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 Sh(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).