Talpaert Y. Differential Geometry with Applications to Mechanics and Physics

Издательство Marcel Dekker, 2001, -475 pp.
Differential geometry is a mathematical discipline which in a decisive manner contributes to modem developments of theoretical physics and mechanics; many books relating to these are either too abstract since aimed at mathematicians, too quickly applied to particular physics branches when aimed at physicists.
Most of the text comes from Master's-level courses I taught at several African universities and aims to make differential geometry accessible to physics and engineering majors.
The first seven lectures rather faithfully translate lessons of my French book "Géométrie Différentielle et Mécanique Analytique," but contain additional examples. The last three lectures have been completely revised and several new subjects exceed the Master's degree. The text sets out, for an eclectic audience, a methodology paving the road to analytical mechanics, fluid-dynamics, special relativity, general relativity, thermodynamics, cosmology, electromagnetism, stellar dynamics, and quantum physics.
The theory and the 133 solved exercises will be of interest to other disciplines and will also allow mathematicians to find many examples and concepts. The introduced notions should be known by students when beginning a Ph.D. in mathematics applied to theoretical physics and mechanics.
The chapters illustrate the imaginative and unifying characters of differential geometry. A measured and logical progression towards (sometimes tricky) ideas, gives this book its originality. All the proofs and exercises are detailed. The important propositions and the formulae to be framed are shown by * and glasses.
Two introduced methods (in fluid-mechanics and calculus of variations) deserve further study.
There is no doubt that engineers could overcome difficulties by using differential geometry methods to meet technological challenges.
Topology and Differential Calculus Requirements.
Manifolds.
Tangent Vector Space.
Tangent Bundle - Vector Field - One-Parameter Group Lie Algebra.
Cotangent Bundle - Vector Bundle of Tensors.
Exterior Differential Forms.
Lie Derivative - Lie Group.
Integration of Forms. Stokes' Theorem, Cohomology and Integral Invariants.
Riemannian Geometry.
Lagrangian and Hamiltonian Mechanics.
Symplectic Geometry - Hamilton-Jacobi Mechanics.