Introducción al cálculo tensorial. by Machado, Luis M. Series: IRAM Edition statement:1a. ed. Physical details: p.: il.; 22 cm. Subject(s). ×. Cálculo tensorial — Problemas y ejercicios. More like this Add tags for “Teoría y problemas de análisis vectorial y una introducción al analisis tensorial”. Be the. Teoría y problemas de análisis vectorial y una introducción al análisis tensorial. [ Murray R Spiegel] vectorial — Problemas, ejercicios, etc. Cálculo de tensores.
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Anonymous survey to sound out the level and interest of the students this one. A tensoriall dictionary between geometry and mechanics through the double pendulum example.
Geometría diferencial (máster) 2014/2015
The problem about “avoidance of crossing” is proposed. Some comments about the kinetic energy. Euclidean Tensors, Einstein summation convention. Tensors on a mainfold. Transformation law for tensors. The cotangent bundle as the phase space. Equivalence with the Newton’s second law for free particles. Conservation of the energy.
The simple pendulum using vector mechanics and Lagrangian mechanics. Examples of applications to the geometry of surfaces: The vector mechanics formulation of the planetary motion. Integral curves on manifolds.
One-parameter groups of diffeomorphisms.
Introducción al cálculo tensorial
Lagrangians invariant by transformations. Conservation of the linear momentum for interactions depending on the distance. Conservation of the angular momentum for interactions with spherical symmetry.
Examples of flow and group law. Lie derivative and its geometrical interpretation. Equality between Lie bracket and Introdccion derivative naive approach.
Distributions and integral submanifolds. Local expression of a differential forms as a combination of determinants. The invariance of the exterior derivative by coordinate changes.
Properties of the exterior derivative. Statement of the Stokes theorem. Maxwell equations and the Stokes theorem. Definition of the de Rham cohomology. The cohonology groups of the spheres. The cohomology groups of the torus. Proof of that cohomology is homotopy invariant. Underlying ideas in Riemannian geometry. Mechanical interpretation of the geodesics.
Newtonian approximation for weak fields. Motivation for the wl of connection. Parallel transport “absolute parallelism”.
Geodesics, covariant derivative and the computation of the Christoffel symbols. Quick definitions of the Riemann tensor, the Ricci tensor and the scalar curvature.
Gauss definition of the curvature. Motivation and meaning of the Riemann tensor. Geometric meaning of Ricci tensor. Exponential map and Gauss Lemma. Some results relating curvature and topology. This web page is not fully translated but following this icon, in each part there are some comments in likely broken English.
The topics covered in each lecture are listed here. Here you can find a brief approximate dictionary Mechanics-Geometry.
Campos de vectores y sus flujos. Corchete de Lie y el teorema de Frobenius.
El teorema de Stokes en variedades. El horario de las clases es de In each part of the course it is intended to use mechanics and other topics on Physics as motivation and to provide examples and applications. The idea is to mention analytical mechanics in the first part, electrodynamics in the second part and general relativity in the third part.
No prior knowledge on Physics is assumed. The lectures are scheduled on Tuesday and Thrusday Son parte del curso. Ejercicios The following sheets are home assignments that contribute to the grading. The grading is as follows: The topic can be chosen freely up to reasonable restrictions. The limitations regarding the format will be announced. More details about grading: E is max Final exam, Composition.
Note that neither the exam nor the composition is mandatory. The final examination is scheduled for January 20th. The composition is a short essay about any topic related to differential geometry. I recommend to ask me for confirmation after choosing the topic. The advisable length is pages double-spaced, 11pt or 12pt. The deadline is January 15th Some other sources of participation can be also counted. The contribution of other activities e.
MathworldPlanetMathWikipedia Mathematics portal. MacTutor History of Mathematics archive. Do wrinkles make me stronger?
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Curso de “Applied Topology” de Robert Ghirst. Content of the lectures Part 1 Date. Plan of the course. Definition of manifold, tangent vector and tangent map. Calculations with the tangent map referred to the double pendulum. Kinetic energy in generalized coordinates for the double pendulum. Interpretation of Lorentz transformations. Conservation of the energy and other observations with regards to the simple pendulum.
Introduccoon and global flows of vector fields. Motivation for Frobenius theorem. Proof of the equality between Lie bracket and Lie derivative. Tensirial forms as fields of alternating forms. The wedge product as an operation between differential forms. Definition of manifold with boundary. Historical comments about the fundamental group, homology and cohomology. Cohomology ans differential topology: Scheme of the proof of the Mayer-Vietoris exact sequence.
The hairy ball theorem and other topological results. Motivation and notation for the concept of metric. Computation of a geodesic for the Schwarzschild metric. The modern view of the curvature tensor. Proof of the minimizing property of geodesics using the Gauss lemma.