MATH 241 SPRING 2006- PROBLEM LISTS and TOPICS (planned)

List 1
vector-valued functions of 1 variable and parametrized curves/ tangent line/ reparametrization/
arc length and arc-length parameter/ unit tangent and unit normal; curvature/ velocity and acceleration;
tangential and normal components/ integration of functions along curves/mean-value inequality.

List  2
functions of 2 and 3 variables/partial derivatives/ chain rule/differentials and tangent-plane approximation/
mean-value theorem

List 3
Directional derivatives/ gradient vector/ level curves and level surfaces; quadric surfaces/ tangents to curves or surfaces defined implicitly

List 4
Functions defined implictly/ coordinate changes and mappings/ parametrized surfaces

List 5
Solutions of equations and systems/Newton's method

List 6
critical points/ the Hessian, local max, local min and saddle points/ global max/min in bounded regions/ constrained max/min
and Lagrange multipliers

List 7
multiple integrals: computation by iteration/ change of variable formula/improper integrals/surface area and scalar functions on surfaces

List 8
calculus of vector fields I:  line integrals, work/ independence of path, potentials

List 9
calculus of vector fields II:  Green's theorem and the divergence theorem in the plane/ Stokes' theorem in R^3/ flux and
the divergence theorem

List 10
The Laplacian/ Maxwell's equations/ potentials and vector potentials