Question? Leave a message!




Lecture notes on Mathematical Methods

advanced mathematical methods lecture notes and mathematical methods physics lecture notes | pdf free download
ShawnPacinocal Profile Pic
ShawnPacinocal,United States,Researcher
Published Date:09-07-2017
Website URL
Comment
LECTURENOTESON MATHEMATICALMETHODS Mihir Sen Joseph M. Powers Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, Indiana 46556-5637 USA updated 29 July 2012, 2:31pm2 CC BY-NC-ND. 29 July 2012, Sen & Powers.Contents Preface 11 1 Multi-variable calculus 13 1.1 Implicit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2 Functional dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3.1 Jacobian matrices and metric tensors . . . . . . . . . . . . . . . . . . 22 1.3.2 Covariance and contravariance . . . . . . . . . . . . . . . . . . . . . . 31 1.3.3 Orthogonal curvilinear coordinates . . . . . . . . . . . . . . . . . . . 41 1.4 Maxima and minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 1.4.1 Derivatives of integral expressions . . . . . . . . . . . . . . . . . . . . 44 1.4.2 Calculus of variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 1.5 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2 First-order ordinary differential equations 57 2.1 Separation of variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.2 Homogeneous equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 Exact equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 2.4 Integrating factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 2.5 Bernoulli equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 2.6 Riccati equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.7 Reduction of order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.7.1 y absent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.7.2 x absent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 2.8 Uniqueness and singular solutions . . . . . . . . . . . . . . . . . . . . . . . . 71 2.9 Clairaut equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 3 Linear ordinary differential equations 79 3.1 Linearity and linear independence . . . . . . . . . . . . . . . . . . . . . . . . 79 3.2 Complementary functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.1 Equations with constant coefficients . . . . . . . . . . . . . . . . . . . 82 34 CONTENTS 3.2.1.1 Arbitrary order . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.1.2 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.1.3 Second order . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.2 Equations with variable coefficients . . . . . . . . . . . . . . . . . . . 85 3.2.2.1 One solution to find another . . . . . . . . . . . . . . . . . . 85 3.2.2.2 Euler equation . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 Particular solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.3.1 Method of undetermined coefficients . . . . . . . . . . . . . . . . . . 88 3.3.2 Variation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . 90 3.3.3 Green’s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 3.3.4 Operator D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4 Series solution methods 103 4.1 Power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.1.1 First-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.1.2 Second-order equation . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.2.1 Ordinary point . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.1.2.2 Regular singular point . . . . . . . . . . . . . . . . . . . . . 108 4.1.2.3 Irregular singular point . . . . . . . . . . . . . . . . . . . . 114 4.1.3 Higher order equations . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2 Perturbation methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.2.1 Algebraic and transcendental equations . . . . . . . . . . . . . . . . . 115 4.2.2 Regular perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 120 4.2.3 Strained coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 4.2.4 Multiple scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.2.5 Boundary layers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 4.2.6 WKBJ method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 S(x) 4.2.7 Solutions of the type e . . . . . . . . . . . . . . . . . . . . . . . . 