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San Francisco State University Department of Physics and Astronomy August 12 2010 Vector Spaces in Physics Notes for Ph 385 Introduction to Theoretical Physics I R Bland TABLE OF CONTENTS Chapter I Vectors A The displacement vector B Vector addition and other properties 1 Equality 2 Vector addition 3 Properties of vector addition 4 Multiplication of a vector by a scalar 5 The zero vector 6 Negative of a vector 7 The scalar product 8 The vector product C Vectors in terms of components D Vector addition and multiplication by a scalar E The zero vector and the negative vector F Properties of a vector space G Other vector quantities H Verifying properties of a vector space in component form I The scalar product J Metric spaces K The cross product L Dimensionality of a vector space and linear independence M Components in a rotated coordinate system Chapter 2 The special symbols ij and ijk the Einstein summation convention and some group theory A The Kronecker delta symbol ij B The Einstein summation convention C The Levi Civita totally antisymmetric tensor Groups The permutation group The Levi Civita symbol D The cross Product E The triple scalar product F The triple vector product The epsilon killer Chapter 3 Linear equations and matrices A B C D E F G Linear independence of vectors Definition of a matrix The transpose of a matrix The trace of a matrix Addition of matrices and multiplication of a matrix by a scalar Matrix multiplication Properties of matrix multiplication ii H The unit matrix I Square matrices as members of a group J The determinant of a square matrix K The 3x3 determinant expressed as a triple scalar product L Other properties of determinants Product law Transpose law Interchanging columns or rows Equal rows or columns M Cramer s rule for simultaneous linear equations N Condition for linear dependence O Eigenvectors and eigenvalues Chapter 4 Practical Examples A Kirchhoff s circuit laws Neuron cells connected in series parallel A two loop example B Coupled oscillations masses and springs A system of two masses Three interconnected masses Systems of many coupled masses C The triple pendulum Chapter 5 The Inverse Numerical Methods A The inverse of a square matrix Definition of the inverse Use of the inverse to solve matrix equations The inverse matrix by the method of cofactors B Time required for numerical calculations C The Gauss Jordan method for solving simultaneous linear equations D The Gauss Jordan method for inverting a matrix Chapter 6 Rotations and Tensors A Rotation of axes B Some properties of rotation matrices Orthogonality Determinant C The rotation group D Tensors E Coordinate transformation of an operator on a vector space F The Conductivity Tensor Chapter 7 The Wave Equation A Qualitative properties of waves on a string B The wave equation Partial derivatives Wave velocity iii C Sinusoidal solutions D General traveling wave solutions E Energy carried by waves on a string Kinetic energy Potential energy F The