Math 560/561  -  Applied Math I/II *

Upon completion of the course students will have an understanding of the following:Inner product spaces: Bases, coordinates, Gram Schmidt orthogonalization, Least squares approximations Matrices: adjoints, diagonalization, maximum principle, projections, Least squares solutions, singular value decomposition, normal equations and the Fredholm alternative Introductory theory of Hilbert spaces including an overview of the properties of bounded, unbounded, and compact operators and their adjoints. Completeness and spectral properties of self adjoint operators. Fredholm alternative and applications. Operator inverses. Introduction to distributions on test function spaces and their properties. Green's functions and Sturm Liouville theory for Boundary Value Problems. Calculus of Variations introduction: functionals, admissible sets, natural boundary conditions for Lagrangians of single and several dependent variables. Euler Lagrange equation derivations Intermediate Calculus of Variation topics: geodesics, isoperimetric problems, transversality, Lagrangians with several independent variables, convexity and sufficiency conditions for minima. Lagrangian and Hamiltonian mechanics and the principle of Least action. Regular perturbation theory for algebraic and differential systems. Implicit function Theorem. Asymptotic expansions and series with applications to root approximation. Matching Theory for asymptotic expansions, overlap domains, intermediate variables. Matched asymptotic expansions for boundary value problems. Perturbation techniques for oscillatory systems: Multiple scales approaches with different temporal straining, the method of averaging and adiabatic invariants.