Math 2101 Calculus III
The vector concept and the development of a whole mathematics of vector quantities have proved indispensible to the development of the science of mechanics. --Grant R. Fowles, Analytical Mechanics, 4th Ed. Saunders College Publishing, Philadelphia, 1986.
Course Prerequisites
To succeed in this course you will need to have mastered basic calculus (Calculus I and II).
Most students in the class will have some experience with the computer algebra system Mathematica in Calculus I and II. If you have never used Mathematica before, do not despair! I will be using it occasionally in class (as will you, but probably not to the same level it was used in class in Calculus I and II), and I will provide you with resources to help you use Mathematica when you need it on Assignments. Mathematica will never do our thinking for us. It will help us understand concepts and answer questions that would be difficult to answer if we were working the solution out solely by hand. The vast majority of the work in this class will be done using our brains, pencils, and paper!
Goals
A student taking this course should expect to be able to
- Visualize and draw three dimensional surfaces; intersections of these three dimensional surfaces; determine volumes, surface areas, and be comfortable with multidimensional integrals.
- See the extensions of what they have learned in Calculus I and II to three dimensions.
- Be comfortable working in, and converting between, different coordinate systems (mainly cartesian, cylindrical, and spherical).
- Be comfortable working with vector functions and performing vector calculus.
Beyond the curriculum, you should also expect to
- develop skill at presenting solutions to problems,
- think beyond technique, and understand the problems studied in some depth,
- develop confidence in your problem solving skills,
- understand how the methods we develop are used in other fields, notably physics and chemistry,
- see the benefit of computers to aid calculation, but also see the absolute necessity of understanding the theory completely before using a computer.
Textbook
Textbook for Spring 2008: Thomas Calculus Media Upgrade Part Two, 11th edition by Thomas, Weir, Hass, & Giordano.
Time Commitment
University policy says ``one credit is defined as equivalent to an average of three hours of learning effort per week (over a full semester) necessary for an average student to achieve an average grade in the course''. Our course is a four-credit course, meeting approximately three hours per week: 4 credits times 3 hours/week/credit - 3 hours/week in lecture = 9 hours/week outside class. Thus, you are expected to spend 9 hours per week working outside of class, reading the textbook and working problems.
Please make the most of my office hours! The content of the course can be difficult at times and I expect to see you all in my office at some time or other. To get the most out of the course you should
- do homework every day,
- allot time to think about what it is we are doing,
- discuss the techniques we are studying and their implementation with your classmates,
- discuss any difficulties with me during office hours.
Course Components
Homework. The homework questions (included in the course calender below) for the previous week will be collected each Monday and graded. Worked solutions to the homework will be available on the course web page after the due date. Hand drawn sketches or Mathematica may prove useful in finding your solutions to problems, so include those as needed. Visualizing three dimensional space is an important skill to acquire in multivariable calculus, so hand drawn figures play an important role in explaining and understanding solutions. Make sure your sketches are neatly drawn and well labeled.
Some of the homework will be completed online (and graded) using WebWorK. There will be links to such homework in the course calender below. The deadline for WebWorK homework will typically be the Monday the following week when any other homework is due.
Some homework will not be collected for grading--such homework will typically have solutions already provided.
The purpose of the homework is to focus your attention on the important lessons of the day, and to serve as review for tests. Tests may include question that were not asked in the homework, so you should do as much extra homework as you deem necessary to enhance your understanding of a topic. I can not stress enough how important it is that you work problems! Falling behind in this course, as in any university course, can lead to disaster, so it is important that you keep up with the homework.
Exams. The exams will consist of five problems--possibly some short answer or true/false questions, along with questions dealing with the application of the techniques we have learned. You will not be allowed any outside material on your desks during the exams (calculators may or may not be be allowed, but even if they are they should not be necessary to solve the problems). The exam should not be significantly harder than the homework; my tests tend to be long, so do not be alarmed if you require the entire class time to complete the test. There will be no Mathematica component to the exams.
Final Exam. The final exam will be similar in format to the exams, except slightly longer, and cover only the material since the last exam.
Grading
Here is the University-wide uniform grading policy.
- A: Represents achievement that is outstanding relative to the level necessary to meet course requirements.
- B: Represents achievement that is significantly above the level necessary to meet course requirements.
- C: Represents achievement that meets the course requirements in every respect.
- D: Represents achievement that is worthy of credit even though it fails to fully meet the course requirements.
- F: Represents failure and indicates that the coursework was completed but at a level unworthy of credit, or was not completed and there was no agreement between the instructor and student that the student would be temporarily given an incomplete.
- I: See the catalog.
