Dave's Web Corner

MTH-264 Week 8 (February 23-March 1)

• Outline
• Assessments
• Handouts
• Videos
• Technology

Outline

3.3 Phase Portraits for Linear Systems with Real Eigenvalues

• recommended: 3, 6, 7, 10, 11, 12, 14, 15, 16, 19, 20, 21, 22
• sink, source, and saddle
• phase planes and the dominant eigenvalue
• qualitative accuracy in tx- and ty-graphs

3.4 Complex Eigenvalues

• recommended: 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 22, 26,
• spiral sink, spiral source, and center
• de moivre's theorem and euler's formula
• growth or decay and oscillation, natural frequency
• efficient calculation with complex numbers

3.5 Special Cases: Repeated and Zero Eigenvalues

• recommended: 3, 5, 6, 7, 8, 17, 18, 19, 24
• almost spiral source and sink
• equilibrium line source and sink
• sunburst source and sink

Assessments

• Deadlines and File Submission
  • Submit your assessment as a single PDF file to the Assessment Folder inside your assigned Google Drive folder by 11:59pm on the date indicated.
  • No late assessments are accepted. If you have a conflict with a deadline, contact me before the deadline.
  • Solutions are posted under these links after the deadline.
• Homework
  • due Wednesday, February 26: 2.4 - 13 alt, 2.5 - 5 alt, 2.7 - 10 alt, 3.1 - 25 alt
  • due Wednesday, March 5: (none)
  • due Wednesday, March 12: 3.2 - 23 alt, 3.3 - 20 alt, 3.4 - 12 alt, 3.5 - 7 alt, 17 alt, 23 alt
• Exams
  • due Wednesday, March 26: Exam 2
    • Exam 2 (preview) (solutions)
    • Exam 2 covers sections 2.4, 2.5, 2.7, 3.1-3.5, 3.7, 5.1, 5.2
    • A preview will be released by 10:59pm on Wednesday, March 19 and Exam 2 will be written in class on Wednesday, March 26.

Handouts

• Announcements
  • Schmidt Award
  • Math Graduate Award
  • Honors Program introduction
  • Tutoring Center services
• Technology Tips
  • Download and Install Mathematica
  • Install DETools
  • TI-89 Slope Fields
  • TI-89 Direction Fields
• Second-Order Linear Equations
  • Second Order Linear Equations
  • Forced Damped Harmonic Oscillator Graphs
• Linear Systems
  • Scaling as Differentiation
  • Solutions to Linear Systems
  • Linear Systems Handout
  • Introduction to Linear Systems
  • Eigenvectors
  • Constructing Eigenvectors
  • Comparing Second Order Equation to First Order System
  • Sink, Source, Saddle
  • Solution Independent of Complex Eigenvalue Choice
  • Spiral Sink, Spiral Source, Center
  • Almost Spiral Sink and Source
  • Sunburst Sink and Source
  • Equilibrium Line Sink and Source
  • The Mystery Point US-UK-La La Land
  • Classification of Linear Systems
  • The Trace Determinant Plane
  • Linear Systems Practice
  • Families of Two-Dimensional Linear Systems

Videos

• Linear Systems: Basics
  • Matrices and Vectors
  • Matrix Multiplication
  • Trace and Determinant
  • The Characteristic Equation of a Matrix
  • Quadratic Formula Shortcut
  • A Formula of Euler
  • The Adjusted Matrix
• Linear Systems: Real, Non-zero Eigenvalues
  • Linearly Independent Solutions
  • General Solutions to a Linear System
  • Specific Solutions
  • Basic Eigenvalues and Eigenvectors
  • The Hooked Exponential
  • Straight Line Solutions
  • The Dominant Eigenvalue
  • Phase Portraits - Sink, Saddle, Source
  • The Five Major Cases Completed
• Linear Systems: Complex Eigenvalues
  • Complex Eigenvalue Choice
  • Best Practices for Choosing Complex Eigenvectors
  • Decomposing a Complex Solution
  • Frequency and Angular Frequency
  • The Contests Between Frequencey and Decay
• Linear Systems: Bifurcation Cases
  • Calculating the Almost Eigenvector
  • Sunburst and Almost Spiral Cases
  • Equilibrium Line Cases

Technology

• Communication
  • Zoom
  • Joining a Zoom Webinar
• Desmos
  • straight-line solution example
  • sink: a parametric illustration
  • combinations of straight-line solutions
  • exercise 3.3 #22
  • graphs of spiral sink, spiral source, and centers
  • experimenting with phase shift
  • experimenting with beats
• Excel
  • Euler First-Order Equation.xlsx
  • Euler First-Order System.xlsx
• Mathematica
  • First Order Linear Systems.nb
  • First Order Systems.nb
  • StreamPlot[]-Coloring-Example.nb
  • First Order System Phase Portrait.nb