Written Exam Workshop – Fall 2007


Announcements


Information

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

Instructor: Miranda Holmes, office 809

Office Hours: Tuesdays 4-6pm




Handouts


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.)