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 Calculus as geometry

Frank Arntzenius

in Space, Time, and Stuff

Published in print:
2012
Published Online:
May 2012
ISBN:
9780199696604
eISBN:
9780191738333
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199696604.003.0008
Subject:
Philosophy, Metaphysics/Epistemology, Philosophy of Science

This chapter argues that calculus, the theory of differentiation and integration that permeates modern science, can be formulated as a purely geometric theory, which does not entail the existence of ... More


Weyl's “Purely Infinitesimal” Constitution of Field Physics

Thomas Ryckman

in The Reign of Relativity: Philosophy in Physics 1915-1925

Published in print:
2005
Published Online:
April 2005
ISBN:
9780195177176
eISBN:
9780199835324
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0195177177.003.0006
Subject:
Philosophy, Philosophy of Science

It is shown how Weyl intertwined phenomenological analysis and mathematical construction in building the foundation of his “pure infinitesimal geometry” underlying gravitation and electromagnetism. ... More


Anomalies in Quantum Field Theory

Reinhold A. Bertlmann

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507628
eISBN:
9780191706400
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507628.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The anomaly, which forms the central part of this book, is the failure of classical symmetry to survive the process of quantization and regularization. The study of anomalies is the key to a deeper ... More


Topological Invariants and Differential Geometry

Paula Tretkoff

in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691144771
eISBN:
9781400881253
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691144771.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter deals with topological invariants and differential geometry. It first considers a topological space X for which singular homology and cohomology are defined, along with the Euler number ... More


Lorentz Geometry

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/9780199230723.003.0001
Subject:
Mathematics, Applied Mathematics

This chapter presents a survey of the basic definitions of Riemannian and Lorentzian differential geometry used in this book. The first nine sections use the simplest formulations, in local ... 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


Algebraic Surfaces and the Miyaoka-Yau Inequality

Paula Tretkoff

in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

Published in print:
2016
Published Online:
October 2017
ISBN:
9780691144771
eISBN:
9781400881253
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691144771.003.0005
Subject:
Mathematics, Geometry / Topology

This chapter discusses complex algebraic surfaces, with particular emphasis on the Miyaoka-Yau inequality and the rough classification of surfaces. Every complex algebraic surface is birationally ... More


Topology, Manifolds, Dynamical Systems, Measure, and Bundles

Xin Wei Sha

in Poiesis and Enchantment in Topological Matter

Published in print:
2014
Published Online:
September 2014
ISBN:
9780262019514
eISBN:
9780262318914
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262019514.003.0006
Subject:
Philosophy, Aesthetics

This is the core chapter, introducing concepts that articulate continua, continuous substance, and continuous process. These concepts find precise and deep forms in point set topology, topological ... More


Frontiers in Complex Dynamics: In Celebration of John Milnor's 80th Birthday

Araceli Bonifant, Misha Lyubich, and Scott Sutherland

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159294
eISBN:
9781400851317
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159294.001.0001
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, ... More


Electromagnetism and differential geometry

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0040
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + ... More


Differential geometry

Nathalie Deruelle and Jean-Philippe Uzan

in Relativity in Modern Physics

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198786399
eISBN:
9780191828669
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198786399.003.0004
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant ... More


The Decomposition of Global Conformal Invariants (AM-182)

Spyros Alexakis

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153476
eISBN:
9781400842728
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153476.001.0001
Subject:
Mathematics, Geometry / Topology

This book addresses a basic question in differential geometry that was first considered by physicists Stanley Deser and Adam Schwimmer in 1993 in their study of conformal anomalies. The question ... More


A Brief Review of General Relativity

Luciano Rezzolla and Olindo Zanotti

in Relativistic Hydrodynamics

Published in print:
2013
Published Online:
January 2014
ISBN:
9780198528906
eISBN:
9780191746505
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528906.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter provides a brief summary of the basic aspects of Einstein’s theory of general relativity. Although not aimed to be comprehensive, a step-by-step presentation of the fundamental ... More


Introduction

Spyros Alexakis

in The Decomposition of Global Conformal Invariants (AM-182)

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691153476
eISBN:
9781400842728
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691153476.003.0001
Subject:
Mathematics, Geometry / Topology

This introductory chapter first sets out the book's purpose, which is to provide a rigorous proof of the Deser–Schwimmer conjecture. This work is a continuation of the previous two papers of the ... More


Angle‐Action Variables

Oliver Davis Johns

in Analytical Mechanics for Relativity and Quantum Mechanics

Published in print:
2011
Published Online:
December 2013
ISBN:
9780191001628
eISBN:
9780191775161
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780191001628.003.0022
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter introduces the angle-action method and provides several examples of its use, including its essential role in the ‘old’ quantum theory, a precursor to the Schroedinger wave equation that ... More


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