Calculus III - Fall 2011

HOWARD UNIVERSITY

Richard E. Bayne

Table of Contents:

[ Text and Required Materials | Course Objectives | Course Content | Pace Sheet | Pre-requisites | Cooperative Learning Groups | Requirements | Administrative Policies | Office Hours ]

, by Howard Anton, Ira Bivens, and Stephen Davis (Wiley)*Calculus*

This course extends ideas of single variable calculus to higher dimensions and is aimed primarily at students whose majors are science, engineering or mathematics. The focus is on multi-dimensional calculus, including the study of functions of several variables, partial derivatives, and optimization problems using various techniques. We study vectors, vector-valued functions, parametric curves and three dimensional surfaces.

Approximate pacing of topicsBack to TOC

This is the third course in a three-semester sequence. The primary aim of the sequence is to help students learn, understand, explain, and use calculus. In addition, it is desired that students will improve their mathematical skills, further their understanding of mathematics and its applications to the sciences, as well as increase both their intellectual curiosity and their desire to learn more about the value of mathematics in general and calculus in particular. This third course concentrates on functions of several variables, vectors and vector-valued functions. At the completion of this course, the student should be to:

- Identify quadratic functions in three variables with quadric surfaces and sketch their graphs.
- For a function of 2 or 3 variables, find the domain and range; find and sketch the level curves for the function.
- Apply vector operations including addition, scalar multiplication, dot products and cross products.
- Resolve a vector into its coordinate components and find the length and magnitude of a vector.
- Determine the angle between two vectors.
- Find the area of a parallelogram using the cross product.
- Solve applications using vector operations.
- Determine limits and continuity of functions of 2 or 3 variables.
- Find and apply the linearization of a function or 2 or 3 variables.
- Calculate and interpret first and second partial derivatives.
- Find, interpret, and apply the gradient of a function of 2 or 3 variables.
- Find and classify critical points of a function of 2 variables as maxima, minima, or saddle points.
- Find absolute extrema for functions of several variables.
- Optimize a function with given constraints, using the method of LaFrange.
- Evaluate double integrals using rectangular and polar coordinates.
- Evaluate triple integrals using rectangular, cylindrical, and spherical coordinates.
- Be able to change the order of integration for double and triple integrals.
- Transform between rectangular, cylindrical and spherical coordinate systems for triple integrals.
- Use double and triple integrals to solve applied problems.
- Represent a curve in space parametrically.
- Describe motion in space with parametric equations.
- Find the velocity of a moving particle from its position vector.
- Find the acceleration of a moving particle from its position vector.
- Determine the parametric equation of a line through a point in space.
- Find the length of a curve in space.
- Draw a vector field in the plane and draw flow lines of a vector field.
- Evaluate a line integral for a given oriented curve.
- Evaluate a line integral using parameterization.
- Find the potential function for gradient fields.
- Use the Fundamental Theorem of Calculus for line integrals to evaluate a line integral.
- Evaluate a line integral using Green's Theorem.

To be successful in this course,** you should have mastery of college
algebra**. You should have received a satisfactory grade in MATH
156 (Calculus I), and MATH 157 (Calculus II). It is assumed
that students are proficient in standard Calculus I and II topics,
including continuity, taking limits, finding derivatives of complicated
functions, using derivatives, calculating definite integrals, and
integration techniques. It builds tools for describing curves, surfaces,
solids and other geometrical objects in three dimensions.
Please see me if you have any
questions about your preparation for this course.

A part of this course may be run using a cooperative learning approach. Classes can be highly interactive and vigorous class participation is expected. Early in the semester, each student will be assigned to a group of four to five students.

Working well in a group is an important skill. Some of you may enjoy the group work more than others, but all of you will benefit from further developing this skill. After graduation, most of you will be working in jobs which will require you to function as a member of a project team. One objective of group work in this course is to help you to develop skills in working effectively as part of a team. Another is to encourage discussion about the concepts.