The grade for the course will be calculated by the following formula:
| Homework | 20% |
| Three Chapter Exams @ 20% each | 60% |
| Final Exam on Chapter 16 | 20% |
Your numerical grades will be converted to letter grades and finally Grade Points via the following cutoffs (see the UMM Catalog for more on Grades and Grading Policy):
| Numerical | 95% | 90% | 87% | 83% | 80% | 77% | 73% | 70% | 65% | 60% | Below 60% |
| Letter | A | A- | B+ | B | B- | C+ | C | C- | D+ | D | F |
| Grade Point | 4.00 | 3.67 | 3.33 | 3.00 | 2.67 | 2.33 | 2.00 | 1.67 | 1.33 | 1.00 | 0.00 |
Please note that you are not competing against your fellow students. I will adjust the difficulty of the questions and the severity of the grading so that, for example, a B+ score corresponds to what I consider B+ achievement.
Expectations
- Be in class on time. I will begin lecturing at 11:45 am sharp, and neither I nor your fellow classmates enjoy the disruption late arrival causes. I know that situations crop up that will entail late arrival (please come even if you are late!) but try to ensure it is the exception and not the rule.
- Cooperation is vital to your future success,
whichever path you take. I encourage cooperation amongst
students wherever possible, but the act of copying or other
forms of cheating will not be tolerated.
Academic dishonesty in any portion of the academic work for a course is grounds for
awarding a grade of F or N for the entire course.
Any act of plagiarism
that is detected will result in a mark of zero on the entire assignment
or test for both parties.
Please come and talk to me if you are in any way unclear about what constitutes
academic dishonesty.
UMM's Academic Integrity policy and procedures can be found at the following website:
Academic Integrity www.morris.umn.edu/Scholastic/AcademicIntegrity/. - Homework is due Monday in class, and only under exceptional circumstances (which are officially documented) will I accept late work. You will receive a mark of zero if homework is submitted late and you have not obtained an extension from me.
- If you are going to miss an exam, let me know in advance so we can work out alternate plans.
-
If you have any special needs or requirements to
help you succeed in the class, come and talk to me as soon as
possible, or visit the appropriate University service yourself.
Some UMM resources include:
The Academic Assistance Center www.morris.umn.edu/services/dsoaac/aac/
Student Counseling www.morris.umn.edu/services/counseling/
Disability Services www.morris.umn.edu/services/dsoaac/dso/
Multi-Ethnic Student Program www.morris.umn.edu/services/msp/
and of course, your academic advisor!
Mathematica Resources
At UMM the computer algebra system of choice is Mathematica (I typically abbreviate this as MMA), and I use it extensively in most of the courses I teach. FYI, files ending in .nb are Mathematica files.
Mathematica is expensive, and we do not expect our students to purchase it. UMM has a site licence for Mathematica, and it can be found on any computer on campus (PC or MacIntosh). When you need to work with Mathematica outside of class, visit one of the many computer labs on campus.
If you have specific questions about Mathematica while working on homework, bring the file you are working with to office hours on a usb drive and we can look at it together, or email the file to yourself (or me) before you come to office hours. Most questions can be answered in under 10 minutes.
- Mathematica Quick Reference (I will hand this out in class)
- Mathematica Basics (MMA file--a good place to start if you have never used it before)
- Intermediate Mathematica (MMA file--introduction to some of the MMA you see beyond calculus)
Course Calendar
| # | Date | Lecture Topic | Homework | Resources | |
|---|---|---|---|---|---|
| Jan 21 | |||||
| 1 | Jan 23 | Course Introduction & 12.1 Three-Dimensional Coordinate System | 34,52,53,54 | 12.1.nb | MMA (3D coordinate system) | |