139 4.2.8 Repeated substitution . . . . . . . . . . . . . . . . . . . . . . . . . . 140 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5 Orthogonal functions and Fourier series 147 5.1 Sturm-Liouville equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.1.1 Linear oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.1.2 Legendre’s differential equation . . . . . . . . . . . . . . . . . . . . . 153 5.1.3 Chebyshev equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.1.4 Hermite equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.1.4.1 Physicists’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5.1.4.2 Probabilists’ . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1.5 Laguerre equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 5.1.6 Bessel’s differential equation . . . . . . . . . . . . . . . . . . . . . . . 165 CC BY-NC-ND. 29 July 2012, Sen & Powers.CONTENTS 5 5.1.6.1 First and second kind . . . . . . . . . . . . . . . . . . . . . 165 5.1.6.2 Third kind . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 5.1.6.3 Modified Bessel functions . . . . . . . . . . . . . . . . . . . 169 5.1.6.4 Ber and bei functions . . . . . . . . . . . . . . . . . . . . . 169 5.2 Fourier series representation of arbitrary functions . . . . . . . . . . . . . . . 169 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 Vectors and tensors 177 6.1 Cartesian index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 6.2 Cartesian tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2.1 Direction cosines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 6.2.1.1 Scalars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.2.1.2 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.2.1.3 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2.2 Matrix representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.2.3 Transpose of a tensor, symmetric and anti-symmetric tensors . . . . . 187 6.2.4 Dual vector of an anti-symmetric tensor . . . . . . . . . . . . . . . . 188 6.2.5 Principal axes and tensor invariants . . . . . . . . . . . . . . . . . . . 189 6.3 Algebra of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 6.3.1 Definition and properties . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.3.2 Scalar product (dot product, inner product) . . . . . . . . . . . . . . 194 6.3.3 Cross product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.3.4 Scalar triple product . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.3.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 6.4 Calculus of vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 6.4.1 Vector function of single scalar variable . . . . . . . . . . . . . . . . . 196 6.4.2 Differential geometry of curves . . . . . . . . . . . . . . . . . . . . . . 196 6.4.2.1 Curves on a plane . . . . . . . . . . . . . . . . . . . . . . . 199 6.4.2.2 Curves in three-dimensional space. . . . . . . . . . . . . . . 201 6.5 Line and surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.1 Line integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204 6.5.2 Surface integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 6.6 Differential operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 6.6.1 Gradient of a scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 6.6.2 Divergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.6.2.1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.6.2.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.6.3 Curl of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.6.4 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.6.4.1 Scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.6.4.2 Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.6.5 Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 CC BY-NC-ND. 29 July 2012, Sen & Powers.6 CONTENTS 6.6.6 Curvature revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 6.7 Special theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.7.1 Green’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.7.2 Divergence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.7.3 Green’s identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 6.7.4 Stokes’ theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 6.7.5 Leibniz’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7 Linear analysis 229 7.