Math 641-600 Final Exam Review (Fall 2011)

The final exam will be given on Monday, Dec. 12, from 3:30 am to 5:30 pm, in our usual classroom. It will cover sections 2.2.7, 3.2 - 3.6, 4.1 - 4.3.2, and it will also cover the material done in class on the Lax-Milgram theorem and Cea's lemma (see class notes, 11/7/2011 - 11/9/2011). The test will consist of the following: statements and/or proofs or sketches of proofs of theorems; statements of definitions; proofs of short propositions or solutions of problems similar to ones done in the homework or in class. Extra office hours: Wednesday, 12/8, 1-2; Friday, 12/9, 1-4; Monday, 12/12, 10-12. For other times, send me an email to arrange an appointment.

Finite elements

The finite element spaces S^h(k,r)
- Cubic splines
- Interpolation
- Smoothing
- Differential equations

Operators and integral equations

Bounded operators
- Norms of linear operators, unbounded operators, continuous linear functionals, spaces associated with operators
- Integral equations -- Hilbert-Schmidt kernels, Fredholm, Volterra
- The Projection (Decomposition) Theorem
- The Riesz Representation Theorem
- Existence of adjoints of bounded operators
- Lax-Milgram Theorem and Cea's Lemma (notes, 11/7 - 11/9)
Compact operators
- Finite rank operators, Approximation Theorem (Theorem 3.4), and Hilbert-Schmidt kernels/operators
- Spectral theory for compact, self-adjoint operators, K = K*
  - Eigenvalues and eigenspaces
    - Eigenvalues are real; eigenvectors for distinct eigenvalues are orthogonal
    - Eigenspaces are finite dimensional
    - The only limit point of the set of eigenvalues is 0.
    - "Maximum principle" (p. 117)
  - Completeness of eigenfunctions on the closure of the range of K (Theorem 3.6)
  - Application to eigenfunction problems involving integral equations
- Closed Range Theorem (Theorem 3.7), Fredholm alternative, resolvents and kernels
- Contraction Mapping Theorem, Neumann series, solving integral equations

Distributions and differential operators

Test function space D, distribution space D′, examples, δ function, δ sequences, integral representation, derivatives of distributions
Green's functions (2nd order operators)
Unbounded operators, domain of an operator, adjoints (2nd order operators)

Additional Problems (Chapter 4)

Section 4.2: 2, 4
Section 4.3: 2, 3, 5

Updated 12/6/2011 (fjn).