# Mathematics 421 – Advanced Calculus for Engineering (01:640:421) – Spring 2010 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)

## A detailed syllabus will evolve here.  Syllabus

This is a copy of the detailed syllabus that evolved on the Sakai site. It is similar to that used in Spring 2009, except that homework will be due in the class following the one in which it is assigned, and more quizzes will be scheduled.

1. Jan. 20 (first class): Definition of Laplace transform; calculation of transforms and inverse transforms; partial fractions.
• 4.1: 12, 26, 38.
• 4.2: 8, 16, 18.
2. Jan. 25: More calculation of transforms and inverse transforms; application to differential equations.
• 4.2: 26, 32, 38.
• 4.3: 2, 6, 18, 24.
3. Jan. 27: Quiz on 4.1 and 4.2. Piecewise definition of functions.
• 4.3: 40, 46, 56, 58, 64.
A detailed syllabus will evolve here.
4. Feb. 01: Derivatives of transforms.
• 4.4: 2, 6, 8, 12, 14.
5. Feb. 03: Quiz on applications to differential equations. Convolution; the Dirac delta function.
• 4.4: 28, 38, 40.
• 4.5: 4, 8.
6. Feb. 08: Periodic functions; systems.
• 4.4: 52, 54.
• 4.6: 2, 4, 8.
7. Feb. 10: Rutgers closed because of snow; no class.
8. Feb. 15: Quiz on the Heaviside function. Review of Laplace transforms.
9. Feb. 17: Exam 1: Laplace transforms.
10. Feb. 22: Introduction to orthogonality and Fourier series.
• 12.1: 8, 14, 18.
• 12.2: 2, 8.
11. Feb. 24: Quiz on orthogonality. Fourier series; sine series; cosine series; half-range expansions.A detailed syllabus will evolve here.
• 12.2: 10.
• 12.3: 14, 22, 26.
12. Mar. 01: Complex Fourier series; an operational approach.
• 12.4: 4, 10.
• Essay 2: A, B, C.
13. Mar. 03: Quiz on Fourier series. Sturm-Liouville problems; series solutions of differential equations.
• 12.5: 8, 10.
• 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, unless the solution is a polynomial with fewer terms".)
14. Mar. 08: Review of complex Fourier series, Sturm-Liouville problems; series solutions of differential equations.A detailed syllabus will evolve here.
• No new homework.
15. Mar. 10: Quiz on complex Fourier series. Legendre series; series solutions of differential equations based at a singular point.
• 12.6: 16, 22.
• 5.2: 18, 20, 28 (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, unless the solution is a polynomial with fewer terms". 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.)
16. Mar. 22: Review of chapter 12; introduction to partial differential equations; separation of variables.
• 12R: 8, 20, 22.
• 13.1: 8, 14.
17. Mar. 24: Quiz on series solutions of diA detailed syllabus will evolve here.  fferential equations. The Heat Equation.
• 13.2: 2.
• 13.3: 2, 4.
18. Mar. 29: The Wave Equation.
• 13.2: 8.
• 13.4: 2, 8, 14, 16.
19. Mar. 31: Quiz on Fourier-Legendre series. More on the Heat Equation and Wave Equation; Laplace's Equation.
• S4: A, B, C.
• 13.5: 4, 16.
20. Apr. 05: More on Laplace's Equation; nonhomogeneous Boundary Value Problems.
• No new homework
21. Apr. 07: Quiz on heat equation. More oA detailed syllabus will evolve here.  n nonhomogeneous Boundary Value Problems.
• No new homework
22. Apr. 12: Exam 2: Fourier series; Boundary Value Problems; Power series solutions of ordinary differential equations; Separation of Variables; Heat equation; Wave equation.
23. Apr. 14: Discussion of exam 2
• No new homework
24. Apr. 19: Nonhomogeneous Boundary Value Problems. Heat Equation and Wave Equation in rectangular regions.
• 13.6: 10, 14, 18.
• 13.8: 2.
• 13R: 12.
25. Apr. 21: Quiz on Laplace's Equation. Introduction to Vector Calculus.
• S5: A, B, C.
26. Apr. 26: Vector Calculus in general coordinate systems.
• Homework to be announced.
27. Apr. 28: Quiz on nonhomogeneous Boundary Value Problems. Wave Equation and Heat Equation in circular regions.
• No Homework.
28. May 03: General review.
• No Homework.
29. May 11: Final exam: Noon—3PM in SEC-211.

## 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. An 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 2009. The syllabus on that page may be considered to be a good approximation of the plan for this semester.