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.2 Differentiation and integration . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.2.1 Fr´echet derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.2.2 Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 7.2.3 Lebesgue integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 7.2.4 Cauchy principal value . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.3 Vector spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.3.1 Normed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 7.3.2 Inner product spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 7.3.2.1 Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . 247 7.3.2.2 Non-commutation of the inner product . . . . . . . . . . . . 249 7.3.2.3 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . 250 7.3.2.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . 253 7.3.2.5 Gram-Schmidt procedure . . . . . . . . . . . . . . . . . . . 254 7.3.2.6 Projection of a vector onto a new basis . . . . . . . . . . . . 255 7.3.2.6.1 Non-orthogonal basis . . . . . . . . . . . . . . . . . 256 7.3.2.6.2 Orthogonal basis . . . . . . . . . . . . . . . . . . . 261 7.3.2.7 Parseval’s equation, convergence, and completeness . . . . . 268 7.3.3 Reciprocal bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 7.4 Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 7.4.1 Linear operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 7.4.2 Adjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 7.4.3 Inverse operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280 7.4.4 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . 283 7.5 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296 7.6 Method of weighted residuals . . . . . . . . . . . . . . . . . . . . . . . . . . 300 7.7 Uncertainty quantification via polynomial chaos . . . . . . . . . . . . . . . . 310 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 8 Linear algebra 323 8.1 Determinants and rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.2 Matrix algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 CC BY-NC-ND. 29 July 2012, Sen & Powers.CONTENTS 7 8.2.1 Column, row, left and right null spaces . . . . . . . . . . . . . . . . . 325 8.2.2 Matrix multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.2.3 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.3.1 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.3.2 Nilpotent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.3.3 Idempotent . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 8.2.3.4 Diagonal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.2.3.5 Transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.2.3.6 Symmetry, anti-symmetry, and asymmetry . . . . . . . . . . 330 8.2.3.7 Triangular . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.2.3.8 Positive definite . . . . . . . . . . . . . . . . . . . . . . . . . 330 8.2.3.9 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . 331 8.2.3.10 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 8.2.3.11 Similar matrices . . . . . . . . . . . . . . . . . . . . . . . . 333 8.2.4 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 8.2.4.1 Over-constrained systems . . . . . . . . . . . . . . . . . . . 333 8.2.4.2 Under-constrained systems . . . . . . . . . . . . . . . . . . . 336 8.2.4.3 Simultaneously over- and under-constrained systems . . . . 338 8.2.4.4 Square systems . . . . . . . . . . . . . . . . . . . . . . . . . 340 8.3 Eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 8.3.1 Ordinary eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . 