Preface xix 1 Legendre Equation and Polynomials 1 1.1 Second-Order Differential Equations of Physics 1 1.2 Legendre Equation 2 1.2.1 Method of Separation of Variables 4 1.2.2 Series Solution of the Legendre Equation 4 1.2.

3 Frobenius Method - Review 7 1.3 Legendre Polynomials 8 1.3.1 Rodriguez Formula 10 1.3.2 Generating Function 10 1.3.3 Recursion Relations 12 1.

3.4 Special Values 12 1.3.5 Special Integrals 13 1.3.6 Orthogonality and Completeness 14 1.3.7 Asymptotic Forms 17 1.

4 Associated Legendre Equation and Polynomials 18 1.4.1 Associated Legendre Polynomials Pm l (x) 20 1.4.2 Orthogonality 21 1.4.3 Recursion Relations 22 1.4.

4 Integral Representations 24 1.4.5 Associated Legendre Polynomials for m 0 and (m) is an increasing function) 141 8.4.2 Case II (m > 0 and (m) is a decreasing function) 142 8.5 Technique and the Categories of Factorization 143 8.5.1 Possible Forms for k(z,m) 143 8.

5.1.1 Positive powers of m 143 8.5.1.2 Negative powers of m 146 8.6 Associated Legendre Equation (Type A) 148 8.6.

1 Determining the Eigenvalues, l 149 8.6.2 Construction of the Eigenfunctions 150 8.6.3 Ladder Operators for m 151 8.6.4 Interpretation of the L+ and L− Operators 153 8.6.

5 Ladder Operators for l 155 8.6.6 Complete Set of Ladder Operators 159 8.7 Schrödinger Equation and Single-Electron Atom (Type F) 160 8.8 Gegenbauer Functions (Type A) 162 8.9 Symmetric Top (Type A) 163 8.10 Bessel Functions (Type C) 164 8.11 Harmonic Oscillator (Type D) 165 8.

12 Differential Equation for the Rotation Matrix 166 8.12.1 Step-Up/Down Operators for m 166 8.12.2 Step-Up/Down Operators for m' 167 8.12.3 Normalized Functions with m = m' = l 168 8.12.

4 Full Matrix for l = 2 168 8.12.5 Step-Up/Down Operators for l 170 Bibliography 171 Problems 171 9 Coordinates and Tensors 175 9.1 Cartesian Coordinates 175 9.1.1 Algebra of Vectors 176 9.1.2 Differentiation of Vectors 177 9.

2 Orthogonal Transformations 178 9.2.1 Rotations About Cartesian Axes 182 9.2.2 Formal Properties of the Rotation Matrix 183 9.2.3 Euler Angles and Arbitrary Rotations 183 9.2.

4 Active and Passive Interpretations of Rotations 185 9.2.5 Infinitesimal Transformations 186 9.2.6 Infinitesimal Transformations Commute 188 9.3 Cartesian Tensors 189 9.3.1 Operations with Cartesian Tensors 190 9.

3.2 Tensor Densities or Pseudotensors 191 9.4 Cartesian Tensors and theTheory of Elasticity 192 9.4.1 Strain Tensor 192 9.4.2 Stress Tensor 193 9.4.

3 Thermodynamics and Deformations 194 9.4.4 Connection between Shear and Strain 196 9.4.5 Hook''s Law 200 9.5 Generalized Coordinates and General Tensors 201 9.5.1 Contravariant and Covariant Components 202 9.

5.2 Metric Tensor and the Line Element 203 9.5.3 Geometric Interpretation of Components 206 9.5.4 Interpretation of the Metric Tensor 207 9.6 Operations with General Tensors 214 9.6.

1 Einstein Summation Convention 214 9.6.2 Contraction of Indices 214 9.6.3 Multiplication of Tensors 214 9.6.4 The Quotient Theorem 214 9.6.

5 Equality of Tensors 215 9.6.6 Tensor Densities 215 9.6.7 Differentiation of Tensors 216 9.6.8 Some Covariant Derivatives 219 9.6.

9 Riemann Curvature Tensor 220 9.7 Curvature 221 9.7.1 Parallel Transport 222 9.7.2 Round Trips via Parallel Transport 223 9.7.3 Algebraic Properties of the Curvature Tensor 225 9.

7.4 Contractions of the Curvature Tensor 226 9.7.5 Curvature in n Dimensions 227 9.7.6 Geodesics 229 9.7.7 Invariance Versus Covariance 229 9.

8 Spacetime and Four-Tensors 230 9.8.1 Minkowski Spacetime 230 9.8.2 Lorentz Transformations and Special Relativity 231 9.8.3 Time Dilation and Length Contraction 233 9.8.

4 Addition of Velocities 233 9.8.5 Four-Tensors in Minkowski Spacetime 234 9.8.6 Four-Velocity 237 9.8.7 Four-Momentum and Conservation Laws 238 9.8.

8 Mass of a Moving Particle 240 9.8.9 Wave Four-Vector 240 9.8.10 Derivative Operators in Spacetime 241 9.8.11 Relative Orientation of Axes in K and K Frames 241 9.9 Maxwell''s Equations in Minkowski Spacetime 243 9.

9.1 Transformation of Electromagnetic Fields 246 9.9.2 Maxwell''s Equations in Terms of Potentials 246 9.9.3 Covariance of Newton''s Dynamic Theory 247 Bibliography 248 Problems 249 10 Continuous Groups and Representations 257 10.1 Definition of a Group 258 10.1.

1 Nomenclature 258 10.2 Infinitesimal Ring or Lie Algebra 259 10.2.1 Properties of rG 260 10.3 Lie Algebra of the Rotation Group R(3) 260 10.3.1 Another Approach to rR(3) 262 10.4 Group Invariants 264 10.

4.1 Lorentz Transformations 266 10.5 Unitary Group in Two Dimensions U(2) 267 10.5.1 Special Unitary Group SU(2) 269 10.5.2 Lie Algebra of SU(2) 270 10.5.

3 Another Approach to rSU(2) 272 10.6 Lorentz Group and Its Lie Algebra 274 10.7 Group Representations 279 10.7.1 Schur''s Lemma 279 10.7.2 Group Character 280 10.7.

3 Unitary Representation 280 10.8 Representations of R(3) 281 10.8.1 Spherical Harmonics and Representations of R(3) 281 10.8.2 Angular Momentum in Quantum Mechanics 281 10.8.3 Rotation of the Physical System 282 10.

8.4 Rotation Operator in Terms of the Euler Angles 282 10.8.5 Rotation Operator in the Original Coordinates 283 10.8.6 Eigenvalue Equations for Lz, L±, and L2 287 10.8.7 Fourier Expansion in Spherical Harmonics 287 10.

8.8 Matrix Elements of Lx, Ly, and Lz 289 10.8.9 Rotation Matrices of the Spherical Harmonics 290 10.8.10 Evaluation of the dlm'm() Matrices 292 10.8.11 Inv.