The Geometry of Physics: An Introduction

Theodore Frankel

doi:10.1017/CBO9781139061377

This book provides a working knowledge of those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms that are essential for a deeper understanding of both classical and modern physics and engineering. Included are discussions of analytical and fluid dynamics, electromagnetism (in flat and curved space), thermodynamics, the Dirac operator and spinors, and gauge fields, including Yang–Mills, the Aharonov–Bohm effect, Berry phase and instanton winding numbers, quarks and quark model for mesons. Before discussing abstract notions of differential geometry, geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space. The book is ideal for graduate and advanced undergraduate students of physics, engineering or mathematics as a course text or for self study. This third edition includes an overview of Cartan's exterior differential forms, which previews many of the geometric concepts developed in the text.

Review of previous edition:‘… highly readable and enjoyable … The book will make an excellent course text or self-study manual for this interesting subject.'

Source: Physics Today

Review of previous edition:‘This book provides a highly detailed account of the intricacies involved in considering geometrical concepts.'

Source: Contemporary Physics

‘If you're looking for a well-written and well-motivated introduction to differential geometry, this one looks hard to beat.'

Fernando Q. Gouvêa Source: MAA Online

‘… a first rate introductory textbook … the style is lively and exposition is clear which make the text easy to read … This book will be beneficial to students and scientists wishing to learn the foundations of differential geometry and algebraic topology as well as geometric formulations of modern physical theories.'

Source: Pure and Applied Geophysics

‘… this book should not be missing in any physics or mathematics library.'

Source: European Mathematical Society

‘This book is a great read and has a lot to offer to graduate students in both mathematics and physics. I wish I had had it on my desk when I began studying geometry.'

Source: AMS Review

Review of previous edition:‘The layout, the typography and the illustrations of this advanced textbook on modern mathematical methods are all very impressive and so are the topics covered in the text.'

Source: Zentralblatt für Mathematik und ihre Grenzgebiete

'… contains a wealth of interesting material for both the beginning and the advanced levels. The writing may feel informal but it is precise - a masterful exposition. Users of this 'introduction' will be well prepared for further study of differential geometry and its use in physics and engineering … As did earlier editions, this third edition will continue to promote the language with which mathematicians and scientists can communicate.'

Jay P. Fillmore Source: SIAM Review

References

[A, M, R] Abraham, R., Marsden, J., and Ratiu, T.Manifolds, Tensor Analysis, and Applications, Addison Wesley, 1983

[A, S] Aharonov, V., and Susskind, L.Observability of the sign change of spinors under 2π rotations. Phys. Rev. 158 (1967), 1237–38

[A] Arnold, V, I.Mathematical Methods of Classical Mechanics, Springer, 1978

[A2] Arnold, V, I.Ordinary Differential Equations. M.I.T. Press, 1978

[B, S] Bamberg, P. and Sternberg, S.A Course in Mathematics for Students of Physics, vol. 2, Cambridge, 1990

[B] Berry, M.Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. Lond. A 392 (1984), pp. 45–57

[B, K, G] Blank, A., Friedrichs, K., and Grad, H.Notes on Magneto-Hydrodynamics, part V, N.Y.U. notes, (1957)

[Bl] Bleecker, David.Gauge Theory and Variational Principles, Addison Wesley, 1981

[B, T] Bott, R. and Tu, L.Differential forms in Algebraic Topology, Springer, 1982

[Bo] Bott, R.Lectures on Morse theory, old and new. Bul. Amer. Math. Soc. 7 (1982) pp. 331–358

[Bo2] Bott, R.On induced representations. Proc. Symp. in Pure Math. 48 (1988), pp. 1–13.

[Boy] Boyling, J.B.An axiomatic approach to classical thermodynamics, Proc. R. Soc., London, A 329 (1972), pp. 35–70

[Br] Brillouin, L.Tensors in Mechanics and Elasticity, Academic Press, 1964

[Ca] Cartan, E.On Manifolds with an Affine Connection and the Theory of Relativity, Bibliopolis, 1986

[C] Coleman, S.Aspects of Symmetry, Cambridge, 1985

[C, J] Courant, R. and John, F.Introduction to Calculus and Analysis, vol. 2, Wiley-Interscience, 1974

[Do] Do Carmo, M.Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976

[D, F] Driver, B. and Frankel, T.On the growth of waves on manifolds, J. Math. Anal. Appl. 178 (1993), pp. 143–55

[D, S] Duff, G. and Spencer, D.Harmonic tensors on Riemannian manifolds with boundary, Ann. Math. 56 (1952), pp. 128–56

[E] Eckmann, B.Harmonische Funktionen und Randwertaufgaben in einem Komplex. Comm. Math. Helv. 17 (1945), pp. 240–55.

[Fe] Felsager, B.Geometry, Particles and Fields, Odense University Press, 1981

[F] Feynman, R.Q.E.D. Princeton, 1985

[FF] Feynman, R.. Theory of Fundamental Processes, Benjamin, 1961

[F, L, S] Feynman, R., Leighton, R. and Sands, M.The Feynman Lectures on Physics, Vols I, II and III, Addison Wesley, 1964

[F, W] Feynman, R. and Weinberg, S.Elementary Particles and the Laws of Physics, Cambridge, 1987

[Fl] Flanders, H.Differential Forms, Academic Press, 1963

[Fr] Frankel, T.Gravitational Curvature, W.H. Freeman, 1979

[Fr 2] Frankel, T.Critical submanifolds of the classical groups and Stiefel manifolds, in Differential and Combinatorial Topology, Edited by Stewart, Cairns, Princeton, 1965