342 8.3.2 Generalized eigenvalues and eigenvectors in the second sense . . . . . 346 8.4 Matrices as linear mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 8.5 Complex matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.6 Orthogonal and unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . 352 8.6.1 Orthogonal matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 8.6.2 Unitary matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355 8.7 Discrete Fourier transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 8.8 Matrix decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.8.1 L·D·U decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 362 8.8.2 Cholesky decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 365 8.8.3 Row echelon form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 8.8.4 Q·R decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 8.8.5 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 8.8.6 Jordan canonical form . . . . . . . . . . . . . . . . . . . . . . . . . . 379 8.8.7 Schur decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 8.8.8 Singular value decomposition . . . . . . . . . . . . . . . . . . . . . . 382 8.8.9 Hessenberg form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 8.9 Projection matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386 8.10 Method of least squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 8.10.1 Unweighted least squares . . . . . . . . . . . . . . . . . . . . . . . . . 388 8.10.2 Weighted least squares . . . . . . . . . . . . . . . . . . . . . . . . . . 389 CC BY-NC-ND. 29 July 2012, Sen & Powers.8 CONTENTS 8.11 Matrix exponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 8.12 Quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393 8.13 Moore-Penrose inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 9 Dynamical systems 405 9.1 Paradigm problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 9.1.1 Autonomous example . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 9.1.2 Non-autonomous example . . . . . . . . . . . . . . . . . . . . . . . . 409 9.2 General theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 9.3 Iterated maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 9.4 High order scalar differential equations . . . . . . . . . . . . . . . . . . . . . 417 9.5 Linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 9.5.1 Homogeneous equations with constant A . . . . . . . . . . . . . . . . 419 9.5.1.1 N eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . 420 9.5.1.2 N eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . 421 9.5.1.3 Summary of method . . . . . . . . . . . . . . . . . . . . . . 422 9.5.1.4 Alternative method . . . . . . . . . . . . . . . . . . . . . . . 422 9.5.1.5 Fundamental matrix . . . . . . . . . . . . . . . . . . . . . . 426 9.5.2 Inhomogeneous equations . . . . . . . . . . . . . . . . . . . . . . . . 427 9.5.2.1 Undetermined coefficients . . . . . . . . . . . . . . . . . . . 430 9.5.2.2 Variation of parameters . . . . . . . . . . . . . . . . . . . . 431 9.6 Non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 9.6.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431 9.6.2 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433 9.6.3 Lyapunov functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438 9.6.4 Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 440 9.7 Differential-algebraic systems . . . . . . . . . . . . . . . . . . . . . . . . . . 442 9.7.1 Linear homogeneous . . . . . . . . . . . . . . . . . . . . . . . . . . . 443 9.7.2 Non-linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 9.8 Fixed points at infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 9.8.1 Poincar´e sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446 9.8.2 Projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 450 9.9 Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 9.9.1 Cantor set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452 9.9.2 Koch curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 9.9.3 Menger sponge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 9.9.4 Weierstrass function . . . . . . . . . . . . . . . . . . . . . . . . . . . 