This is a very rapid plan of study. A great deal of energy and
determination will be needed to keep up with it. Modifications may be
necessary. Periodic assignments (Maple labs, workshops, etc.) may be
due at times, and additional problems may be suggested.
The text is the fifth edition of Stewart's , Brooks/Cole, 1999,
ISBN 0-534-39321-7.
Calculus: Early
Transcendentals |

Syllabus and suggested textbook homework problems for 640:251:05-10 | |||
---|---|---|---|

Lecture | Topic(s) | Relevant text sections | Suggested problems |

1 | R^{3} and vectors |
12.1 12.2 |
12.1: 3, 7, 9, 11, 17, 23, 31, 35, 37 12.2: 11, 19, 21, 25, 29, 31, 43 |

2 | Dot product Cross product |
12.3 12.4 | 12.3: 1, 5, 7, 9, 17, 19, 25,
27, 37, 41, 51, 53 12.4: 3, 9, 15, 31, 39, 45 |

3 | Lines & planes | 12.5 | 12.5: 3, 13, 17, 19, 25, 33, 35, 37, 39, 47, 53, 59 |

4 | Vector functions, curves Derivatives, integrals | 13.1 13.2 | 13.1: 1, 11, 19, 21, 23, 25 13.2: 5, 11, 17, 23, 31, 39, 47 |

5 | Arc length; curvature Motion |
13.3 13.4 | 13.3:
3, 11, 13, 19, 23, 27, 39 13.4: 13, 21, 25, 35 (13.4 to Kepler's laws) |

6 | Several variable functions Limits; continuity |
14.1 14.2 [12.6] |
14.1: 3, 5, 9, 27, 28, 43, 45, 53, 55, 57, 59 14.2: 3, 7, 13, 27, 35, 37 [12.6: 21, 23, 25, 27] |

7 | Partial derivatives Tan. planes, lin. approx. |
14.3 14.4 |
14.3: 3, 7, 9, 15, 17, 21, 41, 45, 47, 57, 67, 71, 83 14.4: 5, 11, 19, 23, 33 |

8 | The Chain Rule | 14.5 | 14.5: 3, 7, 17, 25, 29, 39, 43, 45 |

9 | Directional derivs Gradient |
14.6 | 14.6: 1, 4, 9, 13, 21, 31, 33, 35, 41 |

10 | Max/min | 14.7 | 14.7: 3, 7, 13, 39, 41, 43 |

11 | Lagrange multipliers | 14.8 | 14.8: 3, 5, 9, 19 |

12 | Exam 1 (timing approximate!) | ||

13 | Double & iterated ints | 15.1 15.2 |
15.1: 8, 13 15.2: 1, 9, 17, 23, 29 |

14 | More double integrals & in polar coordinates | 15.3 15.4 |
15.3: 1, 5, 7, 13, 17, 21, 37, 41, 43 15.4: 5, 9, 17, 21, 23, 25, 31 |

15 | Some applications Changing variables | 15.5 15.9 |
15.5: 3, 7, 12 15.9: 5, 7, 9, 11, 13, 21 (15.9 for double integrals) |

16 | Triple integrals | 15.7 | 15.7: 3, 9, 13, 17, 19, 25, 31, 35 |

17 | Cyl. & sph. coordinates Triple integrals | 12.7 15.8 |
12.7: 7, 21, 29, 35, 37, 39, 43, 55 15.8: 1, 3, 7, 13, 17, 21, 29, 33, 35 |

18 | Vector fields Line integrals |
16.1 16.2 | 16.1: 1, 5, 11, 13, 17, 21, 29, 33 16.2: 1, 5, 15, 17, 19, 21, 31, 37 |

19 | More line integrals | ||

20 | FTC for line integrals | 16.3 | 16.3: 7, 9, 13, 15, 19, 23, 29, 31, 33 |

21 | Green's Theorem | 16.4 | 16.4: 3, 9, 11, 15, 17, 19 |

22 | Exam 2 (timing approximate!) | ||

23 | Curl & divergence | 16.5 | 16.5: 3, 5, 13, 15, 25 |

24 | Parametric surfaces Surface area | 15.6 16.6 |
15.6: 1, 3, 7 16.6: 1, 3, 19, 21, 31, 45 |

25 | Surface integrals | 16.7 | 16.7: 5, 7, 11, 19, 23, 39 |

26 | Stokes Theorem | 16.8 | 16.8: 3, 5, 7, 9, 13, 15 |

27 | Divergence Theorem | 16.9 | 16.9: 3, 7, 11, 15, 27 |

28 | Catch-up; a discussion of other applications if there is time & interest. |

This is a very rapid plan of study. A great deal of energy and
determination will be needed to keep up with it. Modifications may be
necessary. Periodic assignments (Maple labs, workshops, etc.) may be
due at times, and additional problems may be suggested.
The text is the first edition of Rogawski's , W.H.Freeman, 2008, ISBN-10: 0-7167-7267-1. It
has been augmented with some Rutgers "local matter," which is also
available here.
Calculus Early
Transcendentals |

