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VECTOR FIELDS AND FLOW EQUATIONS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on ... More


Artificial likelihoods and estimating functions

Christopher G. Small and Jinfang Wang

in Numerical Methods for Nonlinear Estimating Equations

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198506881
eISBN:
9780191709258
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506881.003.0006
Subject:
Mathematics, Probability / Statistics

This chapter discusses the problem of root selection using artificial objective functions associated with an estimating function; the artificial likelihoods discussed in this chapter share many ... More


Liquid Crystals and Harmonic Maps in Polyhedral Domains

Apala Majumdar, Jonathan Robbins, and Maxim Zyskin

in Analysis and Stochastics of Growth Processes and Interface Models

Published in print:
2008
Published Online:
September 2008
ISBN:
9780199239252
eISBN:
9780191716911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199239252.003.0014
Subject:
Mathematics, Probability / Statistics, Analysis

This chapter is concerned with harmonic maps from a polyhedron to the unit two-sphere, which provide a model of nematic liquid crystals in bistable displays. This chapter looks at the Dirichlet ... More


Vectors and fields in space

Chun Wa Wong

in Introduction to Mathematical Physics: Methods & Concepts

Published in print:
2013
Published Online:
May 2013
ISBN:
9780199641390
eISBN:
9780191747786
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199641390.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Vector calculus is discussed, including vector algebra, vector differentiation and integrations of scalar and vector fields, integral theorems of Green and Stokes, and the Helmholtz decomposition of ... More


Differential Geometry: Bundles, Connections, Metrics and Curvature

Clifford Henry Taubes

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.001.0001
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the ... More


Differential Geometry

Peter Mann

in Lagrangian and Hamiltonian Dynamics

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198822370
eISBN:
9780191861253
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198822370.003.0038
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter is key to the understanding of classical mechanics as a geometrical theory. It builds upon earlier chapters on calculus and linear algebra and frames theoretical physics in a new and ... More


Fundamental Vector Fields

Loring W. Tu

in Introductory Lectures on Equivariant Cohomology: (AMS-204)

Published in print:
2020
Published Online:
January 2021
ISBN:
9780691191751
eISBN:
9780691197487
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691191751.003.0011
Subject:
Mathematics, Educational Mathematics

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, ... More


Maps and vector bundles

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0005
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This discussion of maps and vector bundles covers the pull-back construction; pull-backs and Grassmannians; pull-back of differential forms and push-forward of vector fields; invariant forms and ... More


Cellular Homology

Graham Ellis

in An Invitation to Computational Homotopy

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198832973
eISBN:
9780191871375
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198832973.003.0002
Subject:
Mathematics, Computational Mathematics / Optimization, Geometry / Topology

This chapter introduces more basic concepts of algebraic topology and describes datatypes and algorithms for implementing them on a computer. The basic concepts include: chain complex, chain mapping, ... More


Fields with many components and massive electromagnetism

Tom Lancaster and Stephen J. Blundell

in Quantum Field Theory for the Gifted Amateur

Published in print:
2014
Published Online:
June 2014
ISBN:
9780199699322
eISBN:
9780191779435
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199699322.003.0014
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

The ideas of the previous chapters are extended to multicomponent fields in and we work this through for the case of massive electromagnetism, thereby studying a spin-1 vector field.


Fields

Gerhard Oertel

in Stress and Deformation: A Handbook on Tensors in Geology

Published in print:
1996
Published Online:
November 2020
ISBN:
9780195095036
eISBN:
9780197560792
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195095036.003.0006
Subject:
Earth Sciences and Geography, Geology and the Lithosphere

A scalar determined at every point in a given domain, analytically or otherwise, constitutes a scalar field. Vectors similarly determined constitute a vector field. The defining analytical ... More


Holonomic Poisson Manifolds and Deformations of Elliptic Algebras

Brent Pym and Travis Schedler

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0028
Subject:
Mathematics, Geometry / Topology

This chapter introduces a natural non-degeneracy condition for Poisson structures, called holonomicity, which is closely related to the notion of a log symplectic form. Holonomic Poisson manifolds ... More


Spacetime symmetries

Michael Kachelriess

in Quantum Fields: From the Hubble to the Planck Scale

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198802877
eISBN:
9780191841330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802877.003.0006
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the ... More


A SINGLE ODE OF ARBITRARY ORDER

Francesco Calogero

in Isochronous Systems

Published in print:
2008
Published Online:
May 2008
ISBN:
9780199535286
eISBN:
9780191715853
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199535286.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In Chapter 3 several ω-modified ordinary differential equations (ODEs) are obtained and their isochronous character is discussed. Firstly a large class of such equations is identified and several of ... More


Scalar and Vector Fields

B. K. Ridley

in Hybrid Phonons in Nanostructures

Published in print:
2017
Published Online:
April 2017
ISBN:
9780198788362
eISBN:
9780191830280
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198788362.003.0007
Subject:
Physics, Condensed Matter Physics / Materials

Those quantum structures describable in terms of Cartesian coordinates (e.g. the quantum well) can be treated with the minimum of formality. Turning to quantum structures describable in terms of ... More


Covariant derivatives, connections and curvature

Clifford Henry Taubes

in Differential Geometry: Bundles, Connections, Metrics and Curvature

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199605880
eISBN:
9780191774911
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199605880.003.0012
Subject:
Mathematics, Geometry / Topology, Mathematical Physics

This chapter examines the notion of the curvature of a covariant derivative or connection. It begins by describing two notions involving differentiation of differential forms and vector fields that ... More


Geometric theory of ordinary differential equations

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0006
Subject:
Physics, Atomic, Laser, and Optical Physics

All physics is based on differential equations. Ordinary differential equations appear in mechanics. They can be thought of as the integral curves of a vector field on a manifold, the Phase Space. ... More


Static magnetic fields in vacuum

J. Pierrus

in Solved Problems in Classical Electromagnetism: Analytical and Numerical Solutions with Comments

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198821915
eISBN:
9780191861055
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821915.003.0004
Subject:
Physics, Condensed Matter Physics / Materials

Wherever possible, an attempt has been made to structure this chapter along similar lines to Chapter 2 (its electrostatic counterpart). Maxwell’s magnetostatic equations are derived from Ampere’s ... More


Basic Technical Outline

Philip Isett

in Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Published in print:
2017
Published Online:
October 2017
ISBN:
9780691174822
eISBN:
9781400885428
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691174822.003.0003
Subject:
Mathematics, Computational Mathematics / Optimization

This chapter provides a more technical outline of the construction, starting with a solution to the Euler-Reynolds system and a correction v₁ = v + V, p₁ = p + P. The correction V is a divergence ... More


Vector Fields

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0001
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

The velocity of a fluid at each point of space-time is a vector field (or flow). It is best to think of it in terms of the effect of fluid flow on some scalar field. A vector field is thus a first ... More


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