454 9.9.5 Mandelbrot and Julia sets . . . . . . . . . . . . . . . . . . . . . . . . 454 9.10 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 9.10.1 Pitchfork bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . 456 9.10.2 Transcritical bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 457 CC BY-NC-ND. 29 July 2012, Sen & Powers.CONTENTS 9 9.10.3 Saddle-node bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . 459 9.10.4 Hopf bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 9.11 Lorenz equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 460 9.11.1 Linear stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461 9.11.2 Non-linear stability: center manifold projection . . . . . . . . . . . . 463 9.11.3 Transition to chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 10 Appendix 481 10.1 Taylor series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481 10.2 Trigonometric relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 10.3 Hyperbolic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 10.4 Routh-Hurwitz criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 10.5 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484 10.6 Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 10.7 Special functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 10.7.1 Gamma function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 10.7.2 Beta function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 10.7.3 Riemann zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 486 10.7.4 Error functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 10.7.5 Fresnel integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 10.7.6 Sine-, cosine-, and exponential-integral functions . . . . . . . . . . . . 488 10.7.7 Elliptic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489 10.7.8 Hypergeometric functions . . . . . . . . . . . . . . . . . . . . . . . . 490 10.7.9 Airy functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 10.7.10Diracδ distribution and Heaviside function . . . . . . . . . . . . . . . 491 10.8 Total derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 10.9 Leibniz’s rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 10.10Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 10.10.1Euler’s formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 10.10.2Polar and Cartesian representations . . . . . . . . . . . . . . . . . . . 494 10.10.3Cauchy-Riemann equations . . . . . . . . . . . . . . . . . . . . . . . 496 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497 Bibliography 499 CC BY-NC-ND. 29 July 2012, Sen & Powers.10 CONTENTS CC BY-NC-ND. 29 July 2012, Sen & Powers.Preface These arelecture notesforAME60611Mathematical MethodsI,thefirst ofapair ofcourses onapplied mathematicstaught intheDepartment ofAerospace andMechanical Engineering of the University of Notre Dame. Most of the students in this course are beginning graduate students in engineering coming from a variety of backgrounds. The course objective is to survey topics in applied mathematics, including multidimensional calculus, ordinary differ- ential equations, perturbation methods, vectors and tensors, linear analysis, linear algebra, and non-linear dynamic systems. In short, the course fully explores linear systems and con- siders effects of non-linearity, especially those types that can be treated analytically. The companion course, AME 60612, covers complex variables, integral transforms, and partial differential equations. These notes emphasize method and technique over rigor and completeness; the student should call on textbooks and other reference materials. It should also be remembered that practice is essential to learning; the student would do well to apply the techniques presented by working as many problems as possible. The notes, along with much information on the course, can be found at http://www.nd.edu/∼powers/ame.60611. At this stage, anyone is free to use the notes under the auspices of the Creative Commons