# Math 346: Mathematical Analysis 2

## Contents

## Catalog Information

### Title

Linear and Nonlinear Analysis 2

### (Credit Hours:Lecture Hours:Lab Hours)

(3:3:0)

### Offered

W

### Prerequisite

Math 344; concurrent with Math 347

### Description

Theory of Riemann-Darboux integration, calculus on curves and surfaces, complex integration, spectral calculus, generalized inverses of matrices, basic matrix perturbation theory, groups of permutations and matrices. Time permitting, exterior calculus and differential forms.

## Desired Learning Outcomes

### Prerequisites

Math 344; concurrent with Math 347

### Minimal learning outcomes

Students will have a solid understanding of the concepts listed below. They will be able to prove theorems that are central to this material, including theorems that they have not seen before. They will understand connections between the concepts taught, and be able to relate them to other mathematical material that they have studied. They will be able to perform the related computations on small, simple problems.

- Riemann-Darboux Integration
- Definition and Examples
- Integrability of continuous functions
- Iterated integrals and Fubini’s Theorem
- The Jacobian and change of variables
- Polar and spherical coordinates

- Calculus on Curves and Surfaces
- Line and surface integrals
- Stokes’ Theorem
- Applications

- Complex Integration
- Contour Integrals on C
- Laurent Series
- Residues
- Cauchy Integral Formula
- Important Theorems (Liouville, Rouche’s, Maximum Modulus, Fund. Thm. Algebra)

- Exterior Calculus and Differential Forms (time permitting)
- Tensors and Alternating Forms
- Differential Forms
- Calculus of Forms (Poincare Lemma)
- The Generalized Stokes’ Theorem
- Applications

- Spectral Calculus
- The Resolvent
- Local properties
- Spectral Resolution
- Spectral Decomposition
- The Spectral Mapping Theorem
- Positive and Nonnegative Matrices (Perron-Frobenius)

- Generalized Inverses
- Moore-Penrose Inverse
- Drazin Inverse
- Other Inverses

- Basic Matrix Perturbation Theory
- Groups of Permutations and Matrices
- Permutations and Groups
- Homomorphisms
- Canonical Groups
- Matrix Groups and Representation Theory
- Symmetries and Applications

### Textbooks

Possible textbooks for this course include (but are not limited to):