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Unlike other standard textbooks in mathematical methods, this second Volume covers topics in mathematical techniques needed in advanced upper-level courses in theoretical physics that include General relativity, electromagnetism, fluid dynamics, quantum information theory and quantum field theory, solid mechanics, crystallography, continuum mechanics, plasma physics, and thermodynamics. The topics are divided into four chapters: The first chapter introduces manifolds, and the second chapter focuses on vector calculus on Manifolds with several application examples from the special theory of relativity. The third chapter discusses tensor calculus on manifolds, followed by the application of tensor calculus, specifically to relativistic electrodynamics in the four-dimensional Minkowski space-time manifold in chapter four. These topics are built upon and related vector calculus (in three dimensions) covered in the first Volume.
Key Features:
- Unlike other standard books on Mathematical methods, all the topics in Volume II are intended to provide the mathematical techniques needed in advanced physics or engineering courses that heavily depend on a solid background in tensor calculus and its applications.
- Provides several worked examples, in many cases, supported with modern 3D graphics.
- Includes clear, brief, concise, and simplified discussions to make it easy to understand by readers with English as a second language.
- The author will provide a separate solutions manual for all homework problems listed in each chapter.
Preface
Acknowledgements
Author biography
Introduction
1 Manifolds
1.1 What is a manifold?
1.2 Curves and surfaces in a manifold
1.3 Coordinate transformations and summation convention
1.4 The local geometry of a manifold
1.5 Length, area, and volume 1.6 Local Cartesian coordinates and tangent space
1.7 The signature of a manifold
1.8 N-dimensional volume without a constraint
1.9 Homework assignment
Reference
2 Vector calculus on manifolds
2.1 Scalars and vectors on a manifold
2.2 The tangent vector
2.3 The basis vectors
2.4 The metric function and the coordinate basis vectors
2.5 Basis vectors and coordinate transformations
2.6 Components of a vector in coordinate transformations
2.7 The inner product of vectors and the metric tensor
2.8 The inner product and the null vectors
2.9 The affine connections
2.10 The affine connection under coordinate transformation
2.11 The affine connection and the metric tensor
2.12 Local geodesic and Cartesian coordinates
2.13 The gradient, the divergence, and the curl on a manifold
2.14 Intrinsic derivative of a vector along a curve
2.15 Null curves, non-null curves, and affine parameter
2.16 Parallel transport
2.17 The geodesic
2.18 The Euler–Lagrange equation
2.19 Stationary property of the non-null geodesic
2.20 Homework assignments
3 Tensor calculus on manifolds
3.1 Tensors and rank of a tensor
3.2 Components of a tensor
3.3 Permutations and symmetries in tensors
3.4 Associated tensors
3.5 Mapping tensors onto tensors
3.6 Tensors and coordinate transformations
3.7 Tensor equations and the quotient theorem
3.8 Covariant derivatives of a tensor
3.9 Intrinsic derivative
3.10 Homework assignment
4 Tensor application: relativistic electrodynamics
4.1 The Lorentz force and the electromagnetic field tensor
4.2 The four-current density and the continuity equation
4.3 Maxwell’s equations and the electromagnetic field tensor
4.4 The scalar, the vector potentials, and the electromagnetic field tensor
4.5 Gauge transformation
4.6 Maxwell’s equations in a Lorentz gauge
4.7 Charged particle equation of motion
4.8 Homework assignment