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- Mathematics 341 Download Page (Spring 2004)
- Differential Equations and Linear Algebra 4th Edition Book
- Differential Equations and Linear Algebra, 4th Edition
- Mathematics 341 Download Page (Spring 2004)
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Mathematics 341 Download Page (Spring 2004)
For use in this course, see below. Textbook coverage: Chapters The following core sections are recommended; optional sections are indicated with the square bracket notation [ First-Order Differential Equations. Mathematical Models and Numerical Methods. Linear Systems and Matrices. Vector Spaces. Linear Equations of Higher Order. Eigenvalues and Eigenvectors. Linear Systems of Differential Equations. Exercise problem numbering is the same for editions Edition text differences are not very significant.
Coverage of the syllabus is a tricky problem here because combining the two topics of differential equations and linear algebra together in one semester requires cutting interesting parts of both.
However, this is a terminal course so it is perhaps more important that those topics which are covered convey the enthusiasm of the instructor and accomplish student learning, even at the sacrifice of giving less attention to other parts of the syllabus particularly those which occur at the end of the semester when time is running out. Thus the individual instructor must decide how to streamline certain parts of the syllabus in order to compensate for the extra attention given to others.
Sections which are marked as optional can be mined for whatever interesting example that appeals to a given instructor, if desired, but sparingly.
One can easily overspend time in the first two chapters which is full of applications and optional material, so one must take care to pick wisely. One can streamline the first three sections of chapter 3, emphasizing the rref reduction and relying on MAPLE optionally graphing calculators to perform the reduction in practice.
Interpretation of the rref form is more important than the distinction between Gauss and Gauss-Jordan reduction, and hand computing determinants can also be de-emphasized. Section 4. An inconsistent system means this is not possible. When the columns are linearly independent, one finds a unique solution if one exists. In chapter 5, one can give section 2 light treatment, omitting Wronskians or explaining them in the context of the matrix of coefficients in solving the initial condition system , and one can lighten the undetermined coefficient section 5 by not dwelling too much on the most general case.
A very nice interactive Duke University on-line applet really drives home what eigenvectors and eigenvalues mean visually in the 2x2 matrix case. In chapter 7, one can omit the subsection "simple 2-D systems" in section 1 and only expose students to the idea of reduction of order being necessary to reduce coupled damped oscillator systems to first order form.
No need to worry about Wronskians in 7. Note that students have trouble with complex arithmetic algebra! The second order undamped multiple spring systems should be covered as the final topic, since this unites both chapter 5 and the eigenvector technique, tying together the major concepts of the course and provides a toy system with some connection to everyday intuition.
In my mind chapters 3 and 4 for matrix manipulations and then 5, 6 and 7 for second order DEs and systems of DEs are the meat of this course. Chapters 1 and 2 overlap with most Calc2 courses where first order differential equations are usually already covered, although the one new idea encountered immediately there that students have a hard time digesting is that to check a solution of a DE, one must replace the unknown everywhere in the DE by the expression for it in terms of the independent variable, and then simplify till the left and right hand sides agree; if they are not equivalent using algebra, it is not a solution.
Chapter 2 allows some useful practice at modeling problems. For the new section 7. Because of the increasing number of freshmen with advanced credit, little Maple knowledge can be assumed but this is not a significant problem with the increasingly user friendly "clickable calculus" interface philosophy which sidesteps syntax and commands.
Introducing MAPLE example and template worksheets associated with textbook homework problems to use MAPLE as a minimal support tool is a good idea, allowing graphing calculators or Maple to substitute for some required mechanical steps in quizzes and tests, as well as to check any hand calculations requested.
Most calculators can now do row reductions. It is more important to have students use a limited number of MAPLE evaluation tasks on regular homework problems assigned from the textbook. In particular, the solution of any set of DEs plus initial conditions should be always available as a check for all students.
This course is not about differentiation or integration but what to do with the derivative and integral in the context of differential equations. Every student should learn how to solve a differential equation with initial condition s immediately to be able to quickly check any hand solutions see below for syntax , and should also be using Maple or a graphing calculator to check every antiderivative, and indeed provide the antiderivative if the integration is anything but trivial.
This is very easy in Standard Maple with its clickable calculus interface. In chapter 1, DEplot for directionfields and dsolve for exact solutions Project 1. Technology should be emphasized for doing the row operations, since it is extremely difficult to do all the arithmetic in row reduction by hand correctly and arithmetic is not the point: the sequence of row operations is. Later in the course solving the linear system is not the main point, and the complete row reduction should be done at once with technology.
