but first fundamental steps in EUCLIDIAN differential geometry. And so at the opposite « end », so to say, from PSEUDO RIEMANNIAN geometry. First the configuration space is here 3D, when GR lives in 4D. And adding one dimension is not a trivial add. Already crucial changes occur between ZeroD and 1D, as between 1D and 2D. No instrinsic geometry exist in 1D whereas it does in 2D as Gauss showed with fundamental curvatures. Then in such presentation of differential geometry, « time » doesn’t really exist as a physical « concept ». Whereas in Physics, whatever « time » actually means, and it’s a very subtile, non trivial and problematic concept, it’s nevertheless mainly deeply related to « energy » concept trough invariance by « time translation ». In other words, non of this high difficulties of GR are here addressed. And one of the most important is the signature of the metric. Here the metric of euclidian framework is definite positive. Which is no more the case in SR, and even worse in GR where the « coefficiant » of the metric become FUNCTIONS of the space time « position ». And such 4D situation, with non positive definite metric of PSEUDO RIEMANNIAN geometry, and last but not least, « time » intrinsically mixt with « space » changes hugely all what’s going on in such model. So you surely need to master the basic euclidian 1D curve immersed in 3D space, but it’s only the first step toward PSEUDO RIEMANNIAN geometry of GR. And as one example of occurring problem is the mixing of « time » and « space » that brings the need to define the concept of Killing vector to cope with the loss of « speed » concept as a « derivative wrt time »; even the concept « proper time » of SR. vanishes. Not talking of the fact that nobody knows how to solve GR differential equations, unless in very problematically oversimplified extremely simplified symmetric context. For instance the bogus « theory of black holes » which is based on mathematical mistakes… Finally here the backgroung space is fixed, in which curves live. While in GR there is no fixed backgroung space; since space time is itself an intrinsic hyper curve…
I didn't understand the "philosophical" discussion at 31:35 . He said there is no intrinsic geometric in 1-d curve embedded in R^3, only extrinsic, ok I sorta get that I guess.
It seems like at 1:12:13 he is missing the "point" from the equation of a plane in point normal form. Maybe should have read $\langle \alpha(s) - \alpha(0), v angle $. What he has done works in one direction but not the other.
The wedge product between two vectors forms a oriented plane (i.e., an bivector in the exterior algebra) and it's dual is an oriented vector, which is obtained by the Hodge star operator. But this coincides with the cross product. See www.wikiwand.com/en/Cross_product#Cross_product_as_an_external_product.
What does it mean for the norm of the tangent vector not be equal to zero? Is it analogous to the limit delta approaches 0, in the fundamental theorem of calculus: that delta approaches 0 but is never really=0, or "there always is a secant". Secants are linear and hence imply the differentiation in R^n defined with linear mapping; does that make any sense?
And if the length of the tangent vector is 0, we're basically not adding anything, and so the length stops growing after a certain point or "breaks", and is discontinuous.
I doubt that you arer still looking for a reply but I just started this lecture series. To ensure that there is a diffeomorphism to a unit speed path, the norm of the tangent vector cannot be zero. If it were the case that the tangent vector went to zero at some point, then there would be no way to construct the diffeomorphism, since the norm would not be continuous and you would not necessarily be able to reparameterize the curve alpha, to that of the unit speed path. There is no direct analogy to the fundamental theorem of calculus. This fact that delta cannot equal zero there is because the only function that satisfies the epsilon-delta proof of continuity on R^n with delta equals zero, would be the constant function on the domain, f(v)=c whichs allows for the choice of any arbitrary epsilon>0.
One way of what i would like to think about this, is the fact that if u take the gradient of norm(asssume L2), then we would have grad(|f|) = some terms/(|f|)(u may derive it yourself or google it). from here, if |f| = 0, then further derivatives could not be well defined. hence our original arc length will be at most C^1. I hope it makes sense , and im saying the correct interpretation of this.
@@saikatgoswami9516 My maths background is not perfect. Assuming this question is sincere, here's two sets of differential geometry notes in the public domain that I have found useful. homepages.warwick.ac.uk/~masgak/cas/notes/casnotes.pdf raw.githubusercontent.com/quan14/MathsNotesUCL/master/3113%20M113%20Differential%20Geometry%20Notes.pdf
Claudio is an excellent teacher: thorough, detailed and always takes the time to make every detail clear
Regular curves 8:00; P arametrization with arc length 14:30; Curvature 38:00; Normal and Binormal vector and Frenet Basis 43:30; Torsion 51:00
Thanks a lot man , good job.
Thank you so much, man! Exactly what I needed. :)
He used his left hand to define the direction of binormal and got the direction that is opposite to his analytical prescription. Great Lecture !!!
As geometers we ask...Is it all?...loved the lecture...the idea of 0 dim objects in 1 dim universe was awesome...can’t wait how this generalises to infinite dimensions
Awesome lecture and lecturer!
Awesome lecturer indeed.
Thank you very much! I'm really enjoying the course
Idea: instead of regarding arc length as a function of some function, alpha, from I into R3, let's regard arc length as a function of the Image of alpha into R3.
This seems more natural, since, from the point of view of arc length, the set of functions, alpha, having the same image forms an equivalence class. We then let each equivalence class be represented by the one curve, alpha*, for which the arc length grows linearly.
In effect, we have changed the domain for arc lengths to be the set of curves traces out by points moving at constant speed :-).
