Search results “Foundations of differential geometry”

Differential geometry arises from applying calculus and analytic geometry to curves and surfaces. This video begins with a discussion of planar curves and the work of C. Huygens on involutes and evolutes, and the related notions of curvature and osculating circle. We discuss involutes of the catenary (yielding the tractrix), cycloid and parabola. The evolute of the parabola is a semi-cubical parabola. For space curves we describe the tangent line, osculating plane, principle normal and binormal.
Surfaces were studied by Euler, who investigated curvatures of planar sections and by Gauss, who realized that the product of Euler's two principal curvatures gave a new notion of curvature intrinsic to a surface. Curvature was ultimately extended by Riemann to higher dimensions, and plays today a major role in modern physics, due to the work of Einstein.
If you like this topic, and want to learn more, make sure you don't miss Wildberger's exciting new course on Differential Geometry! See the Playlist DiffGeom, at this channel.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 103406
njwildberger

We introduce the approach of C. F. Gauss to differential geometry, which relies on a parametric description of a surface, and the Gauss - Rodrigues map from an oriented surface S to the unit sphere S^2, which describes how a unit normal moves along the surface.
The first fundamental form describes the Euclidean quadratic form in terms of the parametrization, and the second fundamental form is determined by the derivative of the Gauss - Rodrigues map.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 18114
njwildberger

We introduce surfaces, which are the main objects of interest in differential geometry. After a brief introduction, we mention the key notion of orientability, and then discuss the division in the subject between algebraic surfaces and parametrized surfaces. It is very important to have a balanced view between these two aspects; most texts are oriented, following Gauss, to the parametrical side: we will at least initially compensate by providing more detail on the algebraic surfaces.
Important examples of surfaces include quadrics, such as spheres, or more generally ellipsoids, or hyperboloids. We also mention some more unusual examples, including the Oloid, discovered by Paul Schatz in 1929.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 11302
njwildberger

This is a gentle introduction to curves and more specifically algebraic curves. We look at historical aspects of curves, going back to the ancient Greeks, then on the 17th century work of Descartes.
We point out some of the difficulties with Jordan's notion of curve, and move to the polynumber approach to algebraic curves.
The aim is to set the stage to generalize the algebraic calculus of the previous few lectures to algebraic curves.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
A screenshot PDF which includes MathFoundations46 to 79 can be found at my WildEgg website here: http://www.wildegg.com/store/p101/product-Math-Foundations-screenshot-pdf

