Difference between revisions of "Math 346: Mathematical Analysis 2"

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(Desired Learning Outcomes)
(Minimal learning outcomes)
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# Riemann-Darboux Integration
 
# Riemann-Darboux Integration
Definition and Examples
+
#* Definition and Examples
Integrability of continuous functions
+
#* Integrability of continuous functions
Iterated integrals and Fubini’s Theorem
+
#* Iterated integrals and Fubini’s Theorem
The Jacobian and change of variables
+
#* The Jacobian and change of variables
Polar and spherical coordinates
+
#* Polar and spherical coordinates
 
# Calculus on Curves and Surfaces
 
# Calculus on Curves and Surfaces
Line and surface integrals
+
#* Line and surface integrals
Stokes’ Theorem
+
#* Stokes’ Theorem
Applications
+
#* Applications
 
# Complex Integration
 
# Complex Integration
Contour Integrals on C
+
#* Contour Integrals on C
Laurent Series
+
#* Laurent Series
Residues
+
#* Residues
Cauchy Integral Formula
+
#* Cauchy Integral Formula
Important Theorems (Liouville, Rouche’s, Maximum Modulus, Fund. Thm. Algebra)  
+
#* Important Theorems (Liouville, Rouche’s, Maximum Modulus, Fund. Thm. Algebra)  
 
# Exterior Calculus and Differential Forms (time permitting)
 
# Exterior Calculus and Differential Forms (time permitting)
Tensors and Alternating Forms
+
#* Tensors and Alternating Forms
Differential Forms
+
#* Differential Forms
Calculus of Forms (Poincare Lemma)
+
#* Calculus of Forms (Poincare Lemma)
The Generalized Stokes’ Theorem
+
#* The Generalized Stokes’ Theorem
Applications
+
#* Applications
 
# Spectral Calculus
 
# Spectral Calculus
The Resolvent
+
#* The Resolvent
Local properties
+
#* Local properties
Spectral Resolution
+
#* Spectral Resolution
Spectral Decomposition
+
#* Spectral Decomposition
The Spectral Mapping Theorem
+
#* The Spectral Mapping Theorem
Positive and Nonnegative Matrices (Perron-Frobenius)  
+
#* Positive and Nonnegative Matrices (Perron-Frobenius)  
 
# Generalized Inverses
 
# Generalized Inverses
Moore-Penrose Inverse
+
#* Moore-Penrose Inverse
Drazin Inverse
+
#* Drazin Inverse
Other Inverses
+
#* Other Inverses
 
# Basic Matrix Perturbation Theory
 
# Basic Matrix Perturbation Theory
 
# Groups of Permutations and Matrices
 
# Groups of Permutations and Matrices
Permutations and Groups
+
#* Permutations and Groups
Homomorphisms
+
#* Homomorphisms
Canonical Groups
+
#* Canonical Groups
Matrix Groups and Representation Theory
+
#* Matrix Groups and Representation Theory
Symmetries and Applications
+
#* Symmetries and Applications
 
+
  
 
=== Textbooks ===
 
=== Textbooks ===

Revision as of 12:42, 6 June 2012

Catalog Information

Title

Linear and Nonlinear Analysis 2

(Credit Hours:Lecture Hours:Lab Hours)

(3:3:1)

Offered

W

Prerequisite

Math 344; concurrent with Math 347

Description

The 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 Form

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.

  1. 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
  2. Calculus on Curves and Surfaces
    • Line and surface integrals
    • Stokes’ Theorem
    • Applications
  3. Complex Integration
    • Contour Integrals on C
    • Laurent Series
    • Residues
    • Cauchy Integral Formula
    • Important Theorems (Liouville, Rouche’s, Maximum Modulus, Fund. Thm. Algebra)
  4. Exterior Calculus and Differential Forms (time permitting)
    • Tensors and Alternating Forms
    • Differential Forms
    • Calculus of Forms (Poincare Lemma)
    • The Generalized Stokes’ Theorem
    • Applications
  5. Spectral Calculus
    • The Resolvent
    • Local properties
    • Spectral Resolution
    • Spectral Decomposition
    • The Spectral Mapping Theorem
    • Positive and Nonnegative Matrices (Perron-Frobenius)
  6. Generalized Inverses
    • Moore-Penrose Inverse
    • Drazin Inverse
    • Other Inverses
  7. Basic Matrix Perturbation Theory
  8. 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):

Additional topics

Courses for which this course is prerequisite