Math 5A, Winter 2008

Lecture Notes

Thu, 13 Mar 2008 21:35:52 GMT

These are the notes from class:

Self-Assessment Questions

Wed, 16 Jan 2008 00:16:00 GMT

Try to answer the following questions without looking in the book. If you do not feel comfortable with them, please review the corresponding chapters in the book, and them attempt to solve them again.

Self-Assessment 1.

SiteUrl

Wed, 09 Jan 2008 19:06:00 GMT

http://www.math.ucsb.edu/~cgarcia/Courses/Math5A_W08/

Teaching Assistant

Wed, 09 Jan 2008 17:58:00 GMT

Pat Plunkett

South Hall, Room 6432X.
Office Hours: Friday 12-1pm.
Mathlab: Monday 2-4pm.

Daniel Rife

South Hall, Room 6432 S
Office Hours: Tuesday 1-2pm.
Mathlab Hours: Tuesday 3-5pm.

Homework

Tue, 08 Jan 2008 03:01:00 GMT

Homework will be assigned on Thursdays, and will be due the following week. We will be using WebWork. You will need to familiarize yourselves with it. Late homework will not be accepted.

Syllabus

Mon, 07 Jan 2008 19:29:00 GMT

Textbook: Differential Equations and Linear Algebra, by Jerry Farlow, James E. Hall, Jean Marie McDill, and Beverly H. West.

Course description This is the second part of the Differential Equations and Linear Algebra course, the first part being Math 3C. Math 3C is a prerequisite. For this course, you are expected to know the following from Math 3C:

A working knowledge of Linear Algebra:
1. How to solve linear systems of equations.
2. How to determine if a given set is a vector space.
3. How to determine the dimension of a vector space, and how to find a basis.
4. To be able to determine whether a matrix has an inverse, and to be able to compute such inverse.
5. To be able to operate with matrices, i.e., perform additions and multiplications.
To be able to solve first order linear differential equations.
To be able to determine whether a given function is a solution to a given differential equation.

The focus of the course will be on second-order linear differential equations. We will learn how to explicitly solve equations with constant coefficients, and some of the underlying mathematical theory. We will also cover some new concepts in Linear Algebra, such as Linear Transformations, Eigenvalues and Eigenvectors, and Diagonalization.

I will not hand out practice exams or anything like that. As part of the course, you should develop the necessary skills to study this subject.
In order to help you study, I will periodically post Self-Assessment questions. You do not need to hand them in. Those questions will help you know whether you are studying effectively or not, and whether you need to review some parts of the material covered in class.

Assignments and Grading Policy: Homework will be assigned on Thursdays, and will be due the following week. We will be using WebWork. You will need to familiarize yourselves with it. Late homework will not be accepted.

There will be a midterm and a final exam. Your final grade for the course will be decided according to the following formula:

Final Grade = 40% Homework + 25% Midterm + 35% Final

Syllabus: During this course we will try to follow the following schedule. However, much like everything said earlier, this is subject to change.

Week 1:

The Harmonic Oscillator.
Real Characteristic Roots.
Complex Characteristic Roots.

Week 2:

Undetermined Coefficients.
Variation of Parameters.
Forced Oscillations.

Week 3:

Conservation and Conversion.
Linear Transformations.

Week 4:

Properties of Linear Transformations.
Eigenvalues and Eigenvectors.

Week 5:

Eigenvalues and Eigenvectors (cont.).
Coordinates and Diagonalization.

Week 6:

Midterm Exam.
Linear Systems of Differential Equations.
Linear Systems with Real Eigenvalues.

Week 7:

Linear Systems with Non-Real Eigenvalues.
Stability and Linear Classification.
Decoupling a Linear System of Differential Equations.

Week 8:

Matrix Exponential.
Nonhomogeneous Linear Systems.

Week 9:

Nonhomogeneous Linear Systems.
Nonlinear Systems.

Week 10:

Nonlinear Systems (cont.)
Linearization.

Exams

Sat, 05 Jan 2008 01:25:00 GMT

We will have a Midterm exam on Tuesday, February 12th, 2008. The Final exam will be on Friday, March 21st 2008, from 8-11am.