Views: 16621
njwildberger

Differential Geometry (PSI 14/15, Front End) - Matthew Johnson (University of York, UK)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNZzCRyVcC0BfKPNn5WUoX7
Lecture 1 Special relativity; Lorentz transformations; Tensors
Lecture 2 Dual tensors; Metric job description; Gravity and Geometry
Lecture 3 Covariant derivative, Riemann and Ricci tensors; Einstein equations
Lecture 4 Important metrics, inflation, cosmological constant problem
《PSI 14/15 Mathematics Review & Front End Courses》
Full Programme:
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNMmBNo4FJ3I50PDJijr4jJ
1. Complex Analysis - Tibra Ali (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbPVYaAik-XlaPqnTiUOWZyT
2. Classical Mechanics - David Kubiznak (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMjq6UHk6KtI-n3q_H__T48
3. Distributions & Special Functions - Dan Wohns (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNVuijYAmwW-PB8LlRQLD1D
4. Quantum Mechanics - Agata Branczyk (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbO2xOeYHiPphGCeSAbwaAi_
5. Lie Groups & Lie Algebras - Gang Xu (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbOc5cWKgiqK4x5t2yaiBp1S
6. Differential Geometry - Matthew Johnson (York)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNZzCRyVcC0BfKPNn5WUoX7
7. Green's Functions - Denis Dalidovich (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbM5krT20wcR4a3kIHfNIEBF
8. Mathematica, Computational Methods - Erik Schnetter (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMQQz6sLA7Qg4ZnDSl1eAHs
《Perimeter Scholars International (PSI) 2014-2015》
Full Programme:
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMv_x1IuxfJVdLSSbDN2iZK
Core Topics (1)-(7):
1. Relativity (PHYS 604) - Neil Turok (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMWuR0KyALZo-Nk5TX30QNZ
2. Quantum Theory (PHYS 605) - Joseph Emerson (Waterloo)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbOp3GY-wp7HO7ejNIkmutEE
3. Quantum Field Theory I (PHYS 601) - Dan Wohns & Tibra Ali (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNcchvYh2ZwhV7Ru1G5Dy4E
4. Statistical Mechanics (PHYS 602) - Anton Burkov (Waterloo)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMyM4XtY0O3jDiniO7xcKsH
5. Conformal Field Theory (PHYS 609) - Jaume Gomis, Pedro Vieira, Freddy Cachazo (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNTRe3ncNjnB16u5Rc6d7wj
6. Quantum Field Theory II (PHYS 603) - Francois David (CEA, Saclay)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbN6Bwu_QrZNaVzKeGOBns6I
7. Condensed Matter I (PHYS 611) - Marcel Franz (British Columbia)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbN8C2MUgzkB8wcvU7XirzVn
Reviews (8)-(16):
8. Standard Model (PHYS 622) - Gordan Krnjaic (Kavli Institute), Stefania Gori (Chicago)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbN9-MkRNr_pR2ZivSLPBfbr
9. Gravitational Physics (PHYS 636) - Ruth Gregory (Durham)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNmJvn1KhWyMkFBIDIJbQrZ
10. Foundations of Quantum Mechanics (PHYS 639) - Lucien Hardy (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbOGqo_3AY2uc1qN00s6nEzm
11. Condensed Matter II (PHYS 637) - Alioscia Hamma (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMZFs8t5VNcLz7D59CKRaUn
12. String Theory (PHYS 623) - Davide Gaiotto (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbOG6zGRfIc6RajzYRUSFamL
13. Cosmology (PHYS 621) - David Kubiznak (Perimeter), Kurt Hinterbichler (Case Western)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMPV6l8PKGpN-8b__8HOYDz
14. Beyond the Standard Model (PHYS 777) - David Morrissey (Victoria)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbMWO2CJL6llrXDl8WbxJOhJ
15. Quantum Gravity (PHYS 638) - Bianca Dittrich (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNPqBLnQfYukJvzIge6AHkl
16. Quantum Information (PHYS 635) - Daniel Gottesman (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbO9x6wLlaWa94nRP7qqS8mI
Explorations (17)-(21):
17. Explorations in Quantum Information (PHYS 641) - David Cory (Waterloo)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbPkjCov_BmTcG0-aY4wTN9-
18. Explorations in Condensed Matter - Miles Stoudenmire (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbM7M_rVoF3U-gAxceDEyHGT
19. Explorations in Particle Theory (PHYS 646) - Brian Shuve (Harvey Mudd College)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbPpgIwrN9hT5ME6di1mdYFK
20. Explorations in Cosmology (PHYS 649) - Kendrick Smith (Waterloo)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbOY860O3m65iIATsEMmDYF9
21. Explorations in String Theory (PHYS 647) - Pedro Vieira (Perimeter)
▶ https://www.youtube.com/playlist?list=PLFMKfDJ8QzbNcnrGnRafKZg9jDf-8rq_u

Views: 334
Centre for Mathematical Sciences

We introduce the notion of topological space in two slightly different forms. One is through the idea of a neighborhood system, while the other is through the idea of a collection of open sets. While this is all reasonably traditional stuff, regular viewers of this channel will not be surprised to learn that I consider the `infinite set' aspect of these theories highly dubious.
But this is a course at a major university in 2013, so it is still early days for people to be looking more critically at such theories. Of course there are finite versions of these concepts, and we do use some simple examples for illustration, but it is fair to say that these do not capture the true intention of these definitions in trying to set up a theory of what a `continuous space' might actually be.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 19895
njwildberger

With an algebraic approach to differential geometry, the possibility of working over finite fields emerges. This is another key advantage to following Newton, Euler and Lagrange when it comes to calculus!
In this lecture we introduce the basics of finite (prime) fields, where we work mod p for some fixed prime p, and show that our study of tangent conics to a cubic polynomial extends naturally, and leads to interesting combinatorial structures. There are many possible directions for investigation by interested amateurs who have understood this lecture.
After the basics of arithmetic over the field F_p, including a discussion of primitive roots and Fermat's theorem, we discuss polynomial arithmetic and illustrate tangent conics to a particular cubic over F_11. In particular Ghys' lovely observation about the disjointness of such tangent conics (for a cubic) can be illustrated completely here, and some additional patterns visibly emerge from the vertices of the various tangent conics.
One big difference here is that the sub-derivatives and the derivatives are NOT equivalent in general, and we must replace the usual Taylor expansions with one involving sub-derivatives. Some remarks about the useful distinction between polynomials and polynomial functions in this setting are made.
This lecture shows that the calculus is actually a much wider operational tool than is usually appreciated----finite calculus not only makes sense but is a rich source of both combinatorial and algebraic patterns---and questions for further investigations.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 15519
njwildberger

