# Multivariable Mathematics I

**Objectives of the course:** The aim of the
Multivariable Mathematics sequence is to help you achieve proficiency in the
areas of multivariable calculus, linear algebra, and differential equations. You
will also be challenged to improve your mathematical writing skills, your
mathematical reading comprehension and your proficiency with mathematics-related
technology. The course is also designed to help you gain maturity in problem
solving, mathematical intuition and abstract mathematical thinking.

**Material:** Multivariable Mathematics I, covers the
following topics in linear algebra and differential equations:

• vectors and geometry

• linear algebra (vector spaces, linear independence, bases, dimension)

• matrix algebra (systems of equations, row reduction, matrix operations,
inverses, determinants, linear mappings)

• constant coefficient differential equations and applications

Multivariable Mathematics II covers multivariable
calculus, including

• linear systems of differential equations

• more matrix algebra (finding eigenvectors, matrix exponentials)

• differential and integral multivariable calculus with applications

• line integrals, surface integrals, and calculus on vector fields

**Credit for Math 231: **Math 231 counts as 4 credits
toward a mathematics major or minor.

**Prerequisite:** In order to enroll in Math 231, you
must have completed Math 132: Calculus II or the equivalent with a grade of C-
or better. I will assume that you are familiar with standard Calculus I & II
material, including differentiation rules and techniques of integration. See the
last page of this syllabus for more information on what you are expected to
know.

**Work expectation:** The fact that this is a 4-credit
class means that, on average, a student should expect to work 8 to 12 hours
outside of class each week. It is essential that you keep up with the course
material and homework exercises. The best way to do poorly in this class is to
neglect the daily homework assignments and try to ‘cram’ for exams. This course
contains a moderate amount of mathematical rigor and abstraction, in the sense
that it lies between Math 131, Calculus I (nonrigorous) and Math 331, Advanced
Calculus (fully rigorous). Although I will not require you to write mathematical
proofs in this class, we will engage in occasional justifications of theorems in
class, and we will pay careful attention to correctness of mathematical writing
in homework.

**Class format: **The class will be conducted in a
variety of formats. The most common format employed will be lecture, but there
will also be time available for homework question and answer and group work.
Although there is no formal attendance policy, I expect everyone to attend
class. The only way to achieve credit for in class group activities is to come
to class. No make up group work will be allowed.

Occasionally, the Thursday class period will be used as a computer lab. This will occur approximately 5 times during the semester. We will meet in a campus computer lab (Van Zoeren 142) and complete lab work in teams. A written lab report will be required. Details will be given during the first lab period.

**Tools of assessment:** My goals as an instructor are
to help you learn and to assess your progress. Please keep in mind that I am
available and willing to help however I can; please visit my office whenever you
have questions. Your goal will be to attain proficiency in the subject matter in
the following areas:

Course Component |
Assessed Through |

Algebraic manipulation | Homework, exams |

Scientific writing | Homework, lab reports |

Reading comprehension | Homework with daily reading assignments |

Use of technology in mathematics | Lab reports |

Group work, teamwork | In class group work, lab reports |

Graphical and numerical methods | Homework, lab reports, exams |

Problem Solving | Homework, exams |

**Homework:** In almost any mathematics course, real learning takes place
largely through doing and discussing homework. There will be two main types of
homework assignment in this course:

Daily Reading Questions: Each time we begin a new section of the textbook, I

will give a reading assignment along with a list of questions on the most

important ideas. You will need to get a **3-ring binder or a sturdy folder**
in

which to keep your reading questions. Each set of reading questions will have

a due date (usually the class period after it was assigned), and you are
responsible

for keeping your folder up to date and bringing it to class each day. I will not

collect every student’s folder every day; instead, I will collect a subset of
the

folders each time. You will not know ahead of time on what day your folder

is to be collected, so be sure to have it with you every class. If your folder
is

not up to date, or if you fail to turn it in on a day it is requested, you will
lose

some credit on your homework grade.

Weekly Exercise Sets: Each week, I will assign homework exercises that will

take you through the ideas presented. The exercises will range from
straightforward computations to deeper problems that require some level of
problem solving. I will specify a subset of the exercises that are to be turned
in each week. I encourage you to do all of the exercises, even the ones I will
not be collecting. I will leave some class time, and time in the weekly optional
review sessions, for discussion of homework solutions. The only exception to
this is that I will not discuss solutions to exercises that are to be collected
during class (or in the review session) until after the exercises are due. If
you have questions about these exercises, you will need to talk to other
students, attend the Math Lab (details soon), or come and see me individually
for some hints.

