University of Michigan - Shanghai Jiao Tong University Joint Institute

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Vv 556: Methods of Applied Mathematics I


General information

Course description: This course gives an introduction to the theory of bounded linear maps on finite- and infinite-dimensional spaces.In the first part, notions of linear algebra are reviewed and extended to infinite-dimensional vector spaces. This includes concepts such as scalar products, norms and (Schauder-) bases. As an application, Legendre polynomials, introduced as an orthonormalization of the monomials on the interval [−1,1] are introduced, and their role in multipole expansions is explored. Next, Hilbert spaces are intro- duced, leading to spaces of square-integrable functions and Fourier series. A look back and comaprison of the obtained results with the finite-dimensional cases of linear algebra concludes this part.

The second part focuses on bounded linear maps on (infinite- dimensional) spaces, introducing the matrix elements of such opera- tors and using these to define Hilbert-Schmidt operators for square- summable sequences and square-integrable functions. The notions of inverses and adjoints of bounded linear operators are discussed and the spectrum of such operators is introduced. Compact oper- ators are introduced and, motivated by a question from the theory of partial differential equations, the spectral theorem for compact operators is established.

The last part is dedicated to applications of the spectral theory, including the Rayleigh-Ritz method (applied specifically to Sturm- Liouville eigenvalue problems) and the polar and singular value de- compositions of compact operators, which of course includes these decompositions for matrices.

Credit hours: 3 Credits
Required/Elective course: Elective
Pre-requisites: Vv 256 Applied Calculus IV, or Vv 286 Honors Mathematics IV, or consent of instructor.
Terms offered: Fall.
Cognizant faculty: Horst Hohberger
Potential instructors: Horst Hohberger
Textbook/Required material: 
- Kreyszig, E., Introductory Functional Analysis with Applications, Wiley 1989 
- I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems, 3rd Edition, Wiley 2011