We discuss the curvature of planar curves and applications to turning numbers and winding numbers, also called the index. We use this opportunity to talk a little about irrational numbers and transcendental functions: which we treat from an applied point of view: all statements involving integrals, infinite sums etc are to be interpreted in an approximate sense. (For those interests in finding out more about such radical departures from the established dogma, see my MathFoundations series!)
We treat curvatures from a rational turn angle point of view, introduced in the AlgTop series.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 8757
njwildberger

This video begins to lay out proper foundations for planar Euclidean geometry, based on arithmetic. We follow Descartes and Fermat in working in a coordinate plane, but a novel feature is that we use only rational numbers.
Points and lines are the basic objects which need to be defined.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?ty=h .
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 13430
njwildberger

This video looks at the idea of a tangent space at an arbitrary point to any given manifold in which vectors exist. It shows how vectors expressed as directional derivatives form a basis for the tangent space at the given point. This basis has the same dimension as the given manifold.

Views: 4832
Robert Davie

This lecture discusses parametrization of curves. We start with the case of conics, going back to the ancient Greeks, and then move to more general algebraic curves, in particular Fermat's cubic, the Folium of Descartes and the Lemniscate of Bernoulli.
We talk about the 17th century's fascination with motion via Newton's laws, and various interesting mechanisms that generating curves, including the four bar linkage and Watt's linkage.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 31247
njwildberger

Here we go over in some detail three problems that were assigned earlier in the course: the rational parametrization of the cissoid, the parametrization of a particular conic x^2-4xy-2y^2=3, and finding the evolute of the curve y=x^n for a general n.
Note that in my diagram around 14:00 I incorrectly place the point [-1,1], but this has no effect on the algebraic discussion.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 7856
njwildberger

This video looks at the projective Triple quad formula, also known as the Triple spread formula in Rational Trigonometry, and how it relates to curvature. A triangle in the plane has a unique circle through the three points, called the circumcircle of that triangle, and a fundamental question is how to determine the size (for us, quadrance) of that circle from the three side quadrances of the triangle.
The formula has a lovely connection with both the fundamental Triple quad formula and the projective version of it. Quadratic curvature is defined to be the reciprocal of circumquadrance.
Understanding the relation between projective 1-points and the core circle is key to how the projective triple quad formula connects one dimensional and two dimensional geometries.
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 2462
njwildberger

A very brief outline of the contents of the later books in Euclid's Elements dealing with geometry. This includes the work on three dimensional, or solid, geometry, culminating in the construction of the five Platonic solids.
This lecture is part of the MathFoundations series, which tries to lay out proper foundations for mathematics, and will not shy away from discussing the serious logical difficulties entwined in modern pure mathematics. The full playlist is at http://www.youtube.com/playlist?list=PL5A714C94D40392AB&feature=view_all
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/Norman_Wildberger. I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?ty=h .
A screenshot PDF which includes MathFoundations1 to 45 can be found at my WildEgg website here: http://www.wildegg.com/store/p100/product-Math-Foundations-A-screenshot-pdf

Views: 10425
njwildberger

Here we introduce the idea of a curve as a mapping, briefly discuss the need for regularity and show how the velocity acts on functions when we view tangent vectors as derivations.

Views: 3186
James Cook

In this video we introduce projective geometry into the study of conics and quadrics. Our point of view follows Mobius and Plucker: the projective plane is considered as the space of one-dimensional subspaces of a three dimensional vector space, or in other words lines through the origin. In this way we can introduce homogeneous coordinates [X:Y:Z] for the more familiar points [x,y]; the big advantage is that now points at infinity become concrete and accessible: they are simply points of the form [X:Y:0].
A curve like the parabola y=x^2 gets a homogeneous equation YZ=X^2, including now the point at infinity [0:1:0], which corresponds to the direction in the y axis. This gives a uniform view of conics close to Apollonius' view in terms of slices of a cone.
We will see that homogeneous coordinates provide a powerful and useful tool to not only the study of conics and algebraic curves in the plane, but also to quadrics and higher algebraic surfaces in space.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 14001
njwildberger