license below. These notes have appeared in various forms over the past years. An especially general tightening ofnotationand language, improvement offigures, and additionofnumerous small topicswasimplementedin2011. Fall2011studentswerealsoespeciallydiligentinidentifying additional areas for improvement. We would be happy to hear further suggestions from you. Mihir Sen Mihir.Sen.1nd.edu http://www.nd.edu/∼msen Joseph M. Powers powersnd.edu http://www.nd.edu/∼powers Notre Dame, Indiana; USA = CC BY: 29 July 2012 The content of this book is licensed under Creative Commons Attribution-Noncommercial-No Derivative Works 3.0. 11 \12 CONTENTS CC BY-NC-ND. 29 July 2012, Sen & Powers.Chapter 1 Multi-variable calculus see Kaplan, Chapter 2: 2.1-2.22, Chapter 3: 3.9, Here we consider many fundamental notions from the calculus of many variables. 1.1 Implicit functions The implicit function theorem is as follows: Theorem For a given f(x,y) with f = 0 and ∂f/∂y6= 0 at the point (x ,y ), there corresponds a o o unique function y(x) in the neighborhood of (x ,y ). o o More generally, we can think of a relation such as f(x ,x ,...,x ,y) = 0, (1.1) 1 2 N also written as f(x ,y)= 0, n = 1,2,...,N, (1.2) n in some region as an implicit function of y with respect to the other variables. We cannot have ∂f/∂y = 0, because then f would not depend ony in this region. In principle, we can write y =y(x ,x ,...,x ), or y =y(x ), n = 1,...,N, (1.3) 1 2 N n if ∂f/∂y = 6 0. The derivative ∂y/∂x can be determined from f = 0 without explicitly solving for y. n First, from the definition of the total derivative, we have ∂f ∂f ∂f ∂f ∂f df = dx + dx +...+ dx +...+ dx + dy = 0. (1.4) 1 2 n N ∂x ∂x ∂x ∂x ∂y 1 2 n N Differentiating with respect to x while holding all the other x ,m = 6 n, constant, we get n m ∂f ∂f ∂y + = 0, (1.5) ∂x ∂y∂x n n 1314 CHAPTER 1. MULTI-VARIABLE CALCULUS so that ∂f ∂y ∂x n =− , (1.6) ∂f ∂x n ∂y which can be found if ∂f/∂y = 6 0. That is to say, y can be considered a function of x if n ∂f/∂y6= 0. Let us now consider the equations f(x,y,u,v) = 0, (1.7) g(x,y,u,v) = 0. (1.8) Under certain circumstances, we can unravel Eqs. (1.7-1.8), either algebraically or numeri- cally, to form u =u(x,y),v =v(x,y). The conditions for the existence of such a functional dependency can be found by differentiation of the original equations; for example, differen- tiating Eq. (1.7) gives ∂f ∂f ∂f ∂f df = dx+ dy+ du+ dv = 0. (1.9) ∂x ∂y ∂u ∂v Holding y constant and dividing by dx, we get ∂f ∂f ∂u ∂f ∂v + + = 0. (1.10) ∂x ∂u∂x ∂v∂x Operating on Eq. (1.8) in the same manner, we get ∂g ∂g∂u ∂g∂v + + = 0. (1.11) ∂x ∂u∂x ∂v∂x Similarly, holding x constant and dividing by dy, we get ∂f ∂f∂u ∂f ∂v + + = 0, (1.12) ∂y ∂u∂y ∂v∂y ∂g ∂g∂u ∂g∂v + + = 0. (1.13) ∂y ∂u∂y ∂v∂y Equations (1.10,1.11) can besolved for∂u/∂x and∂v/∂x, and Eqs. (1.12,1.13)can besolved 1 for ∂u/∂y and ∂v/∂y by using the well known Cramer’s rule; see Eq. (8.93). To solve for ∂u/∂x and ∂v/∂x, we first write Eqs. (1.10,1.11) in matrix form:      ∂f ∂f ∂f ∂u − ∂u ∂v ∂x ∂x = . (1.14) ∂g ∂g ∂v ∂g − ∂u ∂v ∂x ∂x 1 Gabriel Cramer, 1704-1752, well-traveled Swiss-born mathematician who did enunciate his well known rule, but was not the first to do so. CC BY-NC-ND. 29 July 2012, Sen & Powers.1.1. IMPLICIT FUNCTIONS 15 Thus, from Cramer’s rule we have ∂f ∂f ∂f ∂f − − ∂x ∂v ∂u ∂x ∂(f,g) ∂(f,g) ∂g ∂g ∂g ∂g ∂u − ∂v − ∂(x,v) ∂(u,x) ∂x ∂v ∂u ∂x = ≡− , = ≡− . (1.15) ∂f ∂f ∂f ∂f ∂(f,g) ∂(f,g) ∂x ∂x ∂u ∂v ∂u ∂v ∂(u,v) ∂(u,v) ∂g ∂g ∂g ∂g ∂u ∂v ∂u ∂v In a similar fashion, we can form expressions for ∂u/∂y and ∂v/∂y: ∂f ∂f ∂f ∂f − − ∂y ∂v ∂u ∂y ∂(f,g) ∂(f,g) ∂g ∂g ∂g ∂g − − ∂u ∂v ∂(y,v) ∂(u,y) ∂y ∂v ∂u ∂y = ≡− , = ≡− . (1.16) ∂f ∂f ∂f ∂f ∂(f,g) ∂(f,g) ∂y ∂y ∂u ∂v ∂u ∂v ∂(u,v) ∂(u,v) ∂g ∂g ∂g ∂g ∂u ∂v ∂u ∂v 2 Here we take the Jacobian matrix J of the transformation to be defined as   ∂f ∂f ∂u ∂v J = . (1.17) ∂g ∂g ∂u ∂v This is distinguished from the Jacobian determinant, J, defined as ∂f ∂f ∂(f,g) ∂u ∂v J = detJ = = . (1.18) ∂g ∂g ∂(u,v) ∂u ∂v If J 6= 0, the derivatives exist, and we indeed can form u(x,y) and v(x,y). This is