UNIVERSITY PHYSICS
CONTENTS 1 CALCULUS OF VARIATIONS 1 1 WHAT IS IT GOOD FOR? 1 2 FUNCTIONALS 1 3 LAGRANGIAN MECHANICS 1 4 VARIABLE ENDPOINTS 1 5 LAGRANGE MULTIPLIERS 1 6 MAXIMUM OR MINIMUM? 1 7 FURTHER EXERCISES AND PROBLEMS 2 FUNCTION SPACES 2 1 MOTIVATION 2 2 NORMS AND INNER PRODUCTS 2 3 LINEAR OPERATORS AND DISTRIBUTIONS 2 4 FURTHER EXERCISES AND PROBLEMS 3 LINEAR ORDINARY DIFFERENTIAL EQUATIONS 3 1 EXISTENCE AND UNIQUENESS OF SOLUTIONS 3 2 NORMAL FORM 3 3 INHOMOGENEOUS EQUATIONS 3 4 SINGULAR POINTS 3 5 FURTHER EXERCISES AND PROBLEMS 4 LINEAR DIFFERENTIAL OPERATORS 4 1 FORMAL VS CONCRETE OPERATORS 4 2 THE ADJOINT OPERATOR 4 3 COMPLETENESS OF EIGENFUNCTIONS 4 4 FURTHER EXERCISES AND PROBLEMS 5 GREEN FUNCTIONS 5 1 INHOMOGENEOUS LINEAR EQUATIONS 5 2 CONSTRUCTING GREEN FUNCTIONS CONTENTS 5 3 APPLICATIONS OF LAGRANGE’S IDENTITY 5 4 EIGENFUNCTION EXPANSIONS 5 5 ANALYTIC PROPERTIES OF GREEN FUNCTIONS 5 6 LOCALITY AND THE GELFAND–DIKII EQUATION 5 7 FURTHER EXERCISES AND PROBLEMS 6 PARTIAL DIFFERENTIAL EQUATIONS 6 1 CLASSIFICATION OF PDES 6 2 CAUCHY DATA 6 3 WAVE EQUATION 6 4 HEAT EQUATION 6 5 POTENTIAL THEORY 6 6 FURTHER EXERCISES AND PROBLEMS 7 THE MATHEMATICS OF REAL WAVES 7 1 DISPERSIVE WAVES 7 2 MAKING WAVES 7 3 NONLINEAR WAVES 7 4 SOLITONS 7 5 FURTHER EXERCISES AND PROBLEMS 8 SPECIAL FUNCTIONS 8 1 CURVILINEAR COORDINATES 8 2 SPHERICAL HARMONICS 8 3 BESSEL FUNCTIONS 8 4 SINGULAR ENDPOINTS 8 5 FURTHER EXERCISES AND PROBLEMS 9 INTEGRAL EQUATIONS 9 1 ILLUSTRATIONS 9 2 CLASSIFICATION OF INTEGRAL EQUATIONS 9 3 INTEGRAL TRANSFORMS 9 4 SEPARABLE KERNELS 9 5 SINGULAR INTEGRAL EQUATIONS 9 6 WIENER–HOPF EQUATIONS I 9 7 SOME FUNCTIONAL ANALYSIS 9 8 SERIES SOLUTIONS 9 9 FURTHER EXERCISES AND PROBLEMS 10 VECTORS AND TENSORS 10 1 COVARIANT AND CONTRAVARIANT VECTORS 10 2 TENSORS 10 3 CARTESIAN TENSORS 10 4 FURTHER EXERCISES AND PROBLEMS CONTENTS 11 DIFFERENTIAL CALCULUS ON MANIFOLDS 11 1 VECTOR AND COVECTOR FIELDS 11 2 DIFFERENTIATING TENSORS 11 3 EXTERIOR CALCULUS 11 4 PHYSICAL APPLICATIONS 11 5 COVARIANT DERIVATIVES 11 6 FURTHER EXERCISES AND PROBLEMS 12 INTEGRATION ON MANIFOLDS 12 1 BASIC NOTIONS 12 2 INTEGRATING P-FORMS 12 3 STOKES’ THEOREM 12 4 APPLICATIONS 12 5 FURTHER EXERCISES AND PROBLEMS 13 AN INTRODUCTION TO DIFFERENTIAL TOPOLOGY 13 1 HOMEOMORPHISM AND DIFFEOMORPHISM 13 2 COHOMOLOGY 13 3 HOMOLOGY 13 4 DE RHAM’S THEOREM 13 5 POINCARÉ DUALITY 13 6 CHARACTERISTIC CLASSES 13 7 HODGE THEORY AND THE MORSE INDEX 13 8 FURTHER EXERCISES AND PROBLEMS 14 GROUPS AND GROUP REPRESENTATIONS 14 1 BASIC IDEAS 14 2 REPRESENTATIONS 14 3 PHYSICS APPLICATIONS 14 4 FURTHER EXERCISES AND PROBLEMS 15 LIE GROUPS 15 1 MATRIX GROUPS 15 2 GEOMETRY OF SU(2) 15 3 LIE ALGEBRAS 15 4 FURTHER EXERCISES AND PROBLEMS 16 THE GEOMETRY OF FIBRE BUNDLES 16 1 FIBRE BUNDLES 16 2 PHYSICS EXAMPLES 16 3 WORKING IN THE TOTAL SPACE 17 COMPLEX ANALYSIS 17 1 CAUCHY–RIEMANN EQUATIONS CONTENTS 17 2 COMPLEX INTEGRATION: CAUCHY AND STOKES 17 3 APPLICATIONS 17 4 APPLICATIONS OF CAUCHY’S THEOREM 17 5 MEROMORPHIC FUNCTIONS AND THE WINDING NUMBER 17 6 ANALYTIC FUNCTIONS AND TOPOLOGY 17 7 FURTHER EXERCISES AND PROBLEMS 18 APPLICATIONS OF COMPLEX VARIABLES 18 1 CONTOUR INTEGRATION TECHNOLOGY 18 2 THE SCHWARZ REFLECTION PRINCIPLE 18 3 PARTIAL-FRACTION AND PRODUCT EXPANSIONS 18 4 WIENER–HOPF EQUATIONS II 18 5 FURTHER EXERCISES AND PROBLEMS 19 SPECIAL FUNCTIONS AND COMPLEX VARIABLES 19 1 THE GAMMA FUNCTION 19 2 LINEAR DIFFERENTIAL EQUATIONS 19 3 SOLVING ODES VIA CONTOUR INTEGRALS 19 4 ASYMPTOTIC EXPANSIONS 19 5 ELLIPTIC FUNCTIONS 19 6 FURTHER EXERCISES AND PROBLEMS