We rejuvenate the powerful algebraic approach to calculus that goes back to the work of Newton, Euler and particularly Lagrange, in his 1797 book: The Theory of Analytic Functions (english translation). The idea is to study a polynomial function p(x) by using translation and truncation to create various Taylor approximations to p(x) with respect to a point r on the line. This can all be done with only high school mathematics; in particular NO LIMITS, and NO REAL NUMBERS!! We see that the differential calculus, in its essence, is an elementary theory.
Since this lecture is likely to be of general interest to anyone studying or teaching differential calculus, we begin by reviewing the usual story as found in most modern text books. Then we introduce Lagrange's approach, culminating in identifying clearly the tangent line, tangent conic, tangent cubic etc to a polynomial p(x) at r. These geometric objects associated to a function at a point will play a major role in this course.
With this algebraic calculus, the usual derivatives of p(x) are replaced by simpler renormalizations, which we call the sub-derivatives of p. There are many advantages, and we illustrate some of them with explicit worked out examples.
Prepare to have your confidence in the intrinsic rightness of the standard orthodoxy challenged! In this course we adopt a beginner's mind, so there are many possibilities.
The implications of the wider mathematical community understanding and appreciating this point of view are huge: maths education can turn a new leaf, with a simpler, more elegant and much more logical approach to this important subject. As Euler and Lagrange tried to teach us, more than 200 years ago.
Correction: At 26:00 the formula for D2(pq) should be D2(pq)=D1(D1(pq))=pD2(q)+2*D1(p)*D2(q)+qD2(p) as noted by Faraz Sahba (thanks!)
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 25859
njwildberger

The first lecture of a beginner's course on Differential Geometry! Given by Assoc Prof N J Wildberger of the School of Mathematics and Statistics at UNSW. Differential geometry is the application of calculus and analytic geometry to the study of curves and surfaces, and has numerous applications to manufacturing, video game design, robotics, physics, mechanics and close connections with classical geometry, algebraic topology, the calculus of several variables and mostly notably Einstein's General Theory of Relativity.
This lecture summarizes the basic topics of the course, the unique point of view of the lecturer, and then heads straight into a survey of classical curves, starting with the line, then the conic sections (ellipse, parabola, hyperbola), then moving to classical ways of generating new curves from old ones. These techniques include the Conchoid construction of Nicomedes, the Cissoid construction of Diocles, the Pedal curve construction and the evolute and involute introduced by Huygens. This lecture should be viewed in conjunction with MathHistory16: Differential Geometry.
If your level of mathematics is roughly that of an advanced undergraduate, then please come join us; we are going to look at lots of interesting classical topics, but with a modern, lively new point of view. There will be opportunities for you to contribute to new directions. Prepare to be surprised, for our approach follows that famous Zen saying:
"In the beginner's mind there are many possibilities; in the expert's mind there are few."
The music is by Exchange: a track called Take Me Higher (thanks Steve Sexton!)
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 110517
njwildberger

In this video we extend Lagrange's approach to the differential calculus to the case of algebraic curves. This means we can study tangent lines, tangent conics and so on to a general curve of the form p(x,y)=0; this includes the situation y=f(x) as a special case. It also allows us to deal with situations where the usual tangent is vertical, and so the derivative is undefined.
The case of the lemniscate of Bernoulli is looked at in detail. Since now the tangent conic can be either an ellipse, parabola or hyperbola, we see that the nature of the quadratic approximation at a point allows us to group points on the curve into elliptic, parabolic and hyperbolic type. For the lemniscate, the parabolic points are found, lying on the discriminant conic.
This opens the door to a more differential study of algebraic curves. We also show how this strategy generalizes in a simple and natural way to investigate algebraic surfaces in three dimensional space!
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 13568
njwildberger

A space curve has associated to it various interesting lines and planes at each point on it. The tangent vector determines a line, normal to that is the normal plane, while the span of adjacent normals (or equivalently the velocity and acceleration) is the osculating plane.
In this lecture we start by continuing on from the example of the last lecture--discussing curvature and torsion for a finite segment curve, which is useful in robotics.
Then we go back to G. Monge who introduced geometrical aspects of space curves, such as the idea of the axis, also polar developable and tangent developable surfaces associated to a space curve. An example for the rational helix is illustrated.
We are also interested in algebraic aspects of curves, and give alternate formulas for the curvature and torsion of a curve in terms of the velocity and acceleration. We are particularly interested in the general case where we do not assume a unit speed curve.
We also study how invariants may be considered as quantities which are invariant under reparametrization of the curve.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 10088
njwildberger

