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Representation theory
CONTENTS 26.21 26.22 26.23 26.24 Absolute derivatives along curves Geodesics Exercises Hints and answers 975 976 977 982 27 27.1 Numerical methods Algebraic and transcendental equations 984 985 Rearrangement of the equation; linear interpolation; binary chopping; Newton–Raphson method 27.2 27.3 Convergence of iteration schemes Simultaneous linear equations 992 994 Gaussian elimination; Gauss–Seidel iteration; tridiagonal matrices 27.4 Numerical integration 1000 Trapezium rule; Simpson’s rule; Gaussian integration; Monte Carlo methods 27.5 27.6 Finite differences Differential equations 1019 1020 Difference equations; Taylor series solutions; prediction and correction; Runge–Kutta methods; isoclines 27.7 27.8 27.9 27.10 Higher-order equations Partial differential equations Exercises Hints and answers 1028 1030 1033 1039 28 28.1 Group theory Groups 1041 1041 Definition of a group; examples of groups 28.2 28.3 28.4 28.5 28.6 28.7 Finite groups Non-Abelian groups Permutation groups Mappings between groups Subgroups Subdividing a group 1049 1052 1056 1059 1061 1063 Equivalence relations and classes; congruence and cosets; conjugates and classes 28.8 28.9 Exercises Hints and answers 1070 1074 29 29.1 29.2 29.3 29.4 29.5 Representation theory Dipole moments of molecules Choosing an appropriate formalism Equivalent representations Reducibility of a representation The orthogonality theorem for irreducible representations 1076 1077 1078 1084 1086 1090 xv