**Exams:** The exams will make up the largest part of the grade for this
course. There will be four midterm exams (each during the 80 minute Thursday
period) and a cumulative final. Most problems on exams will be similar to, but
not identical to, homework problems. Some problems may test your understanding
of mathematical ideas by asking you to apply them in a new framework. Doing and
understanding the homework is the best way to study for exams. Do the problems I
recommend, and if you run out, do other problems. Test your understanding of
homework problems by explaining them carefully

to a friend.

**Labs: **In the campus computer labs, the software package Maple 8 is
available. We will spend 5 class periods in the lab this term completing lab
projects. These will chiefly be projects that provide a deeper, computer-aided
exploration of theoretical mathematics. Lab teams will consist of two students
who will remain together for several weeks. You may choose lab partners or ask
that they be assigned to you. A short lab report will be required of each team;
more details on this will be given during the first lab.

**In-class group work: **I believe that you will learn concepts more
quickly by doing than by listening only. For that reason, we will spend some
time each week doing problems in class in small groups. We will make use of this
particularly in the longer Thursday period, but we will probably do group work
on some other days as well. Not only will you learn by doing, but also you will
learn from and teach your classmates. Your teamwork and communication skills
will improve, and this will be especially important to you in the future when
you need to communicate scientific information precisely to a group of people
you are working with.

Group assignments will typically be short writing assignments that either lead you to use material we have previously covered or introduce you to new material. I will often collect one paper from each group and grade it. The largest part of this grade will be for participation.

**Collaboration and Academic Integrity:** Much of the reward in science
comes from being able to work together with others and share ideas. For that
reason, much of the work in this class will be of a collaborative nature. On
homework, you are encouraged to work with another person or in a group. On
problems to be handed in, if you use another person's idea be sure to give her
credit for it. This is not only polite, but it is accepted ethical procedure in
math and science. Also, while it is OK to work together on homework, it is never
OK to simply copy someone else’s work. Work with other students while you are
learning to do the exercises, but work on your own once you begin to prepare
written work to hand in. Lab reports are intended to be a fully collaborative
activity, and each student should contribute toward writing the team’s final lab
report.

On the other hand, exams are to be your own individual work. If you have any questions as to what constitutes academic dishonesty at Hope College, see the 2001-2002 Hope College Catalog, pp. 81—82. Make a conscious decision now to have integrity in all of your academic work so that you can be proud of your accomplishments. Do not develop bad habits that might put you in a position where academic dishonesty may seem attractive, such as waiting until the last minute to study for exams or to begin long assignments.

**Calculators:** Although they are not required for this course,
scientific or graphing calculators may be used on all quizzes and exams. Do not
depend on your calculator to think for you. The exams will not be written so
that you can pass them just by pushing the right buttons. Calculators are a
valuable computational aid, but you should not confuse knowing how to work your
calculator with knowing mathematics.

**Attendance and excused absence: **There is no formal attendance policy
for this class that is directly attached to your grade. **However, late
homework may not be made up for any reason, and missed in-class group work will
count as a zero.** Failing to attend class regularly will have a huge effect
on your understanding of the material and ultimately on your final grade. I
expect that students will attend class unless physically unable to do so. If you
have a valid excuse for missing an exam, check it out with me beforehand to be
sure I'll allow it.

**Exam dates:** There will be 4 midterm exams and a cumulative final exam
in this course.

Midterms: The midterms will be conducted in class on the following dates:

**Thursday, September 12
Thursday, October 10
Thursday, October 31
Thursday, November 21**

These 80-minute exams will be conducted during the usual Thursday class.

