## Syllabus & textbook homework for Math 251, spring 2008

 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 Calculus: Early Transcendentals, Brooks/Cole, 1999, ISBN 0-534-39321-7.

Syllabus and suggested textbook homework problems for 640:251:05-10
LectureTopic(s)Relevant text sectionsSuggested problems
1 R3 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
2Dot 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
3Lines & 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
5Arc 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]
7Partial 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 Rule14.5 14.5: 3, 7, 17, 25, 29, 39, 43, 45
9 Directional derivs
14.6 14.6: 1, 4, 9, 13, 21, 31, 33, 35, 41
10Max/min 14.7 14.7: 3, 7, 13, 39, 41, 43
11 Lagrange multipliers14.8 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 integrals16.3 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.

## Syllabus & textbook homework for Math 251, summer 2008

 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 Calculus Early Transcendentals, W.H.Freeman, 2008, ISBN-10: 0-7167-7267-1. It has been augmented with some Rutgers "local matter," which is also available here.

Syllabus and suggested textbook homework problems for 640:251
LectureTopic(s) and text sectionsSuggested problems
112.1 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
212.3 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
312.5 Planes in Three-Space 12.5: 1, 9, 11, 15, 25, 31, 53
413.1 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
513.3 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
614.1 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
714.3 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
814.5 The Gradient and Directional Derivatives 14.5: 7, 13, 27, 31, 33, 37, 39, 43
914.6 The Chain Rule 14.6: 1, 5, 7, 17, 20, 23, 27, 30
1014.7 Optimization in Several Variables 14.7: 1, 3, 7, 17, 19, 24, 25, 27, 29
1114.8 Lagrange Multipliers: Optimizing with a Constraint 14.8: 2, 7, 11, 13, 15
12Exam 1 (timing approximate!)
1315.1 Integration in Several Variables 15.1: 10, 15, 23, 25, 33, 37, 44
1415.5 Double Integrals over More General Regions 15.2: 3, 5, 11, 25, 32, 37, 43, 45, 49, 59
1515.3 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
1815.5 Change of Variables 15.5: 1, 5, 14, 15, 21, 29, 33, 37
1916.1 Vector Fields 16.1: 1, 3, 10, 17, 23, 27
2016.2 Line Integrals 16.2: 3, 9, 13, 21, 27, 35, 39, 40
2116.3 Conservative Vector Fields 16.3: 1, 5, 9, 13, 17, 19, 21
22Exam 2 (timing approximate!)
2316.4 Parameterized Surfaces and Surface Integrals 16.4: 1, 5, 8, 11, 19, 21, 37
2416.5 Surface Integrals of Vector Fields 16.5: 1, 6, 9, 12, 15, 17, 23
2517.1 Green's Theorem 17.1: 1, 3, 6, 9, 12, 23, 27
2617.2 Stokes' Theorem 17.2: 1, 5, 9, 11, 19, 23
2717.3 Divergence Theorem 17.3: 1, 5, 7, 11, 14, 15, 18
28Catch 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.

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.