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. Blandii 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 δδδδij and εεεεijk, the Einstein summation convention, and some group theory. A. The Kronecker delta symbol, δij 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. Linear independence of vectors. B. Definition of a matrix. C. The transpose of a matrix. D. The trace of a matrix. E. Addition of matrices and multiplication of a matrix by a scalar. F. Matrix multiplication. G. Properties of matrix multiplication.iii 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.iv C. Sinusoidal solutions. D. General traveling-wave solutions. E. Energy carried by waves on a string. Kinetic energy. Potential energy. F. The superposition principle. G. Group and phase velocity. Chapter 8. Standing Waves on a String A. Boundary Conditions and Initial Conditions String fixed at a boundary. Boundary between two different strings. B. Standing waves on a string. Chapter 9. Fourier Series A. The Fourier sine series. The general solution. Initial conditions. Orthogonality. Completeness. B. The Fourier sine-cosine Series. Odd and even functions. Periodic functions in time. C. The exponential Fourier series. Chapter 10. Fourier Transforms and the Dirac Delta Function A. The Fourier transform. B. The Dirac delta function δ(x). The rectangular delta function. The Gaussian delta function. Properties of the delta function. C. Application of the Dirac delta function to Fourier transforms. Basis states. Functions of position x. D. Relation to quantum mechanics. Appendix A. Useful mathematical facts and formulae. A. Complex numbers. B. Some integrals and identities C. The small-angle approximation. D. Mathematical logical symbols. Appendix B. Using the P&A Computer System 1. Logging on. 2. Running MatLab, Mathematica and IDL. 3. Mathematica. Appendix C. Mathematica Routines 1. Calculation of the vector sum using Mathematica. 2. Speed test for Mathematica. ReferencesVector Spaces in Physics 8/12/2010 1 - 1 Chapter 1. Vectors Everyone is familiar with the distinction between things which have a direction and those which don't. When you are asked, "Which way is the wind blowing?" you have to point; when you are asked, "What is the temperature," you don't – and in fact, you can't. The direction of the temperature just doesn't have any meaning. Consider the following list of important and interesting quantities: 1. The price of a ticket to a baseball game. 2. The direction to San Jose. 3. The processor speed of a Mac G4. 4. The depth of the ocean under the Golden Gate Bridge. 5. The love of a child for her dog, 6. How windy it is at Pac Bell park. 7. The location of the student union. Are any of these things vectors? Which ones? To refine the concept of vector, we have to separate physical measurement from sociological context. Here are the physical measurements corresponding to the quantities just listed: 1. Number of dollars paid for a ticket to a baseball game. 2. Compass bearing to follow to go to San Jose. 3. Cycles of Mac G4 clock signal per second. 4. Depth of water under center of Bridge span. 5. No scientific definition (does this mean it's not important?) 6. Speed of wind, compass direction it comes from. 7. Distance and direction to the student union. All but one of these things can be measured with instruments. Some involve direction, some don't. (What about the water depth?) Are any of these quantities vectors? Vectors have something to do with the nature of space. Euclidian geometry (plane and spherical geometry) was a very early way