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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:
Daniel Erenso is a professor of physics at Middle Tennessee State University (MTSU), Murfreesboro, Tennessee, USA. He joined MTSU in 2003 after he received his Ph.D. in theoretical physics from the University of Arkansas, a BSc (1990), and an MSc (1997) in physics from Addis Ababa University, Ethiopia. He also received an Advanced Diploma in Condensed Matter Physics from the Abdul Salam International Center for Theoretical Physics (ASICTP), Trieste, Italy 1999. For more than two decades, Professor Erenso has served in teaching, research, and mentoring at different universities. For the excellence and dedication that Dr. Erenso demonstrated in teaching, he received the MTSU College of Basic & Applied Sciences Excellence in Teaching Award in 2011. Professor Erenso has recently published two books: "Virtual and Real Labs for Introductory Physics II Optics, Modern Physics, and Electromagnetism" and "Studies in Theoretical Physics, Volume 1: Fundamental Mathematical Methods". Professor Erenso also has a strong record of research with extended longevity. He has mentored hundreds of undergraduate and graduate students. His research interests include theoretical and experimental physics, and he has published over 40 and presented over 80 research works at national and international venues. For his outstanding research accomplishment, Dr. Erenso received Sigma Xi the Scientific Research Society Aubrey E Harvey Outstanding Graduate Research Award from the University of Arkansas, the MTSU Foundations Special Project Award, the MTSU College of Basic & Applied Sciences Distinguished Research Award, and a nomination for American Physical Society (APS) Prize for a Faculty Member for Research in an Undergraduate Institution. Professor Erenso is a member of several professional societies, including the American Physical Society, Optical Society of America, American Association of Physics Teachers, Sigma Xi, The Scientific Research Society, and Sigma Pi Sigma, the Physics Honor Society. Professor Erenso is an invited reviewer for several international journals and book publishing companies.
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
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