Math 412 Description and Prerequisites

UM Undergraduate Catalog Course Description

More detailed outlines of some of the topics to covered can be found below and at
http://www.math.umd.edu/undergraduate/courses/syllabi/syllabusMATH410.html .

This course is not open to students who have completed Math 350 and 351.
Credit may not granted for both Math 411 and 412.

Course Prerequisites

• Math 240 and 241 or their equivalents,
• Math 410.

A More Detailed Outline

Summary: Rigorous discussion of fundamental concepts of analysis in several variables combined with computational algorithms such as the Newton 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.

Convergence and Continuity in R^n
     Linear Operations, Vector Norms, Scalar Products
     Convergence of Sequences, Cauchy Criterion
     Convergence of Series, Absolute Convergence
     Interior and Closure, Open and Closed Sets, Dense Sets
     Completeness, Connectedness, and Compactness
     Functions, Continuity, and Limits
     Extreme-Value and Intermediate-Value Theorems
     Cauchy and Uniform Continuity
Differentiability and Derivatives in R^n
     Differentiability and Linear Approximation
     Derivatives and Differentiation
     Local Extrema, Hessians, Concave and Convex Functions
     Mean-Value Theorems, Taylor Approximations
Systems of Equations over R^n
     Linear Systems, Contraction Mapping Theorem
     Inverse Function Theorem, Implicit Function Theorem
     Newton Method
Optimization over R^n
     Unconstrained Quadratic and Convex Optimization
     Steepest Descent and Conjugate Gradient Methods
     Optimization Constrained by Equalities, Lagrange Multipliers
     Optimization Constrained by Inequalities, Karush-Kuhn-Tucker Conditions
Intergration over R^n
     Integrals over Generalized Rectangles and Jordan Domains
     Multivariable Numerical Quadrature
     Change of Variables in Multiple Integrals
     Multivariable Fundamental Theorem of Calculus