[Fk] Friedrichs, K.Differential forms on Riemannian manifolds, Comm. Pure and Appl. Math. 8 (1955), pp. 551–90

[Ga] Galloway, G.A generalization of Myers' theorem and an application to relativistic cosmology, J. Diff. Geom. 14, pp. 105–116, 1979

[G, F] Gelfand, I. and Fomin, S.Calculus of Variations, Prentice-Hall, 1963

[G, H] Greenberg, M. and Harper, J.Algebraic Topology, Benjamin, 1981

[G] Grossman, N.Holonomic measurables in geodesy. J. Geophysical Res. 79. (1974), pp. 689–94

[G, P] Guillemin, V. and Pollack, A.Differential Topology, Prentice-Hall, 1974

[H] Hermann, R.Differential Geometry and the Calculus of Variations, 2nd ed., Mathematical Science Press, 1977

[H, O] Hehl, F.H. and Obukhov, Y.N.Foundations of Classical Electrodynamics, Birkhaüser, 2003

[H, Y] Hocking, J. and Young, G.Topology, Addison-Wesley, 1961

[Hs] Hsiang, W. Y.Lectures on Lie Groups, World Scientific, 2000

[I, Z] Itzykson, C. and Zuber, J-B.Quantum Field Theory, McGraw-Hill, 1980

[Ka] Kato, T.Perturbation Theory for Linear Operators, Springer, 1976

[K] Kobayashi, S.Fixed points of isometries, Nag. Math. J. 13 (1959), pp. 63–68

[K, N] Kobayashi, S. and Nomizu, K.Foundations of Differential Geometry, vols. 1 and 2. Wiley, New York, 1963

[L] Lawson, B.Minimal Varieties in Real and Complex Geometries. Presses de L'Université Montréal, 1974

[L-S, K] Levi Setti, R. and Lasinski, T.Strongly Interacting Particles, University of Chicago Press, 1973

[M, H] Marsden, J. and Hughes, T.Mathematical Foundations of Elasticity, Prentice-Hall, 1983

[Mi] Michel, L.Symmetry defects and broken symmetry. Rev. Mod. Phys. 51 (1980), pp. 617–51

[M] Milnor, J.Morse Theory, Princeton University Press, 1963

[M2] Milnor, J.Topology from the Differentiable Viewpoint, University Press of Virginia, 1965

[M, S] Milnor, J. and Stasheff, J.Characteristic Classes, Princeton, 1974

[M, T, W] Misner, C., Thorne, K., and Wheeler, J.Gravitation, Freeman, 1970

[Mo] Moffat, H.The degree of unknottedness of tangled vortex lines, J. Fluid Mech. 35 (1969), pp. 117–129

[Mu] Murnaghan, F. D.Finite Deformation of an Elastic Solid, Dover, 1951, republished 1967

[Nam] Nambu, Y.Quarks, World Scientific, 1985

[N, S] Nash, C. and Sen, S.Topology and Geometry for Physicists, Academic Press, 1983

[N] Nelson, E.Tensor Analysis, Princeton University Press, 1967

[No] Nomizu, K.Lie Groups and Differential Geometry, Math. Soc. Japan, 1956

[O] Osserman, R.Poetry of the Universe, Anchor Books, 1995

[R] Rabin, J.Introduction to quantum field theory for mathematicians, in Geometry and Quantum Field Theory, Edited by D., Fried and K., Uhlenbeck, Amer. Math Soc. 1995, pp. 183–269

[Ro] Roe, J.Elliptic Operators, Topology and Asymptotic Methods, Longman, 1988

[Sam] Samelson, H.Topology of Lie Groups, Bul. Amer. Math. Soc. 58 (1952), pp. 2–37

[Sam H] Samelson, H.Differential forms, the early days. Amer. Math. Monthly, 108 (2001) pp. 522–30

[S] Simmons, G.Topology and Modern Analysis, McGraw-Hill, 1963

[Si] Simon, B.Holonomy, the quantum adiabatic theorem, and Berry's phase. Phys, Rev. 51 (1983), pp. 2167–70

[Sp] Spivak, M.A Comprehensive Introduction to Differential Geometry, (5 volumes) Publish or Perish Press, 1979

[St] Steenrod, N.Topology of Fiber Bundles, Princeton University Press, 1951

[Sto] Stong, C.L. The amateur scientist, Scientific American 233 (December 1975), pp. 120–5

[Su] Sudbery, A.Quantum Mechanics and the Particles of Nature, Cambridge, 1986

[Sy] Sniatycki, J.Quantization of charge. J. Math. Phys. 15 (1974), pp. 619–20.

['t Hooft] 't Hooft, G.In Search of the Ultimate Building Blocks, Cambridge, 1997

[T] Truesdell, C.The influence of elasticity on analysis: the classic heritage. Bul. Amer. Math. Soc. 9 (1983), pp. 293–310

[T, T] Truesdell, C. and R., Toupin, R. The Classical Field Theories, Handbuch der Physik, III–I, 1960

[Wd] Wald, R.General Relativity, University of Chicago Press, 1984

[Wa] Warner, F.Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman, 1971

[We] Weinberg, S.The Quantum Theory of Fields, Vol II, Cambridge, 1996

[Wy] Weyl, H.The Theory of Groups and Quantum Mechanics, Dover, 1950

[W] Whittaker, E.A History of the Theories of Aether and Electricity, vol. 1, Harper, 1960

[Z] Zhang, D.Yang and contemporary mathematics. Math. Intelligencer 15 (1993), pp. 13–21

The Geometry of Physics

An Introduction

Book description

Reviews

Refine List

Actions for selected content:

Save Search

Contents

Metrics

Altmetric attention score

Full text views

Book summary page views