the condition for existence of implicit to explicit function conversion. Example 1.1 If 6 x+y+u +u+v =0, (1.19) xy+uv =1, (1.20) find ∂u/∂x. Note that we have four unknowns in two equations. In principle we could solve for u(x,y) and v(x,y) and then determine all partialderivatives, such as the one desired. In practice this is not always possible; for example, there is no general solution to sixth order polynomial equations such as we have here. Equations (1.19,1.20) are rewritten as 6 f(x,y,u,v) x+y+u +u+v = 0, (1.21) g(x,y,u,v) = xy+uv−1= 0. (1.22) 2 Carl Gustav Jacob Jacobi, 1804-1851, German/Prussian mathematician who used these quantities, which were first studied by Cauchy, in his work on partial differential equations. CC BY-NC-ND. 29 July 2012, Sen & Powers.16 CHAPTER 1. MULTI-VARIABLE CALCULUS Using the formula from Eq. (1.15) to solve for the desired derivative, we get ∂f ∂f − ∂x ∂v ∂g ∂g ∂u − ∂x ∂v = . (1.23) ∂f ∂f ∂x ∂u ∂v ∂g ∂g ∂u ∂v Substituting, we get −1 1 −y u ∂u y−u = = . (1.24) 5 5 ∂x 6u +1 1 u(6u +1)−v v u Note when 6 v = 6u +u, (1.25) that the relevant Jacobian determinant is zero; at such points we can determine neither ∂u/∂x nor ∂u/∂y; thus, for such points we cannot formu(x,y). At points where the relevantJacobiandeterminant∂(f,g)/∂(u,v)= 6 0 (which includes nearly all of the (x,y) plane), givenalocalvalue of (x,y), we canuse algebrato find a correspondinguandv, which may be multivalued, and use the formula developed to find the local value of the partial derivative. 1.2 Functional dependence Letu =u(x,y) andv =v(x,y). If we can writeu =g(v) orv =h(u), then u and v are said to be functionally dependent. If functional dependence between u and v exists, then we can consider f(u,v)= 0. So, ∂f∂u ∂f ∂v + = 0, (1.26) ∂u∂x ∂v∂x ∂f ∂u ∂f ∂v + = 0. (1.27) ∂u∂y ∂v∂y In matrix form, this is      ∂f ∂u ∂v 0 ∂x ∂x ∂u = . (1.28) ∂u ∂v ∂f 0 ∂y ∂y ∂v Since the right hand side is zero, and we desire a non-trivial solution, the determinant of the coefficient matrix must be zero for functional dependency, i.e. ∂u ∂v ∂x ∂x = 0. (1.29) ∂u ∂v ∂y ∂y CC BY-NC-ND. 29 July 2012, Sen & Powers.1.2. FUNCTIONAL DEPENDENCE 17 T Note, since detJ = detJ , that this is equivalent to ∂u ∂u ∂(u,v) ∂x ∂y J = = = 0. (1.30) ∂v ∂v ∂(x,y) ∂x ∂y That is, the Jacobian determinant J must be zero for functional dependence. Example 1.2 Determine if u = y+z, (1.31) 2 v = x+2z , (1.32) 2 w = x−4yz−2y , (1.33) are functionally dependent. The determinant of the resulting coefficient matrix, by extension to three functions of three vari- ables, is ∂u ∂u ∂u ∂u ∂v ∂w ∂x ∂y ∂z ∂x ∂x ∂x ∂(u,v,w) ∂v ∂v ∂v ∂u ∂v ∂w = , (1.34) = ∂x ∂y ∂z ∂y ∂y ∂y ∂(x,y,z) ∂w ∂w ∂w ∂u ∂v ∂w ∂x ∂y ∂z ∂z ∂z ∂z 0 1 1 = 1 0 −4(y+z) , (1.35) 1 4z −4y = (−1)(−4y−(−4)(y+z))+(1)(4z), (1.36) = 4y−4y−4z+4z, (1.37) = 0. (1.38) 2 So,u,v,w are functionally dependent. In fact w =v−2u . Example 1.3 Let x+y+z = 0, (1.39) 2 2 2 x +y +z +2xz = 1. (1.40) Canx and y be considered as functions ofz? If x =x(z) and y =y(z), then dx/dz anddy/dz must exist. If we take f(x,y,z) = x+y+z = 0, (1.41) 2 2 2 g(x,y,z) = x +y +z +2xz−1 =0, (1.42) ∂f ∂f ∂f df = dz+ dx+ dy = 0, (1.43) ∂z ∂x ∂y CC BY-NC-ND. 29 July 2012, Sen & Powers.18 CHAPTER 1. MULTI-VARIABLE CALCULUS ∂g ∂g ∂g dg = dz+ dx+ dy = 0, (1.44) ∂z ∂x ∂y ∂f ∂f dx ∂f dy + + = 0, (1.45) ∂z ∂xdz ∂y dz ∂g ∂gdx ∂gdy + + = 0, (1.46) ∂z ∂xdz ∂ydz      ∂f ∂f dx ∂f − ∂x ∂y dz ∂z = , (1.47) ∂g ∂g dy ∂g − ∂x ∂y dz ∂z T then the solution matrix (dx/dz,dy/dz) can be obtained by Cramer’s rule: ∂f ∂f − ∂z ∂y −1 1 ∂g ∂g − −(2z+2x) 2y dx −2y+2z+2x ∂z ∂y = = = =−1, (1.48) ∂f ∂f dz 2y−2x−2z 1 1 ∂x ∂y ∂g ∂g 2x+2z 2y ∂x ∂y ∂f ∂f − 1 −1 ∂x ∂z ∂g ∂g 2x+2z −(2z+2x) dy − 0 ∂x ∂z = = = . (1.49) ∂f ∂f dz 2y−2x−2z 1 1 ∂x ∂y ∂g ∂g 2x+2z 2y ∂x ∂y Note here that in the expression for dx/dz that the numerator and denominator cancel; there is no special condition defined by the Jacobian determinant of the denominator being zero. In the second, dy/dz = 0 if y−x−z6= 0, in which case this formula cannot give us the derivative. Now, in fact, it is easily shown by algebraic manipulations (which