Mathematics 421 – Advanced Calculus for Engineering
(01:640:421) – Spring 2009
Section 01 – Professor Bumby
See the main course page for the background of
See the instructor's home page for contact
information and office hours of Prof. Bumby.
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.
Dennis G. Zill and Michael R. Cullen ;
Advanced Engineering Mathematics (third edition);
Jones and Bartlett, 2006;
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).
- Jan. 21: Definition of Laplace transform; calculation of
transforms and inverse transforms; partial fractions.
- 4.1: 12, 26, 38.
- 4.2: 8, 16, 18.
- Jan. 26: Laplace transform of derivatives; application to
differential equations; first shifting theorem.
- 4.2: 26, 32, 38.
- 4.3: 2, 6, 18, 24.
- Jan. 28: Piecewise defined functions.
- Feb. 02: Quiz on Lecture 1. Derivatives of transforms,
Convolution, Volterra equations, periodic functions.
- 4.4: 8, 14, 22, 28, 38, 40, 52, 54.
- Feb. 04: The Dirac delta function, Systems.
- Feb. 09: Quiz on Lectures 2 and 3. Review of Laplace
- 4.R: 10, 20, 34, 36, 38. (due Feb. 18)
- Feb. 11: Further review of Laplace transforms. Introduction to
- 12.1: 8, 14, 18. (due Feb. 23)
- Feb. 16: Exam on Laplace transforms.
- Feb. 18: Direct calculation of Fourier series.
- Feb. 23: Sine series, cosine series, half range expansions.
Complex series. Frequency spectrum.
- 12.3: 14, 22, 26.
- 12.4: 4, 10.
- Feb. 25: Overview of Fourier series.
- Mar. 02: Rutgers closed by snow storm. No class.
- Mar. 04: Quiz on 12.1 and 12.2. Boundary value problems.
- 12.5: 8, 10.
- 12.6: 16, 22.
- Mar. 09: Review of chapter 12.
- 12.R: 8, 14,20, 22 (due March 23).
- Mar. 11: Introduction to linear partial differential equations.
- 13.1: 4, 8, 14 (due March 25).
- 13.2: 2, 8 (due March 25).
- Mar. 23: Separation of variables solution of the heat and
- Mar. 25: Further properties of the wave equations.
- Supplement 4: A, B, C.
- 13.4: 14, 16.
- Mar. 30: Laplace's equation in a rectangle.
- Apr. 01: Quiz on 13.1 and 13.2. Non-homogeneous
boundary value problems.
- Apr. 06: Fourier series in two variables.
- 13.7: NONE.
- 13.8: 2.
- 13.R: 12.
- 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.
- Apr. 13: Series solutions of differential equations at regular
- 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.
- 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..
- 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.
- Apr. 22: Introduction to Vector Calculus.
- 9.5: NONE.
- 9.8: NONE.
- 9.9: NONE.
- 9.12: NONE.
- Supplement 5a: A, B, C.
The following supplements produced during the term have been copied here.
- Overview of Laplace transforms. An overview of
the properties of Laplace transforms.
- 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.
- 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
- 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.
- 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.
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.
Comments on this page should be sent to: bumby AT math.rutgers.edu
This file was last modified on
Tuesday August 01, 2017.