Geometrical Methods in Mathematical Physics

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In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

Reviewer Rating 4:
This is a very enjoyable and clearly written book. From a physics point of view the approach is rather abstract, so although differential geometry is developed from 'scratch', it is probably better to have studied a more elementary text on the theory of 2-surfaces in 3-space first (eg Faber's book Differential Geometry and Relativity Theory ). The first chapter sets the mathematical background expected of the reader. The rudiments of analysis, topology, calculus of many variables and basic linear algebra is reviewed.The ensuing chapters cover differential geometry from a 'modern' viewpoint but the style is quite relaxed and the links to 'co-ordinate approach' are well explained. The exercises concentrate on the abstract approach. Throughout the book the underlying structure of manifolds is concentrated upon. No extra 'structure' eg connections and 'distance' concepts are added until the final chapter on Riemannian spaces. For example the metric tensor throughout the body of the book is merely used as a map between a tangent space and its dual space. It is only used as a 'distance' operator in the final chapter.For the purposes of independent study this is a sound book, there are hints and partial solutions for many of the exercises, which is always a welcome feature for those studying entirely on their own.

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