Final exam: The final exam will take place:

**Tuesday, December 10, 2:00—4:00 p.m.**

**Grading: **Grades will be computed on the basis of the 5 exams, the
written lab reports,

graded homework. The breakdown of grades is flexible; I will choose the system

that gives you the highest possible grade, given the following constraints:

Homework, In class group work and Lab Reports | 180 points |

Required External Participation in Mathematics | 20 points |

Midterm Exam 1 | 50 to 150 points |

Midterm Exam 2 | 50 to 150 points |

Midterm Exam 3 | 50 to 150 points |

Midterm Exam 4 | 50 to 150 points |

Final Exam | 200 to 400 points |

Total |
1000 points |

For example, if you do poorly on Midterm 2, but better on the other midterms
and

the final, Midterm 2 would be worth only 50 points, or 5 percent of the final

grade. The other midterms would be worth 150 and the final 400. On the other

hand, if you do well on the midterms and worse on the final, each midterm will

count 150 points, and the final will only be worth 200 points. In this way, you

can recover from a ‘bad day’ on an exam without too much impact on your final

grade. There will be no further curve in this class beyond this flexible grading

system.

Grades will be determined by the total number of points accumulated according

to the following scale:

A |
900-1000 points |

B |
800-899 points |

C |
650-799 points |

D |
500-649 points |

F |
below 500 points |

Plus and minus grades will be assigned at the instructor's discretion.

**External Participation in Mathematics and Extra Credit:** Each student
is required to engage in some sort of mathematical experience outside of class.
A total of 20 points is required in this area, but additional points can be
accumulated for extra credit. This is the only kind of extra credit available in
this class, and the maximum number of extra credit points available is 20 (for a
total of 40 points in this area). So that students do not put these
opportunities off until the last minute, my policy is that **at most 20 points
total can be attained in this area for material turned in after the Thanksgiving
break.** You can attain points in the following ways:

**• Attend a mathematics colloquium and write a brief report (1 page).**
This is the preferred way for most people to achieve External Participation
credit. Opportunities will be announced.

**• Watch a mathematics-related video or read a mathematics article and write a
brief report (1 page).** This is for people who cannot attend mathematics
colloquia due to conflicts. See me for suggestions of videos or articles.

• Colloquium, video or article points will be distributed as follows: 10 points
for the first two colloquia, videos or articles, and 5 points for each
additional one. (Therefore, to get the full 40 points—20 required plus 20 extra
credit—you must complete a total of 6 colloquium, video, or article reports.)

**Math Lab: **The Math Lab (Van Zoeren 274) is staffed by tutors who can
help you with homework questions. This is a free service provided by the
Academic Support Center. I will announce Math Lab hours during the first week of
the semester. I encourage you to make use of this resource.

**A Few Words about Prerequisites**

Since this class has Calculus I and II as prerequisites, I will assume you
have familiarity with and recollection of material from those two courses. Many
of you have a copy of the Anton/Bivens/Davis book Calculus: Early
Transcendentals, 7^{th} edition, and you should be able to find all of the topics
listed below in that book. If you have a different Calculus I and II book, you
should be able to find equivalent material there. Many mathematics courses
including this one build on what has come before. If you have difficulty with a
certain topic from Calculus I or II it is best to do some review now to avoid
problems later. Below I have listed some of the more important Calculus topics
we will use in Multivariable Mathematics I and II.

**Basic algebra and trigonometry:** By now you should
be pretty comfortable with algebraic manipulation: factoring and expanding
polynomials, working with exponents and logs, and manipulating the basic trig
functions. As you know from Calc I and II, your understanding of this material
is essential in correctly completing calculus problems.

**Derivatives:** I will expect you to know how to take
derivatives of basic functions and combinations thereof using the chain rule,
product rule and quotient rule. You should also recognize the connection between
a function f, its derivative f ′ and its second derivative f ′′. You should also
be able to write the equation of a tangent line to a curve at a given point. If
you don't recall the process of implicit differentiation remind yourself by
doing some problems. (You will need this material right away when we do vector
calculus and partial derivatives.)

**Applications of derivatives: **We will rely heavily
on your ability to find maximum and minimum values. Given a function and a
domain, you should be able to discuss (and often find) local and global maxima
and minima. (This material will be necessary when we begin optimizing functions
of several variables.)

**Integration:** You should know how to evaluate
definite integrals using the Fundamental Theorem of Calculus. You should recall
integration by substitution, integration by parts, and methods of approximating
integrals. You will be expected to know how to find areas of regions and volumes
of solids using integration as in Calc II. (This material will be necessary for
vector calculus and multiple integration.)

The list above gives examples of processes I will expect you to know how to do without me reminding you. There is much other useful information from Calc I and II (such as related rates, other techniques of integration, etc.) that I will try to review for you as needed. If you wonder if you are expected to know something, feel free to ask me.