Syllabus for Applied Mathematics Preliminary Exam
Matrices and systems of linear equations. Vector spaces over general fields, subspaces, linear independence, basis, dimension. Determinants. Linear transformations, associated matrices, change of basis, dimension formula. Dual vectorspaces. Eigenvalues, eigenspaces, diagonalization, Jordan canonical form. Inner product spaces, bilinear, quadratic and hermitian forms. Adjoint, self-adjoint, orthogonal and unitary operators. Diagonalization in Euclidean and unitary spaces. The spectral theorem.
References: Schaum's Outlines: Linear Algebra: Chapters 1--13.
Real numbers: Infimum and supremum, limits of sequences, monotone sequences, Cauchy sequences. Continuity: limits of functions, continuous functions, the intermediate value theorem, maxima and minima, uniform continuity, monotone functions, inverse functions. Differentiation: the derivative, mean value theorem, l’Hospital’s rule, Taylor’s expansion with remainder. Integration: Riemann integrals, the fundamental theorem of calculus, improper integrals. Sequences of functions: pointwise and uniform convergence, continuity and convergence, interchange of limit with derivatives and integrals, Arzela-Ascoli theorem, Weierstrass and Stone-Weierstrass approximation theorems. Differentiation of integrals with parameters. Infinite series: series of numbers and functions, absolute convergence, power series. Elementary functions: rigorous introduction of the exponential, logarithmic, trigonometric and inverse trigonometric functions. Functions of several variables: the derivative as a linear transformation, Taylor’s theorem, the inverse and implicit function theorems. Vector calculus: multiple integrals, path and surface integrals, change of variables theorem for integrals, calculation of areas, volumes and arc-lengths, the integral theorems of vector analysis (Green’s, Stokes’, and Gauss’ theorems). Metric spaces: basic topology, compactness, connectedness, completeness.
References: “Vector Calculus”, Marsden and Tromba
“Principles of Mathematical Analysis”, Walter Rudin
“Elementary Classical Analysis”, J. Marsden and M. Hoffman
Analytic functions, Cauchy-Riemann equations, entire functions, the exponential, trigonometric, and logarithmic functions, Euler’s formula. Line integrals, Cauchy’s theorem, Cauchy’s integral formula, power series representation and consequences, uniqueness theorem, mean value theorem, maximum modulus principle, open mapping theorem. Morera’s theorem, Liouville’s theorem and applications, meromorphic functions, Laurent expansions, residue theorem and applications, fractional linear transformations.
References: “Function Theory of One Complex Variable”, R. Greene and S. Krantz
Ordinary Differential Equations:
Systems of ODE: existence and uniqueness, linear systems, constant coefficient linear systems, Floquet theory, variation of parameters, autonomous systems in the plane, phase portraits, notions of stability, linearized stability,Lyapunov's method, conservation systems, Poincare-Bendixson theorem, limit cycles. .
References: “The Qualitative Theory of ODE", F. Brauer and J. A. Nohel, 1969.
“Nonlinear Differential Equations and Dynamical Systems", Ferdinand Verhulst 1996. Material selected from chapters 1–8, 12.
Partial Differential Equations:
First order PDEs and the method of characteristics. Wave equation in 1D: general soltuion, causality, energy method, reflection at boundary points, sources. Wave equation in 3D: spherical means, Huygen's principle, energy method, Duhamel's formula. Heat equation: fundamental solution, maximum principle, energy method. Laplace and Poisson equations: mean-value theorem, maximum principle, Green's identities and applications, Dirichlet and Neumann problems. Solution of PDEs by eigenfunction expansion. Solution by Fourier transforms. Fundamental solutions, Green's functions, and distributions.
References: "Partial differential equations an introduction", W. Strauss, John Wiley & Sons, 1992. Material selected from chapters 1–7, 9, 11, 12.