## 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

## 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

## 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

## The Lie Derivative and Interior Multiplication

*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/9780691191751.003.0010
- Subject:
- Mathematics, Educational Mathematics

This chapter reviews two operations on differential forms, the Lie derivative and interior multiplication. These are necessary to the definition of invariant forms, horizontal forms, and basic forms ... 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

## Field models

*Iosif L. Buchbinder and Ilya L. Shapiro*

### in Introduction to Quantum Field Theory with Applications to Quantum Gravity

- Published in print:
- 2021
- Published Online:
- April 2021
- ISBN:
- 9780198838319
- eISBN:
- 9780191874666
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198838319.003.0004
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter provides constructions of Lagrangians for various field models and discusses the basic properties of these models. Concrete examples of field models are constructed, including real and ... 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

## 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

## 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

## The Maurer–Cartan Form

*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/9780691191751.003.0015
- Subject:
- Mathematics, Educational Mathematics

This chapter illustrates the Maurer-Cartan form. On every Lie group G with Lie algebra g, there is a unique canonically defined left-invariant g-valued 1-form called the Maurer-Cartan form. The ... 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.

## 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

## 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

## 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

## 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

## 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