**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
01:640:423Laplace 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). |

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:

- Laplace transforms (most of chapter 4 of the text).
- 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.
- 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).

**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.

**Text**

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 Amazon.com 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.

**Technology**

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 x^{3}sin(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.

**Grading**

**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: greenfie@math.rutgers.edu.
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
greenfie@math.rutgers.edu and last modified 8/30/2004.
**