## Written Exam Workshop – Fall 2007

### Announcements

• Office hours are cancelled on November 20

### Information

Thursdays, 7:10-9pm, 613 WWH, Sept 13-Dec 6

Instructor: Miranda Holmes, office 809

Office Hours: Tuesdays 4-6pm

### September 13 – Calculus

Topics: limits, series, sequences and series of functions, uniform convergence, power series, fundamental theorem of calculus, limits of integrals, improper integrals.

References: Buck Ch 4.5, 5, 6.1-6.4, selections from Ch 1-3. See Rudin (or other) for how to take limits of integrals.

Problems: S03#1, S94#3, J05#1, S93#2, S05#3(ii)

### September 20 – Calculus

Topics: same as above.

Problems: J07#2, S99#2, J06#1b, S98#3, S98#4, J98#4, J03#4. Extra (if time): J06#1a.

### September 27 – Linear Algebra

Topics: systems of equations, fundamental subspaces (column space, row space, null space (kernel), left nullspace), rank, determinant, trace, eigenvalues, eigenvectors, characteristic polynomial, matrix diagonolization, Jordan form, similar matrices.

Problems: J03#4, S96#1, S97#5, S93#3, S90#3, J05#2, J04#4, S93#5, J02#2

### October 4 – Linear Algebra

Topics: same as above, plus Matrix norm, Gram-Schmidt.

Problems: J98#5, S02#2, J06#4, J94#3, J90#3, J96#1, J96#4, S02#1.

### October 11 – Complex Analysis

Topics: contour integration, residues, evaluating improper real integrals and infinite sums, Laurent series, radius of convergence, Cauchy Integral Formula, Liouville's theorem, singularities, branch cuts.

Problems: J07#1, S06#1, S04#2, J03#2a, J06#4, J90#2, J90#3, S96#5, S91#4.

### October 18 – Complex Analysis

Topics: Logarithmic derivative, winding number, argument principle, counting zeros and poles, Rouche's Theorem.

Problems: J03#3, S06#5, J05#2, S05#4, J04#1, J90#1, J91#5, S98#5.

### October 25 – Calculus

Topics: Main topics: Lagrange multipliers, Implicit Function Theorem. Other topics: Implicit differentiation, normals/tangents to a surface/line, finding extrema, Taylor's theorem, Taylor's remainder, Mean Value Theorem, Fubini's theorem.

Problems: J95#4, S03#4, S03#5, J95#3, S91#2, S94#4, S96#5, J01#1, J92#2.

### November 1 – Calculus

Topics: Vector calculus. Contour, surface, volume integrals. Divergence theorem, Green's Theorem, Stokes' Theorem. Change of variables, eg polar and spherical coordinates.

Problems: J07#3, J06#2, S02#5, J02#5, J05#4, J92#5, J96#4, J97#2.

### November 8 – Complex Analysis

Topics: Conformal mapping, Fractional linear transformation, Schwartz-Christoffel transformation, Maximum Modulus principle.

Problems: S05#5, J06#3, J04#4, J05#1, J98#4, J00#3, S97#2, S99#4.

### November 15 – Complex Analysis

Topics: Infinite products, canonical products, genus, order, Hadamard's Theorem, Reflection Principle, analytic continuation, branch point, harmonic functions.

Reference: Ahlfors (for infinite products and Hadamard's Theorem), Churchill+Brown (for harmonic functions, proof of Poisson Integral Formula).

Problems: J07#2, J03#4, S95#5, S99#5 (note: *prove* this, don't just state the Poisson Integral Formula!), J02#5, J02#1, J92#5, J93#2.

### November 29 – Linear Algebra

Topics: Projections, reflections, Rayleigh quotient, Courant Max/Min theorem (minimax principle), Cholesky decomposition, symmetric matrices, positive definite matrices.

Problems: J07#1, S95#5, J97#2, J07#4, S94#5, J96#3, S02#4, S96#5.

### December 6 – Linear Algebra

Topics: Singular Value Decomposition, non-Euclidean vector spaces, plus more on topics from Day 3.

Problems: (J96#2), S99#1, J03#1, J07#2, S94#3, J94#4, J07#5, S02#5, J02#5. (The first problem is a warm-up to SVD and we won't do it in class.)