# Rutgers Math 421 – Advanced Calculus for Engineering

## General Information

This page has been revised to reflect a change of textbook. The version used with the previous textbook is also available. Other information about both the history and the future of the course can be found on the main course page.

## Current Semester

• Section 1 meets MTh3 in SEC 216.
• Section 2 meets TTh6 in SEC 216.
• Section 3 meets MW7 in Hill 124.

## Textbook and Syllabus

Textbook: Peter V. O'Neil; Advanced Engineering Mathematics (fifth edition); Brooks/Cole, 2003 (1236+82[Answers]+9[Index] pp.); (ISBN# 0-534-40077-9)

A list of errata appears below the syllabus.

Syllabus: Since this is the first use of the current textbook, this is a tentative syllabus. It is intended mainly to guide instructors in planning. The selection of topics is based on those used with the previous text. There is also a list of suggested homework problems for each section mentioned in the syllabus.

The order of the material may also be changed. Individual Section (or semester) pages should contain current details.

Session Section in Text
or other Activity
Topics
1
3.1
Laplace transforms: Basic Properties.
2
3.2
Initial Value Problems.
3
3.3
Shifting Theorems; Heaviside Functions.
4
3.4
Convolution.
5
3.5
Impulses and Delta Functions.
6
3.6
Solution of Systems.
7
3.7
Equations with Polynomial Coefficients.
8
13.1
13.2
Fourier Series: Introduction.
9
13.4
Cosine and Sine Series.
10
13.5
integration and Differentiation of Fourier Series.
11
Review.
12
First Midterm Exam.
13
16.1
16.2
The wave equation;
Fourier series solution.
14
16.2
16.4
Fourier series and d'Alembert
solutions of the wave equation.
15
16.7
Vibrations of a Rectangular Membrane.
16
17.1
17.2
The heat equation;
Fourier series solution.
17
17.2
17.5
Heat conduction in interval
and rectangle.
18
Review.
19
10.1
10.2
Nonlinear systems;
Phase planes.
20
10.3
Phase portraits of linear systems.
21
10.3 (continued)
More phase portraits of linear systems.
22
Second Midterm Exam
23
10.4
Critical points; stability.
24
10.5
Almost linear systems.
25
Instructor's
Choice
26
27
28

Here are some errors (or infelicities) in O'Neil 5th edition discovered by students and instructors in the course of using the text at Rutgers.

• p 114 In Example 3.2, third line after displayed equations The statement "G(s) is defined for all real s" is misleading or worse, since the calculation is correct only for s positive.
• p 450 4th line from bottom "not stable" should read "unstable".
• p 639 top - Figure 13.10 The leftmost of the five connected pieces of the graph (of which one is an isolated point and the four others are arcs of curves) has noticeably incorrect ordinate at its maximum and at its minimum. Also, the abscissa of the isolated point (which is also the abscissa of ends of two of the arcs) is slightly but not unnoticeably wrong.
• p 868 line after third displayed equation from bottom "T(0)=b=0" should read "T(0)=a=0".
• p 911 third displayed equation from bottom "Y_m(x)" should read "Y_m(y)".
• p A59 in the answers to the odd-numbered problems from 1 to 19 "y(x,t)" should read "u(x,t)".
• p A62 second line from bottom In the answer to problem 3 of section 16.7, "cos" should be replaced by "sin"

Here are similar notes about the accompanying Solutions Manual.
• p 385 lower half of page, in solution to problem 1 of section 16.2 The formulas for b_n and its simplified version for b_{2n} are wrong by an overall sign, for instance they both give b_2 < 0 , whereas b_2 is (I think) manifestly positive; consequently the final formula for y(x,t) is also wrong by an overall sign.
• p 386 in last line of answer to problem 3 (of section 16.2) "cos" should be replaced by "sin"
• p 414 in last line (displayed) of answer to problem 3 of section 16.4 The second occurrence of the fraction 1/2 (right after the second equals sign) should be deleted.

## Archives

This text seems more efficient, so space is left at the end for additional topics, indicated by the phrase "Instructor's Choice". Possible additional topics are

• (A) 4.1 - 4.3: Series solutions of Differential Equations.
• (B) 10.6 - 10.9: Special systems of Differential Equations.
• (C) 14.1 - 14.3: The Fourier Integral.
• (D) 16.5 - 16.6: Vibrations of a circular membrane (requires (A)).
• (E) 17.3: Heat conduction in infinite media (may require (C)).
• (F) 18.1 - 18.2: Dirichlet's Problem