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
• 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:
|Homework, lab reports
|Homework with daily reading assignments
|Use of technology in mathematics
|Group work, teamwork
|In class group work, lab reports
|Graphical and numerical methods
|Homework, lab reports, 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
|Required External Participation in Mathematics
|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
|200 to 400 points
For example, if you do poorly on Midterm 2, but better on the other midterms
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
Grades will be determined by the total number of points accumulated according
to the following scale:
|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, 7th 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.