**Dr. M. Carchidi**

**Mon.&Wed., 4:30-6; Moore 225**

_________________________________________________________________

**2) Class Notes by Michael A. Carchidi**

**3) symbolic/numerical software, e.g. Maple or Mathematica**

**Week Topics Covered**

_________________________________________________________________

**1 Review of Matrices, Matrix Algebra and
Elementary Matrices. Left and Right Identities and Left and Right Inverses.
Gaussian Elimination and Solving Systems of Algebraic Equations. Determinants,
The Adjoint Matrix and Cramer's Rule.**

*Applications*: The Construction of Curves and Surfaces That
Pass Through Specified Points, Finite Regular Markov Chains, Computer Graphics,
Chemical Equilibrium and Mixing of Solutions, Electric Circuit Analysis.

**2 Generalized Vector Spaces, Subspaces, Linear Independence, Basis,
Dimensions and Coordinates Vectors. The Row, Column, and Null Space of
a Matrix. Generalized Inner Products, Abstract Angles, Distances, and Orthogonality.**

*Applications*: Velocity Transformations in Special Relativity.

**3 Linear Operators.**

*Applications*: Legendre Polynomials and Least-Squared Fitting
of Curves to Polynomial Functions.

**General Transformations, Linear Transformations and Operators. The
Kernel and Range of a Transformation, Matrix Representations of Linear
Transformations. Eigenvalues and Eigenvectors of a Linear Operator. Singular-Value
Decomposition.**

*Applications*: Rotations, Reflections and More Computer Graphics,
The Generation of Fractals, Study of Cryptography.

**4 Diagonalization of Matrices. Jordan Conanical Forms, The Cayley-Hamilton
Theorem and Functions of Matrices, The Spectral Theorem.**

*Applications*: Normal Modes of Oscillations in Mechanical Systems,
Moments of Inertia, Solving Systems of Differential and Difference Equations,
Quadratic Forms and Rotation of Axes.

**5 One-Dimensional Flows, Fixed Points and Stability Analysis, Potentials.**

*Applications*: Population Growth, Autocatalysis, Tumor Growth,
The Growth of Organisms and the Allee effect, Chemical Kinetics, Logistics
and Population Growth.

**6 Saddle Node, Transcritical and Pitchfork Bifurcations in One-Dimensional
Flows.**

*Applications*: Laser Dynamics, Mechanics, The Outbreak of Insects,
Fish Population in a Fishery, Biological Patterns Formation in Animals,
A Study of Epidemics.

**7 One-Dimensional Flows on a Circle and Oscillations. Two-Dimensional
Linear Flows, The Phase Plane and the Classification of Linear Systems.**

**8 Two-Dimensional Nonlinear Flows, Fixed Points and Linearization,
Conservative Systems, Reversible Systems, and Index Theory.**

*Applications*: Electric Circuits, Love Affairs, Predator-Prey
Modeling,

**Modeling Epidemics, General Relativity and Planetary Orbits, The
Aerodynamics of a Glider.**

**9 Limits Cycles and the Van der Pol Oscillator, The Study of Closed
Orbits. Hopf Bifurcation.**

*Applications*: A Model for Cell Division, Competing Population
Models, Electric Circuits

**10 One-Dimensional Maps, Fixed Points and Cobwebs, The Logistics
Map and Chaos, The Cantor Set and Fractals.**

**11 First-Order Differential Equations. Special Types of Nonlinear
Differential Equations: Separable, Exact First-Order, and Homogeneous Equations**

*Applications*: Orthogonal Trajectories, Chemical Mixing.

**12 Linear Equations. Nonlinear Equations: Bernoulli's Equation and
Ricatti's Equation. Second-Order Equations Lacking One Variable.**

*Applications*: Motion in a Gravitational Field.

**13 Homogeneous Second-Order Linear Differential Equations and The
Initial-Value Problem and Linear Differential Equations Revisited. Transformation
Methods. Equidimensional Linear Differential Equations. The Algebra of
Linear Operators.**