Weekly Readings

Sat, 05 Jan 2008 01:22:00 GMT

Week 1: Sections 4.1, and 4.2.
1. Objectives for Section 4.1:
  1. Introduction to second order linear differential equations: the harmonic oscillator is a simple model problem.
  2. Develop intuition about the solution to the equations: Oscillatory motion, amplitude, period, and phase shift.
  3. Introductory theoretical aspects via the example of the harmonic oscillator:
    1. Existence of solutions.
    2. Uniqueness: How much data are required to guarantee uniqueness?
    3. Notion of general solution and particular solution.
  4. Phase space.
2. Objectives for Section 4.2:
  1. Notion of characteristic polynomial for a linear differential equation.
  2. To be able to find the general and particular solutions to a homogeneous second order linear differential equation with constant coefficients.
  3. Qualitative behavior of solutions when the roots are real.
  4. Theoretical aspects:
    1. General existence and uniqueness theorem: Generalizes the results for the harmonic oscillator.
    2. The family of solutions forms a vector space:
      1. The linear combination of general solutions is a general solution.
      2. Any solution can be written as a linear combination of the elements in a basis.
For a review of Linear Algebra, please read chapter 3.
Week 3: Sections 4.3, and 4.4.
1. Objectives for Section 4.3:
  1. Exponential of a complex number.
  2. Solution of second order linear equations when the roots of the characteristic polynomial are complex.
2. Objectives for Section 4.4:
  1. Superposition principle as a consequence of the linearity of the equation.
  2. Non-homogeneous principle.
  3. Method of undetermined coefficients.
Week 4: Sections 4.5, 4.6, and 4.7.
1. Objectives for Section 4.5:
  1. How to find particular solutions using variation of parameters.
2. Objectives for Section 4.6:
  1. A review of what we have done with the non-homogeneous equation, in the context of the harmonic oscillator.
3. Objectives for Section 4.7:
  1. Relation between the differential equations we are studying, and an associated energy.
  2. Phase space: What the energy tells us about the solution.
  3. How to convert an n-th order scalar equation into a first order system of equations of dimension n.
Week 5: Sections 5.1.
1. Objectives for Section 5.1:
  1. Definition of linear transformation.
  2. Geometry of linear transformations: How space is transformed.
  3. Matrix associated to a linear transformation between finite-dimensional spaces.
  4. Compositions of linear transformations: The matrix associated to the composition is the product of the corresponding matrices.
Week 6: Sections 5.2, 5.3.
1. Objectives for Section 5.2:
  1. Injectivity, surjectivity, and their relation to the kernel and image of a linear transformation. To be able to compute the image and the kernel of a linear map.
  2. Dimension theorem: As a consequence of this theorem, we recover the known result that a linear system of equations has either no solution, a unique solution, or infinitely many solutions.
  3. Non-homogeneous principle for linear transformations: In terms of linear systems of equations, it explains that if there are infinitely many solutions, they can be written as a particular solution, plus a vector in the kernel. The same result follows for a linear, non-homogeneous, differential equation.
2. Objectives for Section 5.3:
  1. Definition of eigenvalues and eigenvectors.
  2. How to compute the eigenvalues and eigenvectors.
  3. We will also see how knowing all the eigenvectors and eigenvalues of a matrix helps understand the associated linear map.
Week 7: Sections 5.4, 6.1.
1. Objectives for Section 5.4:
  1. How to change basis in a vector space.
  2. Matrix representation of a liner transformation in a given basis.
  3. How the matrix changes when the basis changes.
2. Objectives for Section 6.1:
  1. Existence and uniqueness results in the context of linear systems of equations.
  2. Fundamental solution matrix: What is it useful for?

ColorPalette

Thu, 03 Jan 2008 17:26:00 GMT

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GettingStarted

Thu, 03 Jan 2008 03:03:00 GMT

Welcome to the Math 5A Wiki!

To navigate this page, simply follow the links on your left.

Instructor: Who I am, and how to contact me.
Teaching Assistant: There will be one or more TAs for this course.
Location: Where the class will meet.
Syllabus: Information about the course.
Weekly Readings: Here I will post the readings for each week, and the lesson objectives. Please take a look at it before the class.
Homework: I will post the homework here.
Lecture Notes: I will post the notes from class here.
Self-Assessment Questions: This will help you study for the midterm and the final exam.
Exams: Information about the midterm and final exams.

Once you open a tiddler, you can close it by clicking on its link again. Alternatively you can use the close button over on the right.

This page is experimental, and hopefully more services will be added as the course progresses, so come back often.

PageTemplate

Thu, 03 Jan 2008 02:58:00 GMT

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Location

Thu, 03 Jan 2008 02:58:00 GMT

Tuesday, & Thursday, 8:00-9:15am, CHEM 1179

SiteSubtitle

Thu, 03 Jan 2008 02:57:00 GMT

Differential Equations and Linear Algebra
Tuesday & Thursday, 8:00-9:15am, CHEM 1179

SiteTitle

Thu, 03 Jan 2008 02:57:00 GMT

Math 5A, Winter 2008

MainMenu

Wed, 31 Oct 2007 01:36:00 GMT

Instructor
Teaching Assistant
Location
Syllabus
Weekly Readings
Homework
Lecture Notes
Self-Assessment Questions
Exams

Instructor

Tue, 25 Sep 2007 01:44:00 GMT

Carlos J. García-Cervera.

South Hall, Room 6707.
Office Phone: (805) 893 3681
Office Fax: (805) 893 2385
Office Hours: Tuesday & Thursday, 11am-12:30pm.
Main Website: http://www.math.ucsb.edu/~cgarcia

Introduction

Mon, 13 Aug 2007 21:51:00 GMT

In this course we will study mostly second order linear ordinary differential equations of the form
ay' '+by'+cy = f(x).

Since the equation is linear, it turns out that the set of solutions forms a finite dimensional vector space, which explains why we study Linear Algebra in this course.

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Sat, 11 Aug 2007 17:00:00 GMT

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