## X-ray wave-fields in free space

*David M. Paganin*

### in Coherent X-Ray Optics

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198567288
- eISBN:
- 9780191717963
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567288.003.0001
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter considers the theory of classical X-ray wave-fields in free space, taking the Maxwell equations as a starting point. Vacuum wave equations (d’Alembert equations) are developed for X-rays ... More

## NUMERICAL METHODS

*Miguel Alcubierre*

### in Introduction to 3+1 Numerical Relativity

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199205677
- eISBN:
- 9780191709371
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199205677.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

There are many ways to solve partial differential equations numerically. The most popular methods are: finite differencing, finite elements, and spectral methods. This chapter describes the main ... More

## Global Existence Theorems: Asymptotically Euclidean Data

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0015
- Subject:
- Mathematics, Applied Mathematics

This chapter shows how the Penrose transform can be used to prove global existence of solutions of various semilinear field equations. It outlines the foundation points of Friedrich's conformal ... More

## Green's functions and retarded potentials

*Mauro Fabrizio and Morro Angelo*

### in Electromagnetism of Continuous Media: Mathematical Modelling and Applications

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198527008
- eISBN:
- 9780191713316
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527008.003.0002
- Subject:
- Mathematics, Applied Mathematics

This chapter gives a concise and self-contained presentation of the use of Green’s functions to solve linear problems with a given point source. Specific functions are determined for the harmonic ... More

## Other Hyperbolic-Elliptic Well-Posed Systems

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0008
- Subject:
- Mathematics, Applied Mathematics

This chapter presents well-posed hyperbolic or hyperbolic-elliptic systems that lead to the same local existence andgeometric uniqueness theorems as the wave gauge choice. However, these different ... More

## Central Equation of Thermoelasticity with Finite Wave Speeds

*Józef Ignaczak and Martin Ostoja‐Starzewski*

### in Thermoelasticity with Finite Wave Speeds

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199541645
- eISBN:
- 9780191716164
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199541645.003.0006
- Subject:
- Mathematics, Applied Mathematics, Mathematical Physics

This chapter focuses on a central equation of thermoelasticity with finite wave speeds, i.e. a partial differential equation which, most remarkably, has a similar form for both theories. Given next ... More

## Review of quantum mechanics

*Alan Corney*

### in Atomic and Laser Spectroscopy

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199211456
- eISBN:
- 9780191705915
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199211456.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter derives the quantum mechanical wave functions which describe the energy levels of simple atoms. Schrödinger's equation is introduced as well and the angular part of the equation is ... More

## Harmonic morphisms between semi-Riemannian manifolds

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0014
- Subject:
- Mathematics, Pure Mathematics

This final chapter discusses how the main definitions and results need to be modified in the semi-Riemannian case. In this context, harmonic maps include the strings of mathematical physics. Weakly ... More

## Relativistic wave equations

*Victor Galitski, Boris Karnakov, Vladimir Kogan, and Victor Galitski, Jr.*

### in Exploring Quantum Mechanics: A Collection of 700+ Solved Problems for Students, Lecturers, and Researchers

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199232710
- eISBN:
- 9780191774973
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232710.003.0015
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter deals with problems related to the Klein-Gordon equation and the Dirac equation.

## Seismic Waves and Sources

*Heiner Igel*

### in Computational Seismology: A Practical Introduction

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198717409
- eISBN:
- 9780191835070
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198717409.003.0002
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

The specific partial differential equation with which the numerical methods are introduced in the remainder of the volume is presented in detail. Fundamental consequences of the elastic wave equation ... More

## Electromagnetic fields and waves 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/9780198821915.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

In previous chapters four experimental laws of electromagnetism were encountered: Gauss’s law in electrostatics, Gauss’s law in magnetism, Faraday’s law and Ampere’s law. Now, in this chapter, these ... More

## Photons

*P. K. Basu*

### in Theory of Optical Processes in Semiconductors: Bulk and Microstructures

- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198526209
- eISBN:
- 9780191706790
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526209.003.0003
- Subject:
- Physics, Atomic, Laser, and Optical Physics

The classical or semiclassical theory presented in Chapter 2 gives a satisfactory account of light-matter interactions in some cases. However, a detailed understanding of different processes requires ... More

## Transformations, matrices and operators

*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/9780199641390.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Linear algebra of matrices is discussed in the context of rotations in space. Determinants are used to find solutions of simultaneous linear equations, to invert matrices and to find matrix ... More

## Relativistic quantum mechanics

*Franco Strocchi*

### in An Introduction to Non-Perturbative Foundations of Quantum Field Theory

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199671571
- eISBN:
- 9780191751073
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199671571.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

The chapter focuses on the problems at the origin of QFT: i) the conflict between locality and energy positivity, which affects relativistic wave equations in general, and ii) the need of describing ... More

## The Energy Critical Wave Equation in 3D

*Carlos Kenig*

*Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)*

### in Advances in Analysis: The Legacy of Elias M. Stein

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159416
- eISBN:
- 9781400848935
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159416.003.0009
- Subject:
- Mathematics, Numerical Analysis

This chapter discusses the energy critical nonlinear wave equation in 3 space dimensions. It mainly focuses on soliton resolution for radial solutions of nonlinear waves. For a long time there has ... More

## Some applications of Maxwell’s equations in matter

*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/9780198821915.003.0010
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter comprises questions of a miscellaneous nature. They mostly have little in common except that all processes are time-dependent and occur within matter. The first few questions introduce ... More

## Systems with Many Degrees of Freedom

*Mohammad Samiullah*

### in A First Course in Vibrations and Waves

- Published in print:
- 2015
- Published Online:
- October 2015
- ISBN:
- 9780198729785
- eISBN:
- 9780191804113
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198729785.003.0004
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter generalizes the normal mode analysis of the previous chapter to systems with many degrees of freedom, including continuous systems with infinitely many degrees of freedom. The classical ... More

## Theoretical Underpinnings: Stellar Oscillations

*Sarbani Basu and William J. Chaplin*

### in Asteroseismic Data Analysis: Foundations and Techniques

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691162928
- eISBN:
- 9781400888207
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691162928.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter shows how the oscillations of a star are related to its structure. It also describes some of the important properties of stellar oscillations that allow us to use seismic data to ... More

## Relativistic Wave Equations

*Laurent Baulieu, John Iliopoulos, and Roland Sénéor*

### in From Classical to Quantum Fields

- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788393
- eISBN:
- 9780191830310
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788393.003.0006
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

Relativistically covariant wave equations for scalar, spinor, and vector fields. Plane wave solutions and Green’s functions. The Klein–Gordon equation. The Dirac equation and the Clifford algebra of ... More

## The Classical Connection

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0015
- Subject:
- Mathematics, Applied Mathematics

This chapter discusses the relationship between the elegance of the classical Lagrangian and Hamiltonian formulations of mechanics and optics. In physics, action is a mathematical functional which ... More