Syllabus and suggested textbook homework problems for 640:251 | |||
---|---|---|---|

Lecture | Topic(s) and text sections | Suggested problems | |

1 | 12.1 Vectors in the Plane 12.2 Vectors in Three Dimensions |
12.1: 5, 9, 11, 15, 21, 40, 47 12.2: 11, 13, 19, 25, 27, 31, 51 | |

2 | 12.3 Dot Product and the Angle Between Two
Vectors 12.4 The Cross Product |
12.3: 1, 13, 21, 29, 31, 52, 57, 63 12.4: 1, 5, 13, 20, 25, 26, 43, 44 | |

3 | 12.5 Planes in Three-Space | 12.5: 1, 9, 11, 15, 25, 31, 53 | |

4 | 13.1 Vector-Valued Functions 13.2 Calculus of Vector-Valued Functions |
13.1: 5, 13, 15, 18 13.2: 4, 14, 30, 31, 33, 41, 49 | |

5 | 13.3 Arc Length and Speed 13.4 Curvature 13.5 Motion in Three-Space |
13.3: 3, 9, 13, 14 13.4: 1, 7, 17, 21 13.5: 3, 6, 32 | |

6 | 14.1 Functions of Two or More Variables 14.2 Limits and Continuity in Several Variables |
14.1: 7, 20, 23, 27, 36, 40 14.2: 5, 15, 27, 35 | |

7 | 14.3 Partial Derivatives 14.4 Differentiability, Linear Approximation and Tangent Planes |
14.3: 3, 19, 21, 39, 47, 50, 53 14.4: 3, 4, 7, 15, 27, 33 | |

8 | 14.5 The Gradient and Directional Derivatives | 14.5: 7, 13, 27, 31, 33, 37, 39, 43 | |

9 | 14.6 The Chain Rule | 14.6: 1, 5, 7, 17, 20, 23, 27, 30 | |

10 | 14.7 Optimization in Several Variables | 14.7: 1, 3, 7, 17, 19, 24, 25, 27, 29 | |

11 | 14.8 Lagrange Multipliers: Optimizing with a Constraint | 14.8: 2, 7, 11, 13, 15 | |

12 | Exam 1 (timing approximate!) | ||

13 | 15.1 Integration in Several Variables | 15.1: 10, 15, 23, 25, 33, 37, 44 | |

14 | 15.5 Double Integrals over More General Regions | 15.2: 3, 5, 11, 25, 32, 37, 43, 45, 49, 59 | |

15 | 15.3 Triple Integrals | 15.3: 3, 5, 11, 15, 17, 25, 33 | |

16 | 12.7 Cylindrical and Spherical Coordinates 15.4 Integration in Polar, Cylindrical, and Spherical Coordinates | 12.7: 1, 5, 23, 31, 41, 43, 48, 53 15.4: 1, 5, 9, 19, 23, 27, 31, 37, 39, 42, 47, 51, 59 | |

17 | |||

18 | 15.5 Change of Variables | 15.5: 1, 5, 14, 15, 21, 29, 33, 37 | |

19 | 16.1 Vector Fields | 16.1: 1, 3, 10, 17, 23, 27 | |

20 | 16.2 Line Integrals | 16.2: 3, 9, 13, 21, 27, 35, 39, 40 | |

21 | 16.3 Conservative Vector Fields | 16.3: 1, 5, 9, 13, 17, 19, 21 | |

22 | Exam 2 (timing approximate!) | ||

23 | 16.4 Parameterized Surfaces and Surface Integrals | 16.4: 1, 5, 8, 11, 19, 21, 37 | |

24 | 16.5 Surface Integrals of Vector Fields | 16.5: 1, 6, 9, 12, 15, 17, 23 | |

25 | 17.1 Green's Theorem | 17.1: 1, 3, 6, 9, 12, 23, 27 | |

26 | 17.2 Stokes' Theorem | 17.2: 1, 5, 9, 11, 19, 23 | |

27 | 17.3 Divergence Theorem | 17.3: 1, 5, 7, 11, 14, 15, 18 | |

28 | Catch up; review; possible discussion of some applications of vector analysis. |

**General background**

I worked out the *draft* of this syllabus the week after final
exams were given for spring 2008. This is definitely a first try at
constructing a syllabus for Math 251 from Rogawski's text. Let me
discuss some ideas which I followed in selecting problems.

We do not teach "reformed" calculus (at least, not systematically reformed!) here. So most of the problems suggested are of this type: state a problem involving formulas and then answer the problem, almost always with other formulas or with a number. Most of the problems I selected were of this type. They were selected from the earlier problems in each exercise set (I almost always included problem 1!). I did try to distribute the formula types (polynomial, rational, exponential and logarithmic, trig and sometimes inverse trig). I also tried to select odd-numbered problems, since the answers can then be checked by students. I did, however, include some even-numbered problems which appeared particularly appropriate. There are certainly many similar even-numbered problems to assign as homework to be handed in if this is desired. I suggested some graphical or numerical problems if they seemed very useful but I did not include many. I believe C-level students should be able to successfully complete most of the suggested problems.

