#### 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