In this video we discuss Gauss's view of curvature in terms of the derivative of the Gauss-Rodrigues map (the image of a unit normal N) into the unit sphere, and expressed in terms of the coefficients of the first and second fundamental forms. We have a look at these equations for the special case of a paraboloid, where we can compare with our previous discussion of curvature.
We then look at a discrete analog of curvature which applies to polyhedra, which goes back to Descartes. Involved are formulas for the sum of angles of a spherical polygon. This discrete form gives us an easy justification of Gauss' Theorema Egregium. We have a look at the Gauss Bonnet theorem in this context, where the total curvature of a closed surface with the topology of the sphere is 2 pi times the Euler characteristic.
Is this the final video in this series (but we may very well carry on at some future point!)
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 35925
njwildberger

This video follows on from DiffGeom21: An Introduction to surfaces, starting with ruled surfaces. These were studied by Euler, and Monge gave examples of how such surfaces arose from the study of curves, namely as polar developables. A developable surface is a particularly important and useful kind of ruled surface: it can locally be laid out on a plane. Euler classified developable surfaces, and we discuss his characterizations.
We talk about Mean and Gaussian curvatures, and introduce minimal surfaces and Plateau's problem, studied also by Lagrange. Then we discuss quite a different family of surfaces: the algebraic ones given by a polynomial equation. Examples feature the ellipson, from Rational Trigonometry, a beautiful and symmetrical cubic surface, Cayley's surface and others.
Finally we discuss some projective constructions of surfaces.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 8799
njwildberger

We discuss the intrinsic distance on surfaces as well as the basic theory of isometries for surfaces. This follows section 6.4 of O'neill.

Views: 778
James Cook

Here we continue our study of the works of three important French differential geometers. Today we discuss G. Monge, who is sometimes called the father of the subject. He was the inventor of descriptive geometry (which he developed for military applications), and various theorems in Euclidean geometry, including homothetic centers of three circles, and the Monge point of a tetrahedron. He also studied curves, families of surfaces, edges of regression, and lines of curvature.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 5948
njwildberger

In this video we introduce Brahmagupta's celebrated formula for the area of a cyclic quadrilateral in terms of the four sides. This is an obvious extension of Heron's formula. We are interested in finding a rational variant of it, that will be independent of a prior theory of `real numbers', `square roots' and `lengths'.
For motivation we look at the situation of four affine 1-points. Is there an analog of the Triple Quad Formula? Yes there is: it is the more mysterious and complicated Quadruple Quad Formula. To get it, we will examine some important manipulations for a pair of quadratic equations which are of independent interest.
This lecture has some more serious algebra in it: a great place to practice your manipulation and organizational skills.
If you are teaching college mathematics, please consider doing yourself and your students a favour: teach them some of the material of this lecture carefully and explicitly!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 5574
njwildberger

GeoGebra is a dynamic geometry package, available for free, which allows us to easily make planar geometric constructions which are dynamic (move-able), and investigate associated algebraic formulas and relations. This short lecture gives a brief introduction, since we will be using this software for visualization in this course. We illustrate the program by constructing the nine-point circle of a triangle.
It is highly recommended that you download the software (free) and play around with it if you do not already have it.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 20885
njwildberger

How do we organize metrical geometry? This video gives a broad outline of the subject---using a close parallel with how you might make a pizza. Just as there are quite a lot of different pizzas out there, so there are a lot of mathematical geometries out there, and we can try to be systematic in organizing them!
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 2926
njwildberger

We describe the important classification of compact, oriented 2-manifolds, and the relation with the topological invariant called the Euler characteristic. The idea is to work combinatorially, by decomposing a 2-manifold into polygon pieces which are glued, or identified, along common edges, and then performing cut and paste operations to try to get such a configuration into a normal form.
This was worked out by Dehn and Heegaard around 1910, and the resulting classification is one of the most important foundational results in differentential geometry and algebraic topology. In fact they worked in a more general context which also includes non-orientable surfaces, such as the projective plane or Klein bottle, but for differential geometry it suffices to work with orientable surfaces, which have a consistent notion of positive orientation on small circles around the surface.
Remarkably, the resulting classification shows that the Euler invariant completely determines the compact oriented surface.
NOTE: Due to a mistake in numbering, there is no DiffGeom27 video! So please skip ahead to DiffGeom28.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 9190
njwildberger