This is a first edition. Although it has been checked for errors, some have still not been detected and corrected. Please inform me of errors in problems or text. Also, certainly, please please please tell me where the syllabus does not work so that I can fix it!

Each section of the text has a very short SUMMARY subsection immediately before the EXERCISES. Although I am an experienced calculus instructor, I usually found consulting at least the summary subsection useful before preparing classes in both 151 and 152, and I will continue doing this when I teach 251. We don't need to repeat what is in the book always, but I think we should be consistent (notation, names, most ideas) with the textbook in these courses.

What follows are notes regarding specific topics and lectures,
sometimes comparing the previous text (Stewart) with the new text
(Rogawski). You may also wish to consult the course web page for my
most recent "instantiation" of 251. This has links to detailed lecture
notes, workshops (!), exams and exam solutions, formula sheets, etc. I
hope you will find some of this material useful.

http://www.math.rutgers.edu/~greenfie/mill_courses/math251/math251_index.html

`Maple` labs and workshops

The course has four suggested `Maple` labs
during the standard semester, in addition to a `Maple` lab 0 which is introductory and should be
help in the first week or two. During the summer we've asked
instructors to try to have students do two of the suggested
labs. Maybe labs 2 and 4, about curvature and centers of mass,
respectively, would be the best choices. Lab 3 is maybe not too
interesting except for its discussion of quadric surfaces, and I hope
to write an alternative to it during the coming summer.

In the standard semester, I also had students hand in three workshop
problems. See the web page above for more details. I wanted students
to continue improving their skills in technical writing.

**Note about lectures 1 through 3**

Much of this material is familiar to students who have good physics
and engineering background. Engineers especially may have seen much of
this material in three previous courses! Almost paradoxically,
students with *only* a standard math background will likely have
not seen this before. By the end of the first three lectures, the hope
is that all students will be familiar with basic algebra of vectors
(addition, the multiplications, etc.) and with simple applications to
algebraic descriptions of lines and planes.

Please notice that the syllabus omits section 12.6, A Survey of
Quadratic Surfaces. The ideas concerning quadratic surfaces are
actually addressed in the third `Maple` lab, and
certainly some knowledge of quadratic surfaces is useful when
considering the graphs of functions of several variables and studying
critical points. Although I will omit this section formally, I am
certain that I would introduce appropriate examples and names early in
the course.

**Note about lecture 5**

The familiarity of arc length and some other material builds on
examples in Math 152. Covering all of the material in these three
sections in class is nearly impossible. Minimally, students need some
idea of curvature, and they should associate the geometry of curvature
with the tangential and normal components of the acceleration vector
(p.774, Understanding the Acceleration Vector).

**New versus old: lectures 8 and 9**

Rogawski's text introduces gradient
*before* the general chain rule. Therefore students following
Stewart's text should be advised of the difference.

**Lecture 12, the suggested first exam**

Your own class's circumstances will certainly determine the timing of
exams. This syllabus is aimed at standard semesters, not summer
semesters, which are far more crowded and busy. In a standard
semester, I like to have a first exam near the end of the first 5
weeks. The first exam grade helps me assign warning grades, and I
assume that grading the first exam will take 3 to 5 days.

**New versus old: lectures 13 through 18**

The Stewart syllabus spends 5 lectures on multiple integrals. 6
lectures are devoted to this topic in the Rogawski syllabus and there
are slight changes in order. A `Maple` lab
computes centers of mass, so perhaps one or two examples of this
computation could be done "by hand" during these lectures. Some of the
material about cylindrical and spherical coordinates could be
discussed earlier if there is time (joke!).

I would systematically discuss change of variables for multiple
integrals only for double integrals and mention that these
considerations are also valid for "higher" multiple integrals
(yielding, for example, the spherical coordinates Jacobian).

Surface integrals also have similar (related) formulas, of course.

One additional lecture is given here about multiple integrals. What's
lost from the Stewart syllabus? One lecture (#23) is devoted to curl
and divergence which are, more or less, included as they are needed in
the lectures covering chapter 17, Fundamental Theorems of Vector
Analysis. I think this will work.

**Lecture 22, the suggested second exam**

Again this timing is approximate. In a standard (non-summer) semester,
I try to give the second exam far enough away (!) from the final so
that I can grade and return the exam, and the students would be able
to not repeat errors.

**Lecture 28**

I've taught Math 251 many times. I've found the huge number of ideas
(o.k., darn it, *plethora*!) difficult to cover adequately. Many
times I've had to reduce coverage of the vector analysis results or
change of variables for multiple integrals or ... something. When I've
had time, I've tried to give people some idea how the big results are
used. So I've been able to discuss a derivation of the heat equation a
few times. Sometimes I've discussed Gauss's Law (flux from a point
source), and once I discussed Archimedes' Principle and buoyancy. Some
sort of payoff, if possible, is nice.

**
Maintained by
greenfie@math.rutgers.edu and last modified 5/22/2008.
**