for more general functions are not possible) that √ 2 x(z) = −z± , (1.50) 2 √ 2 y(z) = ∓ . (1.51) 2 This forms two distinct lines in x,y,z space. Note that on the lines of intersection of the two surfaces √ that J = 2y−2x−2z =∓2 2, which is never indeterminate. The two original functions and their loci of intersection are plotted in Fig. 1.1. It is seen that the surface representedby the linear function, Eq. (1.39), is a plane, and that representedby the quadratic function, Eq. (1.40), is an open cylindrical tube. Note that planes and cylinders may or may not intersect. If they intersect, it is most likely that the intersection will be a closed arc. However, when the plane is aligned with the axis of the cylinder, the intersection will be two non-intersecting lines; such is the case in this example. Let us see how slightly altering the equation for the plane removes the degeneracy. Take now 5x+y+z = 0, (1.52) 2 2 2 x +y +z +2xz = 1. (1.53) Can x and y be considered as functions of z? If x = x(z) and y = y(z), then dx/dz and dy/dz must exist. If we take f(x,y,z) = 5x+y+z = 0, (1.54) 2 2 2 g(x,y,z) = x +y +z +2xz−1= 0, (1.55) CC BY-NC-ND. 29 July 2012, Sen & Powers.1.3. COORDINATE TRANSFORMATIONS 19 xx -1 -1 22 00 11 11 yy 00 11 -1 -1 0.5 0.5 -2 -2 00 z 11 -0.5 -0.5 0.5 0.5 z -1 -1 00 0.5 0.5 -0.5 -0.5 00 yy -0.5 -0.5 -1 -1 -1 -1 -0.5 -0.5 00 xx 0.5 0.5 11 2 2 2 Figure1.1: Surfaces ofx+y+z = 0 andx +y +z +2xz = 1, and their loci of intersection. T then the solution matrix (dx/dz,dy/dz) is found as before: ∂f ∂f − ∂z ∂y −1 1 ∂g ∂g − −(2z+2x) 2y dx −2y+2z+2x ∂z ∂y = = = , (1.56) ∂f ∂f dz 5 1 10y−2x−2z ∂x ∂y ∂g ∂g 2x+2z 2y ∂x ∂y ∂f ∂f − 5 −1 ∂x ∂z ∂g ∂g dy − 2x+2z −(2z+2x) −8x−8z ∂x ∂z = = = . (1.57) ∂f ∂f dz 5 1 10y−2x−2z ∂x ∂y ∂g ∂g 2x+2z 2y ∂x ∂y The two original functions and their loci of intersection are plotted in Fig. 1.2. Straightforward algebra in this case shows that an explicit dependency exists: √ √ 2 −6z± 2 13−8z x(z) = , (1.58) 26 √ √ 2 −4z∓5 2 13−8z y(z) = . (1.59) 26 Thesecurvesrepresenttheprojectionofthecurveofintersectiononthex,z andy,z planes,respectively. In both cases, the projections are ellipses. 1.3 Coordinate transformations 3 Many problems are formulated in three-dimensional Cartesian space. However, many of these problems, especially those involving curved geometrical bodies, are more efficiently 3 Ren´e Descartes, 1596-1650,French mathematician and philosopher. CC BY-NC-ND. 29 July 2012, Sen & Powers.20 CHAPTER 1. MULTI-VARIABLE CALCULUS 22 11 x -0.2 00 11 0.2 yy 00 0.5 0.5 y 00 -1 -1 -0.5 -0.5 -1 -1 -2 -2 11 11 0.5 0.5 zz 00 z 00 -0.5 -0.5 -1 -1 -1 -1 -1 -1 -0.5 -0.5 00 xx 0.5 0.5 11 2 2 2 Figure1.2: Surfacesof5x+y+z = 0andx +y +z +2xz = 1, andtheir lociofintersection. posed in a non-Cartesian, curvilinear coordinate system. To facilitate analysis involving such geometries, one needs techniques to transform from one coordinate system to another. 4 For this section, we will utilize an index notation, introduced by Einstein. We will take 1 2 3 untransformed Cartesian coordinates to be represented by (ξ ,ξ ,ξ ). Here the superscript i is an index and does not represent a power of ξ. We will denote this point by ξ , where 5 i = 1,2,3. Because the space is Cartesian, we have the usual Euclidean distance from 6 Pythagoras’ theorem for a differential arc length ds:    2 2 2 2 1 2 3 (ds) = dξ + dξ + dξ , (1.60) 3 X 2 i i i i (ds) = dξ dξ ≡dξ dξ . (1.61) i=1 Here we have adopted Einstein’s summation convention that when an index appears twice, a summation from 1 to 3 is understood. Though it makes little difference here, to strictly adhere to the conventions of the Einstein notation, which require a balance of sub- and superscripts, we should more formally take 2 j i i (ds) =dξ δ dξ =dξdξ , (1.62) ji i 4 Albert Einstein, 1879-1955, German/American physicist and mathematician. 5 Euclid of Alexandria,∼ 325 B.C.-∼ 265 B.C., Greek geometer. 6 Pythagoras of Samos, c. 570-c. 490 BC, Ionian Greek mathematician, philosopher, and mystic to whom this theorem is traditionally credited. CC BY-NC-ND. 29 July 2012, Sen & Powers.