One of the primary objectives of this course is to help you learn to think about problems mathematically and to solve the problems on your own. Working with your colleagues in this class and talking about problems with your group members are strategies to help you better understand a problem situation from several points of view. Experience has shown that those students that actually do work with their groups not only do better in the course, they also learn more. Those who for one reason or another refuse to fully participate in their cooperative group invariably do worse.

**Regular attendance, participation, homework:****EXAMS:**- There will be
**four hour exams**each worth 100 points. The material to be covered on each test will be announced in advance of the scheduled test date. Tests are tentatively scheduled for the following dates, but please note that these dates may change depending on class progress and unforseen circumstances. **Final Exam: 200 points**- The
*final exam is cumulative*and is listed in the University's class schedule.

*Regular attendance is expected.* Some material will be presented
in class from a different perspective than that given in the text. "Getting
someone's notes" is a poor substitute for being present and involved
in class discussion. However, if you must miss a class, it is your
responsibility to find out what you missed.** Make a friend! **

Each student will be expected to do the following: 1. Attend every class. 2. Devote a minimum of 12 hours of study per week to the course. 3. Come to each class on time and ready to participate. 4. Be willing to help your classmates. 5. Be able to explain concepts to the instructor or to other students. 6. Meet with group members at least twice each week to review and discuss course material. 7. Do all class activities and homework assignments. Please note that only under the most unusual circumstances will class activities or homework assignments be accepted after the due date.

Exam #1 week 4 Exam #2 week 8 Exam #3 week 12 Exam #4 week 15 Final Exam DEC 6Please note:

Ordinarily, there are no make-up tests; exceptions to this policy will
be considered on a case-by-case basis. You must determine **BEFORE
the exam date** whether your excuse will be acceptable.

Generally, incomplete grades will not be given. If there is an emergency which causes a student to be unable to finish course requirements, the emergency must be documented by the student's advisor or by the advisory center.

If you have concerns about your progress or ability to keep up with course assignments, please discuss these with me as soon as possible. DO NOT WAIT until late in the semester.

**GRADES **Grades will be determined as follows:

A85% - 100% of total points availableB75% - 84% of total points availableC62% - 74% of total points availableD50% - 61% of total points availableF0% - 49% of total points available

Students who cheat violate their own integrity and the integrity of the university by claiming credit for work they have not done and knowledge they do not possess. All students are expected to recognize and to abide by the policy on academic integrity found in the Student Handbook. Because you will be asked to do a lot of work in collaboration with your group members, I will ask you to sign all homework assignments attesting to the fact that you have actively participated in the work.

- Mon 2:00 pm - 3:00 pm
- Wed 2:00 pm - 3:00 pm
- Thu 1:00 pm - 2:00 pm

If you need to reach me between classes:

- you may send an email message to rbayne@howard.edu
- you may leave a voice message at 202-806-7673, or
- you may drop by my office and see if I am free.

I regularly check email several times a day both from home and at school,
but I check voice messages only when I am on campus.

The easiest way to contact me is to
send an e-mail message to Richard Bayne.

I. Intro & Review of Previous Material | weeks 1 & 2 |
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A. Goals for course | ||||

B. Working in Groups | ||||

C. Review of Ideas from Calc I & II | ||||

II. Curves & Surfaces | weeks 3, 4 & 5 |
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A. Polar Coordinates | ||||

B. Parametric Equations | ||||

C. Vectors | ||||

III. Functions of Several Variables | weeks 6, 7 & 8 |
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A.Surfaces & Level Curves | ||||

B. Partial Differentiation | ||||

C. Applications of Partial Derivatives | ||||

IV. Multiple Integration | weeks 9 & 10 |
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A.Double Integrals | ||||

B. Iterated Integrals | ||||

C. Applications of Double Integrals | ||||

D. Triple Integrals | ||||

E. Transformation of Coordinates | ||||

V. Line & Surface Integrals | weeks 11 & 12 |
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A. Line Integrals in R2 | ||||

B. Line Integrals in R3 | ||||

C. Surface Integrals | ||||

VI. Some Classic Theorems | weeks 13 & 14 |
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A. Green's Theorem | ||||

B. Stokes' Theorem | ||||

C. Gauss' Theorem | ||||

This page was updated August 2011.