Description

Rigorous discussion of fundamental concepts of analysis in several variables combined with computational algorithms such as Newton's method and the method of steepest descent. Application to problems in many areas with a view to both computing solutions and deriving qualitative conclusions about the models. (This course is not open to students who have completed Math 350 and 351. Credit may not granted for both Math 412 and 411.)

Prerequisites

A C- or better in MATH 410

Topics

The basics

Vector norms on Rn
Open sets
Closed sets
Compactness
Connectedness
Continuous functions
Max and min
Uniform continuity
Differentiable functions (linear approximation)
Mean value theorem
Hessian matrix
Positive definite matrices
Second derivative test
Taylor expansions for functions of several variables

Solving equations

Matrix norms
Perturbations of invertible liner maps
Contraction mapping principle
Inverse function theorem
Newton's method
Implicit function theorem

Optimization

Method of steepest descent
Constrained optimization: method of Lagrange multipliers
Kuhn-Tucker formulation of inequality constraints

Integration in several variables

Extensions of trapezoid and Simpson's rule to higher dimensions
Change of variable in multiple integrals
Applications of change of variable in numerical calculation and statistics
Derivation of the Euler equations of fluid flow