Students should know how to compute determinants with technology so they can use their values to draw conclusions, after having at least one technology experience using row reduction without MultiplyRow to evaluate a determinant. Then in chapter 6 the eigenvector tutor and right click eigenvector evaluation should be introduced. In chapter 5, solve and fsolve or right click access to them should be used for higher order even quadratic! In chapter 6 and 7, the DEplot command should be extended from chapter 1 to include the phase plane plots for 2-D linear systems and used to motivate and visualize eigenvectors using 2x2 matrices.
Maple specific hints:. For stating differential equations using prime notation, the default differentiation variable x is assumed. Don't waste time using subscripted variables like x 1 with prime notation, just call it x1 although the subscripted variable names will work. Matrices can be entered with the Matrix palette. A superscript of -1 will produce the inverse of a square matrix, while a space " " between matrices will multiply them, without loading the LinearAlgebra or Student[LinearAlgebra] packages.
Right-clicking on a matrix and selecting Standard Operations allows the determinant to be evaluated. Selecting Eigenvalues, etc allows one to get the eigenvalues and eigenvectors from the single choice Eigenvectors , or the preliminary characteristic polynomial. Right-clicking on a matrix and selecting Solvers and Forms , then Row Echelon Form , then Reduced will give the row reduced echelon form of a matrix.
This eliminates the mistakes from many arithmetic steps which are not what humans are meant to do except for very small matrices. Students must be reminded to explicitly type an asterisk or leave a space to imply multiplication of two quantities in Maple, and that Euler's number is not obtained by typing the letter "e", but the Expression palette or exp x notation must be used to insert an exponential expression.
For more useful interface hints see Maple Examples and Tips. Villanova University was founded in by the Order of St. MAT Syllabus. Syllabus Comments by Course Coordinator Coverage of the syllabus is a tricky problem here because combining the two topics of differential equations and linear algebra together in one semester requires cutting interesting parts of both.
Maple Comments. Maple specific hints: For stating differential equations using prime notation, the default differentiation variable x is assumed.
Differential Equations and Linear Algebra 4th Edition Book
Henry Edwards is emeritus professor of mathematics at the University of Georgia. He earned his Ph. Sloan Research Fellow. He has received numerous teaching awards, including the University of Georgia''s honoratus medal in for sustained excellence in honors teaching , its Josiah Meigs award in the institution''s highest award for teaching , and the statewide Georgia Regents award for research university faculty teaching excellence. His scholarly career has ranged from research and dissertation direction in topology to the history of mathematics to computing and technology in the teaching and applications of mathematics. In addition to being author or co-author of calculus, advanced calculus, linear algebra, and differential equations textbooks, he is well-known to calculus instructors as author of The Historical Development of the Calculus Springer-Verlag,
Differential Equations and Linear Algebra, 4th Edition
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Mathematics 341 Download Page (Spring 2004)
For use in this course, see below. Textbook coverage: Chapters The following core sections are recommended; optional sections are indicated with the square bracket notation [ First-Order Differential Equations. Mathematical Models and Numerical Methods.
by C. Henry Edwards and David E. Penney. Differential Equations: Computing and Modeling, Fourth Edition. Pearson Custom Publishing treatment of the linear algebra that is needed, and then presents the eigenvalue approach to linear.
Henry Edwards, David E. Penney, David Calvis. Now available, Expanded Applications, an online companion manual containing expanded applications and programming tools.
Edwards and D. Penney and D. You may opt out by following the instructions here. This is a hybrid course which teaches the allied subjects of linear algebra and differential equations. Engineering students interested in pursuing further mathematics courses can elect to earn a minor in Mathematics by taking additional courses.
The right balance between concepts, visualization, applications, and skills Differential Equations and Linear Algebra provides the conceptual development and geometric visualization of a modern differential equations and linear algebra course that is essential to science and engineering students. It balances traditional manual methods with the new, computer-based methods that illuminate qualitative phenomena — a comprehensive approach that makes accessible a wider range of more realistic applications. The book combines core topics in elementary differential equations with concepts and methods of elementary linear algebra.
Updated: Mar 15, Henry Edwards is emeritus professor of mathematics at the University of Georgia. October 21,
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