General information about Math 421, fall 2004

Math 421
This is the catalog description of the course:

01:640:421. Advanced Calculus for Engineering (3)
Primarily for mechanical engineering majors. Prerequisite: CALC 4. Credit not given for both this course and 01:640:423
Laplace transforms, numerical solution of ordinary differential equations, Fourier series, and separation of variables method applied to the linear partial differential equations of mathematical physics (heat, wave, and Laplace's equation).
The course develops an assortment of topics which are necessary for advanced courses in the current Mechanical Engineering (650) undergraduate curriculum. The assortment of topics chosen this semester will be different from the catalog description but similar to what was done last spring. Further discussions with Professor A. Norris and Professor A. Cuitino, who are, respectively, the chair and vice-chair for the undergraduate program of Mechanical Engineering, have endorsed this temporary change.

Very recently Math 421 has also been made a required course for Chemical Engineering (155). Professor Davidson, who is teaching the transport sequence (303-304), usually taken in the junior year, has urged students to take Math 421 no later than the semester in which 303 is taken. Math 421 is also useful for Process Control.

The course will have three parts:

  1. Laplace transforms (most of chapter 4 of the text).
  2. Further topics from linear algebra (most of chapter 8 of the text). Although there is some coverage of linear algebra in Math 244 (the CALC 4 course usually taken by engineering majors) experience has shown that this is insufficient for Mechanical Engineering students. They need to take advantage of symmetries (eigenvalues, etc.) and to know when and how to solve systems of linear equations. For example, this material is useful to know when applying the Finite Element Method.
  3. Fourier series and applications to boundary value problems, including separation of variations for the heat and wave equations (principally material selected from chapters 12 and 13 of the text).
Professors Norris and Cuitino remarked that knowledge of these topics is quite important for the required courses 650:443 (Vibrations and Controls) and 650:481 (Heat Transfer), as well as other optional courses in the 650 curriculum. Projects in the Senior Design course are also likely to use these topics. Some previous instructors of Math 421 and some former students were also consulted about the current syllabus of the course.

Further changes ...
It is likely that there will be further changes in the Mechanical Engineering curriculum in the next year or so. At that time, this course will also change in order to better support the education of Mechanical Engineering students.

The text, new for this semester, is Advanced Engineering Mathematics (second edition) by Dennis G. Zill and Michael R. Cullen. It is published by Jones and Barlett, 2000 and has 926+95 [Appendices, Answers and Index] pages (ISBN# 0-763-71065-2).
The book is for sale at the Rutgers University bookstore for $133. It can be bought at with free shipping for $110.
This is a very large book. Only a few of its 20 chapters will be covered. Other sections of the book will be useful in other courses, and in other parts of students' careers.
Warning As with all long and technical texts, there are misprints. Please read the book carefully.

Many of the computations needed to apply the techniques of this course are quite elaborate. Therefore such software packages as Matlab and Maple (and others) include many special functions designed to handle these techniques. While we (strongly!) encourage students to use these programs, course exams and most homework should be done by hand. The exams will be designed to avoid elaborate and tedious computation. Appropriate use of technology is important, and, just as students should recognize that the antiderivative of x3sin(5x) is not likely to be exp(17x) (!), enough facility with "hand computation" should be developed so that students can check (approximately and appropriately) Laplace transform, Fourier series, and linear algebra computations.

Formal exams Several formal exams will be given during classes. These exams will be announced in advance. There will be a three-hour final exam. Some formula sheets may be used during portions of the exams. The times of the exams and the format will be assigned in advance.
Homework Students should do homework. Several problems will be collected each week. While we encourage students to work together studying the material, homework should be written up independently.
Informal quizzes Informal quizzes may be given in any class. The results of these quizzes will not be major components of the course grade, but may be useful to both the instructor and the student regarding progress in the course.
A precise formula? I don't have an exact formula for grades yet. Last semester (spring 2004) I gave three exams which counted for equal weight and got another score from homework and class quizzes which was averaged with the three exams.

Office hours
My office is in Hill Center: Hill 542, telephone number: (732) 445-3074. I will have formal office hours, to be announced. I usually check e-mail several times a day so it is probably the best way to communicate with me: I also encourage you to ask questions via e-mail or after almost any class or or at office hours or to make an appointment at a mutually convenient time.

Other references
Much of the material covered in this course has been an important part of scientific and engineering education for a century. The amount of literature available is extraordinary. For example, on 1/19/2004 Google reported about 48,300 web pages in response to the query Laplace transform while Amazon had 889 results under books and Laplace transform. Students who learn of useful references (especially interactive web pages) are encouraged to report them to their instructor.

Maintained by and last modified 8/30/2004.