The image of alpha is a curve, therefore your function is defined on the curve. The advantage of defining functions on the intervals is that we already built Calculus and therefore we have all the apparatus of calculus to study the properties of such functions. What you are suggesting is to throw away that apparatus; functions defined on the curve are completely different from functions defined on intervals. In a sense, the whole point of differential geometry is to build the study of functions defined on curved spaces (like curves) using what we already know about functions defined on flat spaces (like intervals). So your idea is the aim of the course, not something we should imbue in definitions in the beginning of the course
1:08:23 exercise should be curvature = |a|/sqrt(a^2+b^2) and torsion = -b/sqrt(a^2+b^2)
should the binormal vector B pointing in the opposite direction @44:20?
yes
Good explanations👏.He needs more space for sure
Nice lecture except in the definition of Frenet (direct) triedron, correctly defined by the cross product, but falsely determined on the picture by use of the LEFT HAND!
Does this course satisfy what you need to learn general relativity
This is very important, but first fundamental steps in EUCLIDIAN differential geometry. And so at the opposite « end », so to say, from PSEUDO RIEMANNIAN geometry. First the configuration space is here 3D, when GR lives in 4D. And adding one dimension is not a trivial add. Already crucial changes occur between ZeroD and 1D, as between 1D and 2D. No instrinsic geometry exist in 1D whereas it does in 2D as Gauss showed with fundamental curvatures. Then in such presentation of differential geometry, « time » doesn’t really exist as a physical « concept ». Whereas in Physics, whatever « time » actually means, and it’s a very subtile, non trivial and problematic concept, it’s nevertheless mainly deeply related to « energy » concept trough invariance by « time translation ». In other words, non of this high difficulties of GR are here addressed. And one of the most important is the signature of the metric. Here the metric of euclidian framework is definite positive. Which is no more the case in SR, and even worse in GR where the « coefficiant » of the metric become FUNCTIONS of the space time « position ». And such 4D situation, with non positive definite metric of PSEUDO RIEMANNIAN geometry, and last but not least, « time » intrinsically mixt with « space » changes hugely all what’s going on in such model.
So you surely need to master the basic euclidian 1D curve immersed in 3D space, but it’s only the first step toward PSEUDO RIEMANNIAN geometry of GR. And as one example of occurring problem is the mixing of « time » and « space » that brings the need to define the concept of Killing vector to cope with the loss of « speed » concept as a « derivative wrt time »; even the concept « proper time » of SR. vanishes. Not talking of the fact that nobody knows how to solve GR differential equations, unless in very problematically oversimplified extremely simplified symmetric context. For instance the bogus « theory of black holes » which is based on mathematical mistakes… Finally here the backgroung space is fixed, in which curves live. While in GR there is no fixed backgroung space; since space time is itself an intrinsic hyper curve…
I didn't understand the "philosophical" discussion at 31:35 . He said there is no intrinsic geometric in 1-d curve embedded in R^3, only extrinsic, ok I sorta get that I guess.
Wonderful : )
It seems like at 1:12:13 he is missing the "point" from the equation of a plane in point normal form. Maybe should have read $\langle \alpha(s) - \alpha(0), v
angle $. What he has done works in one direction but not the other.
Why the wedge product for vector cross product? Am I missing something?
The wedge product between two vectors forms a oriented plane (i.e., an bivector in the exterior algebra) and it's dual is an oriented vector, which is obtained by the Hodge star operator. But this coincides with the cross product. See www.wikiwand.com/en/Cross_product#Cross_product_as_an_external_product.
What does it mean for the norm of the tangent vector not be equal to zero? Is it analogous to the limit delta approaches 0, in the fundamental theorem of calculus: that delta approaches 0 but is never really=0, or "there always is a secant". Secants are linear and hence imply the differentiation in R^n defined with linear mapping; does that make any sense?
And if the length of the tangent vector is 0, we're basically not adding anything, and so the length stops growing after a certain point or "breaks", and is discontinuous.
I doubt that you arer still looking for a reply but I just started this lecture series. To ensure that there is a diffeomorphism to a unit speed path, the norm of the tangent vector cannot be zero. If it were the case that the tangent vector went to zero at some point, then there would be no way to construct the diffeomorphism, since the norm would not be continuous and you would not necessarily be able to reparameterize the curve alpha, to that of the unit speed path.
There is no direct analogy to the fundamental theorem of calculus. This fact that delta cannot equal zero there is because the only function that satisfies the epsilon-delta proof of continuity on R^n with delta equals zero, would be the constant function on the domain, f(v)=c whichs allows for the choice of any arbitrary epsilon>0.
One way of what i would like to think about this, is the fact that if u take the gradient of norm(asssume L2), then we would have grad(|f|) = some terms/(|f|)(u may derive it yourself or google it). from here, if |f| = 0, then further derivatives could not be well defined. hence our original arc length will be at most C^1. I hope it makes sense , and im saying the correct interpretation of this.
total focus on definitions
At 44:00 he used his left hand....
Examples would have been helpful
Look like class of philosophy
Ezzz claap
Where is geometry
There's lots of geometry in this. The lectures cover a differential geometry syllabus commonly taught in 3rd year undergraduate maths.
The entire lecture? What are you looking for?
@@konev13thebeast Angles and protractors and compasses. ;-)
@@billguastalla1392 Can you give me a link to your detailed undergraduate maths syllabus
@@saikatgoswami9516 My maths background is not perfect. Assuming this question is sincere, here's two sets of differential geometry notes in the public domain that I have found useful.
homepages.warwick.ac.uk/~masgak/cas/notes/casnotes.pdf
raw.githubusercontent.com/quan14/MathsNotesUCL/master/3113%20M113%20Differential%20Geometry%20Notes.pdf