A Student's Guide to Vectors and Tensors (Student's Guides)

Description

Adopting the same approach used in his highly popular A Student's Guide to Maxwell's Equations, Fleisch explains vectors and tensors in plain language to give undergraduate and beginning graduate students a better understanding of how to use vectors and tensors to solve problems in physics and engineering. Cover A Student’s Guide to Vectors and Tensors Title Copyright Contents Preface Acknowledgments 1 Vectors 1.1 Definitions (basic) 1.2 Cartesian unit vectors 1.3 Vector components 1.4 Vector addition and multiplication by a scalar 1.5 Non-Cartesian unit vectors 1.6 Basis vectors 1.7 Chapter 1 problems 2 Vector operations 2.1 Scalar product 2.2 Cross product 2.3 Triple scalar product 2.4 Triple vector product 2.5 Partial derivatives 2.6 Vectors as derivatives 2.7 Nabla – the del operator 2.8 Gradient 2.9 Divergence 2.10 Curl 2.11 Laplacian 2.12 Chapter 2 problems 3 Vector applications 3.1 Mass on an inclined plane 3.2 Curvilinear motion 3.3 The electric field 3.4 The magnetic field 3.5 Chapter 3 problems 4 Covariant and contravariant vector components 4.1 Coordinate-system transformations 4.2 Basis-vector transformations 4.3 Basis-vector vs. component transformations 4.4 Non-orthogonal coordinate systems 4.5 Dual basis vectors 4.6 Finding covariant and contravariant components 4.7 Index notation 4.8 Quantities that transform contravariantly 4.9 Quantities that transform covariantly 4.10 Chapter 4 problems 5 Higher-rank tensors 5.1 Definitions (advanced) 5.2 Covariant, contravariant, and mixed tensors 5.3 Tensor addition and subtraction 5.4 Tensor multiplication 5.5 Metric tensor 5.6 Index raising and lowering 5.7 Tensor derivatives and Christoffel symbols 5.8 Covariant differentiation 5.9 Vectors and one-forms 5.10 Chapter 5 problems 6 Tensor applications 6.1 The inertia tensor 6.2 The electromagnetic field tensor 6.3 The Riemann curvature tensor 6.4 Chapter 6 problems Further reading Index