# Mathematics 421 – Advanced Calculus for Engineering (01:640:421) – Spring 2009 Section 01 – Professor Bumby

## General Information

See the main course page for the background of the course.

See the instructor's home page for contact information and office hours of Prof. Bumby.

## Current Semester:

The course will use Sakai for all material during the semester. All enrolled students should have automatic access to the site after logging in to Sakai. Current information about syllabus and homework will be found there. Selected material from the Sakai site will be transferred to this page for archival purposes at the end of the course.

## Textbook

Dennis G. Zill and Michael R. Cullen ; Advanced Engineering Mathematics (third edition); Jones and Bartlett, 2006; (ISBN# 0-763-74591-X)

## Syllabus

This is a copy of the detailed syllabus that evolved on the Sakai site. Each entry shows the lecture date and sections discussed with a few homework problems. Homework was due two class meetings following the assignment date, allowing for questions about the assignment to be discussed in one class meeting between assignment and collection. Only a few problems were assigned to be handed in, but students were encouraged to do similar exercises for practice (and questions about these practice exercises may be raised in class).

1. Jan. 21: Definition of Laplace transform; calculation of transforms and inverse transforms; partial fractions.
• 4.1: 12, 26, 38.
• 4.2: 8, 16, 18.
2. Jan. 26: Laplace transform of derivatives; application to differential equations; first shifting theorem.
• 4.2: 26, 32, 38.
• 4.3: 2, 6, 18, 24.
3. Jan. 28: Piecewise defined functions.
• 4.3: 40, 46, 56, 58, 64.
4. Feb. 02: Quiz on Lecture 1. Derivatives of transforms, Convolution, Volterra equations, periodic functions.
• 4.4: 8, 14, 22, 28, 38, 40, 52, 54.
5. Feb. 04: The Dirac delta function, Systems.
• 4.5: 2, 6, 8.
• 4.6: 2, 8.
6. Feb. 09: Quiz on Lectures 2 and 3. Review of Laplace transforms.
• 4.R: 10, 20, 34, 36, 38. (due Feb. 18)
7. Feb. 11: Further review of Laplace transforms. Introduction to Fourier series.
• 12.1: 8, 14, 18. (due Feb. 23)
8. Feb. 16: Exam on Laplace transforms.
9. Feb. 18: Direct calculation of Fourier series.
• 12.2: 2, 8, 10, 16.
10. Feb. 23: Sine series, cosine series, half range expansions. Complex series. Frequency spectrum.
• 12.3: 14, 22, 26.
• 12.4: 4, 10.
11. Feb. 25: Overview of Fourier series.
• Supplement 2: A, B, C.
12. Mar. 02: Rutgers closed by snow storm. No class.
13. Mar. 04: Quiz on 12.1 and 12.2. Boundary value problems.
• 12.5: 8, 10.
• 12.6: 16, 22.
14. Mar. 09: Review of chapter 12.
• 12.R: 8, 14,20, 22 (due March 23).
15. Mar. 11: Introduction to linear partial differential equations.
• 13.1: 4, 8, 14 (due March 25).
• 13.2: 2, 8 (due March 25).
16. Mar. 23: Separation of variables solution of the heat and wave equations.
• 13.3: 2, 4.
• 13.4: 2, 8.
17. Mar. 25: Further properties of the wave equations.
• Supplement 4: A, B, C.
• 13.4: 14, 16.
18. Mar. 30: Laplace's equation in a rectangle.
• 13.5: 4, 10, 16.
19. Apr. 01: Quiz on 13.1 and 13.2. Non-homogeneous boundary value problems.
• 13.6: 10, 14, 18.
20. Apr. 06: Fourier series in two variables.
• 13.7: NONE.
• 13.8: 2.
• 13.R: 12.
21. Apr. 08: Series solutions of differential equations.
• 5.1: 20, 24, 30 (For 20 and 24, solutions should be written as a sum of two special solutions, one solution with y(0)=1 and y'(0)=0, and a second solution with y(0)=0 and y'(0)=1. For all problems, "solution" should be interpreted as "the first four nonzero terms of the series solution", which you may follow by "..." if you like.) Due Apr. 20.
22. Apr. 13: Series solutions of differential equations at regular singular points.
• 5.2: 18, 20, 24. (For these problems, "solution" should be interpreted as "the first four nonzero terms of the series solution", which you may follow by "..." if you like. Showing the series automatically gives the solutions of the indicial equation and allows a trivial verification that the roots do not differ by an integer. You should do those steps, but it is not necessary to say that you did them.) Due Apr. 22.
23. Apr. 15: Exam on chapter 12 and sections 13.1 - 4. Topics are Fourier series, Sturm-Liouville problems, separation of variables applied to Partial Differential Equations with emphasis on heat and wave equations..
24. Apr. 20: A look back at Exam 2.
• Exam problems 1 and 2: Redo problem with lower score.
• Exam problems 4 and 5: Redo problem with lower score.
25. Apr. 22: Introduction to Vector Calculus.
• 9.5: NONE.
• 9.8: NONE.
• 9.9: NONE.
• 9.12: NONE.
• Supplement 5a: A, B, C.

## Supplements

The following supplements produced during the term have been copied here.

1. Overview of Laplace transforms. An overview of the properties of Laplace transforms.
2. An operational view of Fourier series. A (not completely successful) attempt to apply the operational method used for working with Laplace transforms to Fourier coefficients. The text insists on explicit evaluation of the integrals for the Fourier coefficients every time they are encountered, instead of remembering previous results and quting them when a similar quantity is to be expanded in a Fourier series. In particular, the use of integration by parts to find the Fourier coefficients a function in terms of the coefficients of its derivative exposes one to errors that are easily avoided.
3. Boundary Value Problems. An expanded treatment of the Sturm-Liouville theory. The treatment in the text was far too brief, so these notes gave more details about the eigenfunction expansions that would be used in the solution of the classical partial differential equations.
4. Partial Differential Equations. Additional information about the wave equation and heat equation in one dimension. In addition to results obtained by the method of separation of variables, there is a description of d'Alembert's solution.
5. Vector Calculus (part 1). A first attempt at describing "div, grad, curl, and all that" (as the subject is characterized in the title of a book on the subject). This provides the background for the appearance of the Laplacian in other coordinate systems.

Other supplements dealt with solutions of individual homework exercises. They will not be made available outside of the Sakai site.

## History

There is a similar version of the course from Spring 2008. The syllabus on that page may be considered to be a good approximation of the plan for this semester.