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