Differential equations | Fiber bundles | Differential topology
In differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to write differential equations on sections of a fiber bundle in an invariant form. Jets may also be seen as the coordinate free versions of Taylor expansions. Historically, jet bundles are attributed to Charles Ehresmann, and were an advance on the method (prolongation) of Élie Cartan, of dealing geometrically with higher derivatives, by imposing differential form conditions on newly introduced formal variables. Jet bundles are sometimes called sprays, although sprays usually refer more specifically to the associated vector field induced on the corresponding bundle (e.g., the geodesic spray on Finsler manifolds.) Since the early 1980s, jet bundles have appeared as a concise way to describe phenomena associated with the derivatives of maps, particularly those associated with the calculus of variations. Consequently, the jet bundle is now recognized as the correct domain for a geometrical covariant field theory and much work is done in general relativistic formulations of fields using this approach. (Wikipedia).
Introduction to Fiber Bundles part 1: Definitions
We give the definition of a fiber bundle with fiber F, trivializations and transition maps. This is a really basic stuff that we use a lot. Here are the topics this sets up: *Associated Bundles/Principal Bundles *Reductions of Structure Groups *Steenrod's Theorem *Torsor structure on arith
From playlist Fiber bundles
What else goes into building a jet engine? - Part 1
This is part one of a follow up video series on our recent turbojet engine test seen in our previous video - https://youtu.be/EP4Hf63zR6w Check out our jet engine playlist: https://youtube.com/playlist?list=PLzrI14lOlSqfbnnwLgubRqAyzCyOUuCD_ Find us on Patreon and our website: https://ww
From playlist Jet Engines
What is a Manifold? Lesson 12: Fiber Bundles - Formal Description
This is a long lesson, but it is not full of rigorous proofs, it is just a formal definition. Please let me know where the exposition is unclear. I din't quite get through the idea of the structure group of a fiber bundle fully, but I introduced it. The examples in the next lesson will h
From playlist What is a Manifold?
A group of people Jet-skiing in a line at Airlie Beach
A short snapshot of a group of people Jet-skiing in a line at Airlie Beach from a sailing boat on the sea.
From playlist Travel in Australia
The TRUTH about TENSORS, Part 9: Vector Bundles
In this video we define vector bundles in full abstraction, of which tangent bundles are a special case.
From playlist The TRUTH about TENSORS
Mod-01 Lec-04 Turbojet, Reheat Turbojet and Multi-spool Engines
Jet Aircraft Propulsion by Prof. Bhaskar Roy and Prof. A. M. Pradeep, Department of Aerospace Engineering, IIT Bombay. For more details on NPTEL visit http://nptel.iitm.ac.in
From playlist IIT Bombay: Aerospace - Jet Aircraft Propulsion (CosmoLearning Aerospace Engineering)
An animation I put together of a helicopter flying through a mountain range. I did not create the mountain background animation.
From playlist Motion Design Portfolio
What is the connection between vectors and equations of planes? Find out here! Free ebook https://bookboon.com/en/introduction-to-vectors-ebook (updated link) Test your understanding via a short quiz http://goo.gl/forms/ZTQ0pvOq1q
From playlist Introduction to Vectors
Introduction to h-principle by Mahuya Datta
DATE & TIME: 25 December 2017 to 04 January 2018 VENUE: Madhava Lecture Hall, ICTS, Bangalore Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex structure. The moduli space of these curves (
From playlist J-Holomorphic Curves and Gromov-Witten Invariants
D. Brotbek - On the hyperbolicity of general hypersurfaces
A smooth projective variety over the complex numbers is said to be (Brody) hyperbolic if it doesn’t contain any entire curve. Kobayashi conjectured in the 70’s that general hypersurfaces of sufficiently large degree in PN are hyperbolic. This conjecture was only recently proved by Siu. Th
From playlist Complex analytic and differential geometry - a conference in honor of Jean-Pierre Demailly - 6-9 juin 2017
Charles Fefferman : Whitney problems and real algebraic geometry
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Analysis and its Applications
S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (Part 3)
The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every holomorphic map from the complex plane to it
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
Henri Moscovici. Differentiable Characters and Hopf Cyclic Cohomology
Talk by Henri Moscovici in Global Noncommutative Geometry Seminar (Europe) http://www.noncommutativegeometry.nl/... on October 20, 2020.
From playlist Global Noncommutative Geometry Seminar (Europe)
S. Diverio - Kobayashi hyperbolicity of complex projective manifolds and foliations (Part 4)
The aim of this mini course is to highlight some links between the study of the Kobayashi hyperbolicity properties of complex projective manifolds and holomorphic foliations. A compact complex space is Kobayashi hyperbolic if and only if every holomorphic map from the complex plane to it
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
J. Demailly - Existence of logarithmic and orbifold jet differentials
Abstract - Given a projective algebraic orbifold, one can define associated logarithmic and orbifold jet bundles. These bundles describe the algebraic differential operators that act on germs of curves satisfying ad hoc ramification conditions. Holomorphic Morse inequalities can be used to
From playlist Ecole d'été 2019 - Foliations and algebraic geometry
Jean-Pierre Demailly: Improved bounds for the Kobayashi conjecture on generic hyperbolicity
Abstract: A famous conjecture of Kobayashi from the 1970s asserts that a generic algebraic hypersurface of sufficiently large degree d≥dn in the complex projective space of dimension n+1 is hyperbolic. Yum-Tong Siu introduced several fundamental ideas that led recently to a proof of the co
From playlist Algebraic and Complex Geometry
Matthew Harrison-Trainor 12/11/15 Part 2
Title: Differential-Algebraic Jet Spaces and Internality
From playlist Fall 2015
I explain how to build a simple, inexpensive turbojet engine. We cover the selection of the turbocharger, the terminology and design principles for the entire engine as well as the associated instruments, pumps, filters and controls. Find us on Patreon - https://www.patreon.com/techingred
From playlist Jet Engines