# Math 412 Description and Prerequisites

** Math 412, *** Advanced Calculus with Applications * (3 credits)

Analysis in several variables, and applications, from a computational perspective.
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.

While it is not required, it is a good idea to take an advanced linear
algebra course like Math 401 or 405 either before or while taking this
course.

## * 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