MAE 502/APC 506: Mathematical Methods of Engineering Analysis II Course information, Spring 2020 Instructors Clancy Rowley (Lectures) [email protected] Office hours: Tue, 10–11:30am, Equad D232 Udari Madhushani (AI) [email protected] Office hours: Wed, 10–11:30am, Equad D207 Schedule Time: TTh, 8:30–9:50am Place: Friend Center 008 Course summary This course covers a variety of topics in applied mathematics, with applications in physics and engineering. The first part of the course covers topics in complex analysis, including power series, singularities, contour integration, Cauchy’s theorems, and Fourier series. The remainder of the course covers topics in functional analysis, including an introduction to measure theory and the Lebesgue integral, Hilbert spaces and Banach spaces, linear operators and their spectral properties. A detailed list of topics is given below. Prerequisites Undergraduatelevel engineering mathematics: elementary linear algebra, multivariable calcu lus, and ordinary differential equations. Grading There will be approximately weekly homework assignments (35%), a midterm (30%), and a final exam (35%). Exams will be open book/open notes, take home, 3–4 hour total duration. Course text and references The first part of the course will follow the first four chapters from the following book: • E. M. Stein and R. Shakarchi, Complex Analysis, Princeton University Press, 2003. The book is not required, but I recommend purchasing it if you do not already own a book on complex analysis. Copies have been ordered at Labyrinth. For the second part of the course, I will post my lecture notes on the Blackboard site. These are intended to be selfcontained, but if you would like more information, a number of references are given below: 1 • D. G. Luenberger, Optimization by Vector Space Methods, John Wiley and Sons, 1997. • T. Tao. An Epsilon of Room, I: Real Analysis, volume 117 of Graduate Studies in Mathe matics. American Mathematical Society, 2010. • L. Debnath and P. Mikusiński, Introduction to Hilbert Spaces with Applications, Elsevier Academic Press, 3rd Edition, 2005. • A.W. Naylor and G.R. Sell, Linear Operator Theory in Engineering and Science, Springer, 1982. If you are rusty on some of the background material, such as matrix linear algebra and differential equations, the following references might be useful: • R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985, last reprinted 1996. • W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Prob lems, John Wiley and Sons, 5th ed, 1992. • L.N. Trefethen and D.I. Bau, Numerical Linear Algebra, SIAM, 1997. Course topics • Complex analysis (4 weeks) – Complex numbers, convergence, sets in the complex plane, holomorphic functions, power series. – Contour integration, Cauchy’s theorem and its applications – Zeros and poles, the residue formula, singularities and meromorphic functions, the argument principle and applications, the complex logarithm – Fourier series and harmonic functions, the Fourier transform of functions holomor phic in a horizontal strip – Conformal mappings, automorphisms of the disc and upper halfplane, Riemann mapping theorem • Functional analysis (8 weeks) – Metric spaces and topological properties, linear spaces, normed spaces, inner prod uct spaces – Introduction to measure theory, Lp spaces, Hölder and Minkowski inequalities – Hilbert spaces, the projection theorem, orthonormal systems and Fourier series, the Fourier transform on L2 , linear functionals, the Riesz representation theorem, adjoints – Banach spaces, dual spaces, the HahnBanach theorem – Linear transformations, selfadjoint operators, normal operators, the spectrum, com pact operators and the Fredholm alternative, spectral theorem for compact self adjoint operators and applications to SturmLiouville problems, spectral theorem for bounded selfadjoint operators 2
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