Ebook: Differentiable Manifolds: A Theoretical Physics Approach
- Tags: Manifolds and Cell Complexes (incl. Diff.Topology), Mechanics, Mathematical Methods in Physics, Topological Groups Lie Groups
- Year: 2012
- Publisher: Birkhäuser Basel
- Edition: 1
- Language: English
- pdf
This textbook explores the theory behind differentiable manifolds and investigates various physics applications along the way. Basic concepts, such as differentiable manifolds, differentiable mappings, tangent vectors, vector fields, and differential forms, are briefly introduced in the first three chapters. Chapter 4 gives a concise introduction to differential geometry needed in subsequent chapters. Chapters 5 and 6 provide interesting applications to connections and Riemannian manifolds. Lie groups and Hamiltonian mechanics are closely examined in the last two chapters. Included throughout the book are a collection of exercises of varying degrees of difficulty.
Differentiable Manifolds is intended for graduate students and researchers interested in a theoretical physics approach to the subject. Prerequisites include multivariable calculus, linear algebra, differential equations, and a basic knowledge of analytical mechanics.
This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.
The work’s first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.
This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.
The work’s first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.
Content:
Front Matter....Pages I-VIII
Manifolds....Pages 1-28
Lie Derivatives....Pages 29-48
Differential Forms....Pages 49-65
Integral Manifolds....Pages 67-91
Connections....Pages 93-114
Riemannian Manifolds....Pages 115-160
Lie Groups....Pages 161-200
Hamiltonian Classical Mechanics....Pages 201-253
Back Matter....Pages 255-275
This textbook gives a concise introduction to the theory of differentiable manifolds, focusing on their applications to differential equations, differential geometry, and Hamiltonian mechanics.
The work’s first three chapters introduce the basic concepts of the theory, such as differentiable maps, tangent vectors, vector and tensor fields, differential forms, local one-parameter groups of diffeomorphisms, and Lie derivatives. These tools are subsequently employed in the study of differential equations (Chapter 4), connections (Chapter 5), Riemannian manifolds (Chapter 6), Lie groups (Chapter 7), and Hamiltonian mechanics (Chapter 8). Throughout, the book contains examples, worked out in detail, as well as exercises intended to show how the formalism is applied to actual computations and to emphasize the connections among various areas of mathematics.
Differentiable Manifolds is addressed to advanced undergraduate or beginning graduate students in mathematics or physics. Prerequisites include multivariable calculus, linear algebra, differential equations, and (for the last chapter) a basic knowledge of analytical mechanics.
Content:
Front Matter....Pages I-VIII
Manifolds....Pages 1-28
Lie Derivatives....Pages 29-48
Differential Forms....Pages 49-65
Integral Manifolds....Pages 67-91
Connections....Pages 93-114
Riemannian Manifolds....Pages 115-160
Lie Groups....Pages 161-200
Hamiltonian Classical Mechanics....Pages 201-253
Back Matter....Pages 255-275
....