In this video we further develop and extend Lagrange's algebraic approach to the differential calculus. We show how to associate to a polynomial function y=p(x) at a point x=r not just a tangent line, but also a tangent conic, a tangent cubic and so on. Only elementary high school manipulations are needed--no limits or real numbers-- and we efficiently obtain a hierarchy of approximations to a polynomial at a given point.
Instead of derivatives, closely related quantities called sub-derivatives grab the spotlight. They are often simpler and more general quantities!
The quadratic approximation, given by the tangent conic, will be crucially important for us in our development of differential geometry: the key point is that the subject is largely what we get when we look at curves and surfaces quadratically!
Tangent conics (and higher approximations) of polynomial curves is a potentially rich theory that deserves a lot more attention. We highlight a beautiful observation of E. Ghys: that for a cubic polynomial, the various tangent conics are disjoint (this is in my opinion the loveliest theorem in calculus). Should not all undergraduates be exposed to such natural geometric applications in their calculus courses??
The power of this point of view is shown clearly by the ease with which we can extend it to the multivariable situation. A function of two variables z=p(x,y) defines a surface, which may be studied at a point [x,y]=[r,s] in an analogous way, yielding at each point a tangent plane, a tangent quadric, a tangent cubic surface etc.
We explicitly look at the surface associated to the Folium of Descartes, namely z=x^3+y^3+3xy and try to visualize it. This is a powerful but elementary alternative to the usual way of thinking about functions!
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 15713
njwildberger

Projective geometry is a fundamental subject in mathematics, which remarkably is little studied by undergraduates these days. But this situation is about to change---there are just too many places where a projective point of view illuminates mathematics. We will see that differential geometry is no exception.
In this video we show how to view a general conic in a projective way, yielding an important correspondence with 3x3 symmetric projective matrices. This motivates the introduction of projective linear algebra: where the basic objects are invariant under scaling.
We distinguish between projective points as row vectors, projective lines as column vectors, and incidence in terms of the usual matrix product between them. This brings out the all important duality in projective geometry, usually missing from most linear algebra treatments, where the affine view obscures the symmetry between points and lines. We explain Pappus' theorem from this view, and show its dual result.
We then have a look at some advantages in representing points and lines in the plane with projective coordinates--first of all we can use the third coordinate to absorb denominators, meaning that fraction arithmetic can be replaced by integer arithmetic. Secondly the main computations of finding meets and joins both reduce to a single computation: finding a cross product of two vectors.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 12407
njwildberger

Following from the last lecture on the Frenet Serret equations, we here look in detail at an important illustrative example--that of a helix. The Fundamental theorem of curves is stated--that the curvature and torsion essentially determine a 3D curve up to congruence.
We introduce the osculating, normal and rectifying planes, and try to explain the physical meaning of torsion.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 11194
njwildberger

This is the first of three videos that discuss the mathematical lives and works of three influential French differential geometers. We begin with J. Meusnier, who was a soldier, engineer and mathematician. He investigated lines of curvature and discovered a famous result that shows how to compute the sectional curvature of a surface cut by a non-normal plane, which we restate using the language of Rational Trigonometry--of course!
We discuss also elliptic, hyperbolic and parabolic points and umbilic points.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 5788
njwildberger

In this video we use the relation between the circumquadrance of a circle, and the quadrances and quadrea of a triangle circumscribed in that circle to derive a classical formula for the curvature of the standard parabola y=x^2.
The method is quite general, and applies to a wide variety of algebraic curves. This gives a completely algebraic alternative to the definition and calculation of curvature in the planar situation, at least for such kinds of curves. Note that what we get is actually the quadratic curvature-- the square of the usual curvature.
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of each lecture. Great for review, study and summary.
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 4992
njwildberger

The Frenet Serret equations describe what is happening to a unit speed space curve, twisting and rotating around in three dimensional space. This is done with the language of vector valued derivatives.
The idea is to attach to each point of the curve, a triple of unit vectors, called traditionally T, N and B, the tangent, normal and bi-normal unit vectors. These form at every point a mutually perpendicular frame of basis vectors, much like the i,j and k standard unit basis vectors along the x,y and z axes.
The vector T is in the direction of the curve (T standards for tangent), while N is in the direction of the acceleration, which for a unit speed curve must be perpendicular to the tangent. The third vector B can be defined as the cross product of T and N.
The Frenet Serret equations describe what happens as we move along the curve with unit speed s, namely what are the derivatives of T,N and B with respect to s. The curvature k(s) comes into play, as does a new quantity called the torsion, usually denoted tau(s).
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 24595
njwildberger

