LECTURE NOTES ON MATHEMATICAL METHODS

1 LECTURE NOTES ONMATHEMATICAL METHODSM ihir SenJoseph M. PowersDepartment of Aerospace and Mechanical EngineeringUniversity of Notre DameNotre Dame, Indiana 46556-5637 USAupdated29 July 2012, 2:31pm2CC BY-NC-ND. 29 July 2012, Sen & Multi-variable Implicit functions .. Functional dependence .. Coordinate transformations .. Jacobian matrices and metric tensors .. Covariance and contravariance .. Orthogonal curvilinear coordinates .. Maxima and minima .. Derivatives of integral expressions .. Calculus of variations .. Lagrange multipliers .. 50 Problems .. 542 First-order ordinary differential Separation of variables .. Homogeneous equations .. Exact equations .. Integrating factors .. Bernoulli equation.

2 Riccati equation .. Reduction of order .. Uniqueness and singular solutions .. Clairaut equation .. 73 Problems .. 763 Linear ordinary differential Linearity and linear independence .. Complementary functions .. Equations with constant coefficients .. Arbitrary order .. First order .. Second order .. Equations with variable coefficients .. One solution to find another .. Euler equation .. Particular solutions .. Method of undetermined coefficients .. Variation of parameters .. Green s functions .. OperatorD.. 97 Problems .. 1004 Series solution Power series .. First-order equation .. Second-order equation .. Ordinary point .. Regular singular point.

3 Irregular singular point .. Higher order equations .. Perturbation METHODS .. Algebraic and transcendental equations .. Regular perturbations .. Strained coordinates .. Multiple scales .. Boundary layers .. WKBJ method .. Solutions of the typeeS(x).. Repeated substitution ..140 Problems .. 1415 Orthogonal functions and Fourier Sturm-Liouville equations .. Linear oscillator .. Legendre s differential equation .. Chebyshev equation .. Hermite equation .. Physicists .. Probabilists .. Laguerre equation .. Bessel s differential equation .. 165CC BY-NC-ND. 29 July 2012, Sen & First and second kind .. Third kind .. Modified Bessel functions .. Ber and bei functions.

4 Fourier series representation of arbitrary functions .. 169 Problems .. 1766 Vectors and Cartesian index notation .. Cartesian tensors .. Direction cosines .. Scalars .. Vectors .. Tensors .. Matrix representation .. Transpose of a tensor, symmetric and anti-symmetric tensors .. Dual vector of an anti-symmetric tensor .. Principal axes and tensor invariants .. Algebra of vectors .. Definition and properties .. Scalar product (dot product, inner product) .. Cross product .. Scalar triple product .. Identities .. Calculus of vectors .. Vector function of single scalar variable .. Differential geometry of curves .. Curves on a plane .. Curves in three-dimensional space.

5 Line and surface integrals .. Line integrals .. Surface integrals .. Differential operators .. Gradient of a scalar .. Divergence .. Vectors .. Tensors .. Curl of a vector .. Laplacian .. Scalar .. Vector .. Identities .. 213CC BY-NC-ND. 29 July 2012, Sen & Curvature revisited .. Special theorems .. Green s theorem .. Divergence theorem .. Green s identities .. Stokes theorem .. Leibniz s rule .. 223 Problems .. 2247 Linear Sets .. Differentiation and integration .. Fr echet derivative .. Riemann integral .. Lebesgue integral .. Cauchy principal value .. Vector spaces .. Normed spaces .. Inner product spaces .. Hilbert space .. Non-commutation of the inner product.

6 Minkowski space .. Orthogonality .. Gram-Schmidt procedure .. Projection of a vector onto a new basis .. Non-orthogonal basis .. Orthogonal basis .. Parseval s equation, convergence, and completeness .. Reciprocal bases .. Operators .. Linear operators .. Adjoint operators .. Inverse operators .. Eigenvalues and eigenvectors .. Equations .. Method of weighted residuals .. Uncertainty quantification via polynomial chaos .. 310 Problems .. 3168 Linear Determinants and rank .. Matrix algebra .. 325CC BY-NC-ND. 29 July 2012, Sen & Column, row, left and right null spaces .. Matrix multiplication .. Definitions and properties .. Identity .. Nilpotent .. Idempotent.

7 Diagonal .. Transpose .. Symmetry, anti-symmetry, and asymmetry .. Triangular .. Positive definite .. Permutation .. Inverse .. Similar matrices .. Equations .. Over-constrained systems .. Under-constrained systems .. Simultaneously over- and under-constrained systems .. Square systems .. Eigenvalues and eigenvectors .. Ordinary eigenvalues and eigenvectors .. Generalized eigenvalues and eigenvectors in the second sense .. Matrices as linear mappings .. Complex matrices .. Orthogonal and unitary matrices .. Orthogonal matrices .. Unitary matrices .. Discrete Fourier transforms .. Matrix decompositions .. D Udecomposition .. Cholesky decomposition.

8 Row echelon form .. Rdecomposition .. Diagonalization .. Jordan canonical form .. Schur decomposition .. Singular value decomposition .. Hessenberg form .. Projection matrix .. Method of least squares .. Unweighted least squares .. Weighted least squares .. 389CC BY-NC-ND. 29 July 2012, Sen & Matrix exponential .. Quadratic form .. Moore-Penrose inverse .. 396 Problems .. 3999 Dynamical Paradigm problems .. Autonomous example .. Non-autonomous example .. General theory .. Iterated maps .. High order scalar differential equations .. Linear systems .. Homogeneous equations with constantA.. < Neigenvectors .. Summary of method .. Alternative method .. Fundamental matrix.

9 Inhomogeneous equations .. Undetermined coefficients .. Variation of parameters .. Non-linear systems .. Definitions .. Linear stability .. Lyapunov functions .. Hamiltonian systems .. Differential-algebraic systems .. Linear homogeneous .. Non-linear .. Fixed points at infinity .. Poincar e sphere .. Projective space .. Fractals .. Cantor set .. Koch curve .. Menger sponge .. Weierstrass function .. Mandelbrot and Julia sets .. Bifurcations .. Pitchfork bifurcation .. Transcritical bifurcation .. 457CC BY-NC-ND. 29 July 2012, Sen & Saddle-node bifurcation .. Hopf bifurcation .. Lorenz equations .. Linear stability .. Non-linear stability: center manifold projection.

10 Transition to chaos ..468 Problems .. 47310 Taylor series .. Trigonometric relations .. Hyperbolic functions .. Routh-Hurwitz criterion .. Infinite series .. Asymptotic expansions .. Special functions .. Gamma function .. Beta function .. Riemann zeta function .. Error functions .. Fresnel integrals .. Sine-, cosine-, and exponential-integral functions .. Elliptic integrals .. Hypergeometric functions .. Airy functions .. distribution and Heaviside function .. Total derivative .. Leibniz s rule .. numbers .. s formula .. and Cartesian representations .. equations .. 496 Problems .. 497 Bibliography499CC BY-NC-ND. 29 July 2012, Sen & BY-NC-ND. 29 July 2012, Sen & are LECTURE NOTES for AME 60611 MATHEMATICAL METHODS I, the first of a pair of courseson applied mathematics taught in the Department of Aerospace and Mechanical Engineeringof the university of Notre Dame.