In this tutorial we explore the surface z=x^3+y^3+3xy using GeoGebra. The aim is to develop our skills using this dynamic geometry package, at the same time trying to use a two dimensional representation to understand a surface in three dimensions, with its tangent planes and tangent quadrics.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 9364
njwildberger

In this video we extend the discussion of curvature from parabolas to more general conics, and hence to more general algebraic curves. The advantage of basing things on the parabola is that we get nice connections between curvature and the foci, and that once we move to studying surfaces in three dimensional space, the normal paraboloids will play an exactly analogous role as do the normal parabolas here.
This lecture does have quite a lot of formulas in it, because we are interested in laying out technology that will easily allow us to move from one view to another. At some point, we begin to appreciate the power of formulas in pinning down this subject, but it does help to develop some sympathy with algebraic relations!
A point of divergence from the usual formulation: we emphasize the usefulness in thinking about the square of the usual curvature. This quadratic curvature can be formulated without any appeal to square roots.
Correction: That equation at 18:58 should have the factor (mx-ly)^2, not (lx-my)^2 (it can be properly remembered by being a cross product term). Following on, the next equation should also have the same correction. Thanks Thomas Fuhrmann!
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 5999
njwildberger

We review the simple algebraic set-up for projective points and projective lines, expressed as row and column 3-vectors. Transformations via projective geometry are introduced, along with an introduction to quadratic forms, associated symmetrix bilinear forms, and associated projective 3x3 matrices. An important example is the Lorentz/Einstein/Minkowski geometry. Then notions of perpendicularity are closely related to the pole-polar duality between points and lines associated to the unit circle in the plane. This goes back to Apollonius, and is closely related to the developments in the UnivHypGeom series.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 8489
njwildberger

We now extend the discussion of curvature to a general parabola, not necessarily one of the form y=x^2. This involves first of all understanding that a parabola is defined projectively as a conic which is tangent to the line at infinity.
We find the general projective 3x3 matrix for such a parabola with its vertex at [0,0]. We then derive the focus directrix definition of such a parabola and study its local behaviour at the origin, in particular connecting it to a function representation of the form y=alpha x+beta x^2 +...
This lecture has some simple but no doubt unfamiliar formulas which pin down the curvature of a general parabola: the main ingredient to understanding curvature of a general curve, following the philosophy of Zvi Har'El in basing curvature on the approximation by parabolas.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 8324
njwildberger

In this lecture we introduce a general approach to metrical structure, via a symmetric bilinear form in either an affine or projective setting, and then begin studying the crucially important concept of curvature, first of all for a parabola of the form y=ax^2.
Metrical structures are usually associated with quadratic forms, or the corresponding symmetric bilinear forms, also known as dot products or inner products, and are represented mathematically by symmetric matrices. These are also naturally linked to conics. The essential equivalence between these seemingly different objects is an important understanding.
For applications to physics, in particular relativistic geometries, it is useful to have a flexible attitude here about metrical structure--the Euclidean one is not the only one to consider. This is rather a departure from classical courses in differential geometry, but the student will gain much by adopting such a more flexible and general point of view. We discuss two-dimensional relativistic geometries, and explain why circles in these contexts are what we would usually call rectangular hyperbolas.
For those particularly interested in these metrical connections between 2D and 3D relativistic geometries, I suggest the video in my MathSeminars series called
Hyperbolic Geometry is Projective Relativistic Geometry.
In the last part of the lecture we begin studying the parabola y=ax^2 at the point [0,0], with a view of determining the relations between the focus of the parabola, the center and radius of curvature, and the curvature itself. We are following here the approach of Zvi Har'El, explained in a paper called: Curvature of Curves and Surfaces: a Parabolic Approach (1995).
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 9715
njwildberger

Here we introduce a somewhat novel approach to the curvature of a surface. This follows the discussion in DiffGeom23, where we looked at a paraboloid as a function of the form 2z=ax^2+2bxy+cy^2.
In this lecture we generalize the discussion to the important case of a paraboloid, which we define projectively as a quadric which is tangent to the plane at infinity, and whose vertex has been translated to go through the origin. We show how to represent such a paraboloid algebraically using a projective 4x4 matrix., and write down the corresponding Cartesian equations.
Such a quadric will have two curvatures at the origin: both are defined in terms of the coefficients of the characteristic polynomial, suitably renormalized so they are independent of scaling. The first curvature K_1 is up to a factor of 4 the square of the usual mean curvature, while the second curvature K_2 is the usual Gaussian curvature.
So we see that curvature is really coming from linear algebra and the characteristic polynomial of a symmetric matrix. This is an important insight that lends itself pleasantly to higher dimensional generalizations.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/.... I also have a blog at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things, and you can check out my webpages at http://web.maths.unsw.edu.au/~norman/. Of course if you want to support all these bold initiatives, become a Patron of this Channel at https://www.patreon.com/njwildberger?... .

Views: 6231
njwildberger

TabletClass Math http://www.tabletclass.com learn the basics of calculus quickly. This video is designed to introduce calculus concepts for all math students and make the topic easy to understand.

Views: 1803272
TabletClass

We extend our discussion of elementary metrical projective geometry in one dimension to incorporate Einstein's special theory of relativity. This remarkable new understanding of Einstein transformed much of 20th century physics, but its effect on pure mathematics has been surprisingly modest.
In this video we see that even in the one dimensional setting, relativistic geometry, also associated with the names of Lorentz and Minkowski, is a rich alternative framework. This is a useful introduction for physicists to a geometry that figures very prominently in modern physics. One of the key new features is the existence of null points, which are analogous to the situation that we met over F_5 in our previous lecture on the Euclidean setting, and which correspond to the space-time trajectories of photons.
This video is an introduction to a fascinating new world of geometry, much bigger than the one we usually thing about. A key feature is that we are required to think more algebraically, and let go of that real number dreaming that currently distorts our geometrical understanding.
A screenshot PDF which includes MathFoundations122 to 149 can be found at my WildEgg website here: http://www.wildegg.com/store/p103/product-math-foundations-B2-screenshot-pdfs

Views: 1883
njwildberger

Programa de Mestrado 2014: Geometria Diferencial
Professor: Luis Adrián Florit
Aula 04 - 20-03-2014
Página: http://www.impa.br/opencms/pt/ensino/mestrado/disciplinas_mestrado/disciplinas_catalogo_2011/disciplinas_mestrado_geometria_diferencial
Download dos vídeos: http://video.impa.br/index.php?page=mestrado-geometria-diferencial
Pré-requisito: Análise em variedades (variedades e tensores), Teorema fundamental das equações diferenciais ordinárias (podendo ser cursada em paralelo).
Curvas e superfícies no R3. Primeira forma fundamental, área. Aplicação normal de Gauss; direções principais, curvatura Gaussiana e curvatura média, linhas de curvatura. Exemplos clássicos de superfícies. Geometria intrínseca: métrica e derivada covariante, o teorema Egregium; curvatura geodésica; equações das geodésicas, cálculo de geodésicas em superfícies. O teorema de Gauss-Bonnet. Rigidez da esfera em R3 e Teorema de Alexandrov. Fibrados, fibrados vetoriais e fibrados principais.
Conexões em fibrados. Mapas de fibrados, pull-back. Subfibrados e Teorema de Frobenius. Teoria de Chern-Weil. Teorema de Gauss-Bonnet em dimensões superiores. Outros tópicos.
Referências:
CARMO, M. – Differential Geometry of Curves and Surfaces. Englewood Cliffs, Prentice-Hall, 1976.
DUPONT, J. – Fiber Bundles in Gauge Theory, Arhus Universitet, 2003.
DUPONT, J. – Curvature and Characteristic Classes, Springer, 1978.
KOBAYASHI, S. e NOMIZU, K. – Foundations of Differential Geometry, Wiley - Interscience, 1996.
MADSEN, H. – From Calculus to Cohomology: De Rham Cohomology and Characteristic Classes, Cambridge University Press, 1997.
MONTIEL, S. e ROS, A. - Curves and Surfaces, Graduate Studies in Mathematics, vol. 69, AMS, 2005.
POOR, W. A. – Differential Geometric Structures, Dover Publicaitons; Dover Ed edition, 2007.
SPIVAK, M. – A Comprehensive Introduction to Differential Geometry, vol.3, Berkeley, Publish or Perish, 1979.
IMPA - Instituto Nacional de Matemática Pura e Aplicada ©
http://www.impa.br | http://video.impa.br

Views: 362
Instituto de Matemática Pura e Aplicada

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Selling in special circumstances. shares you bought at different times and prices in one company shares through an investment club shares after a company merger or takeover employee share scheme shares. Jointly owned shares and investments. If you sell shares or investments that you own jointly with other people, work out the gain for the portion that you own, instead of the whole value. There are different rules for investment clubs. What to do next. Deduct costs. Apply reliefs.