| 2 | Jan 25 | 12.2 Vectors | WeBWorK: Vectors | MMA (vectors) | Vector Spaces | |
| 3 | Jan 28 | 12.3 The Dot Product | Turn in 12.1 8,18,22,32,48 | 12.3.nb |
||
| 4 | Jan 30 | 12.4 The Cross Product | WeBWorK: Cross Product | MMA (cross product) | Scalar Triple Product | |
| 5 | Feb 1 | 12.5 Lines and Planes in Space | WeBWorK: Lines and Planes 22,30,61,64,70 |
MMA (lines and planes in space) | |
| 6 | Feb 4 | 12.6 Cylinders and Quadric Surfaces | Turn in 12.3 1-6 (use MMA if you like to match, but submit hand drawn versions of the surfaces) and 80a | 12.6.nb |
MMA (quadric surfaces) MMA (quadric traces) |
|
| 7 | Feb 6 | 13.1 Vector Functions | 14,24,38,43,54c,61 | 13.1.nb | MMA (vector functions) | |
| 8 | Feb 8 | 13.3 Arc Length and the Unit Tangent Vector T | 8,10,12,18 | 13.3.nb | MMA (arc length reparameterization) | |
| 9 | Feb 11 | 13.4 Curvature and the Unit Normal Vector N | Turn in 12.6, 13.1 MMA |
MMA (TNB Coordinate Axes) MMA (osculating circle) |
|
| 10 | Feb 13 | 13.5 Torsion and the Unit Binormal Vector B | WeBWorK: TNB Frame | MMA (aT and aN) | |
| 11 | Feb 15 | Review of Chapter 12 and 13 | practice questions | Making Sense of Trig Identities Making Sense of Basic Integrals |
|
| 12 | Feb 18 | Test #1 on Chapters 12 and 13 | 2008 Grade Distribution | |||
| 13 | Feb 20 | 14.1 Functions of Several Variables | WeBWorK: Functions of Several Variables | MMA (functions of several variables) MMA (intersection of surfaces) |
|
| 14 | Feb 22 | 14.2 Limits and Continuity in Higher Dimensions | WeBWorK: Limits and Continuity | MMA (limits and continuity) | |
| 15 | Feb 25 | 14.3 Partial Derivatives | Turn in nothing! WeBWorK: Partial Derivatives PartialDerivativesHW.nb |
||
| 16 | Feb 27 | 14.4 The Chain Rule | WeBWorK: Chain Rule 42,45,46,48 | 14.4.nb |
||
| 17 | Feb 29 | 14.5 Directional Derivatives and Gradient Vectors | WeBWorK: Directional Derivatives and Gradient Vectors 18,24,32,33 |
MMA (directional derivatives) | |
| 18 | Mar 3 | 14.6 Tangent Planes and Differentials | Turn in 14.4, 14.5 WeBWorK: Tangent Plane Differentials |
MMA (tangent planes) | |
| 19 | Mar 5 | 14.6 Tangent Planes and Differentials | 55, 57 | ||
| 20 | Mar 7 | 14.7 Extreme Values and Saddle Points | WeBWorK: Extreme Values Saddle Points | MMA (optimization, linear regression) | |
| 21 | Mar 10 | 14.8 Lagrange Multipliers | Turn in 14.6 WeBWorK: Lagrange Multipliers |
MMA (Lagrange multipliers) | |
| 22 | Mar 12 | Review of 14.1-14.8 | practice questions | ||
| 23 | Mar 14 | Test #2 on Chapter 14.1-14.8 | 2008 Grade Distribution | |||
| Mar 17--21 | Spring Break--no class | ||||
| 24 | Mar 24 | 14.9 Partial Derivatives with Constrained Variables 14.10 Taylor`s Formula for Two Variables |
14.9: 2,3,8 14.10: 4,6 |
MMA (Taylor`s series) | |
| 25 | Mar 26 | 15.1 Double Integral | WeBWorK: Double Integrals | MMA (double integrals) | |
| 26 | Mar 28 | 15.2 Areas, Moments, and Centers of Mass | WeBWorK: Moments2D | ||
| 27 | Mar 31 | 15.3 Double Integrals in Polar Form | Turn in 14.9, 14.10 WeBWorK: DoubleIntegralsPolar |
||
| 28 | Apr 2 | 15.4 Triple Integrals in Rectangular Coordinates Examples |
WeBWorK: TripleIntegrals | MMA (triple integrals) | |
| 29 | Apr 4 | 15.4 Triple Integrals in Rectangular Coordinates | 14,16,42,44 | ||
| 30 | Apr 7 | 15.5 Masses and Moments in Three Dimensions 15.6 Triple Integrals in Cylindrical and Spherical Coordinates |
Turn in 15.4 WeBWorK: Moments3D |
MMA (spherical and cylindrical coordinates) | |
| 31 | Apr 9 | 15.6 Triple Integrals in Cylindrical and Spherical Coordinates | WeBWorK: CylindricalSphericalCoordinates | MMA (triple integrals in spherical and cylindrical coordinates) | |
| 32 | Apr 11 | 15.7 Substitutions in Multiple Integrals | 1,3,6,7,10 | 15.7.nb | ||
| 33 | Apr 14 | 15.7 Substitutions in Multiple Integrals | Turn in 15.7 WeBWorK: Substitutions |
||
| 34 | Apr 16 | Review of Chapter 15 | practice questions | ||
| 35 | Apr 18 | Test #3 on Chapter 15 (excluding 14.9 and 14.10) | 2008 Grade Distribution | |||
| 36 | Apr 21 | 16.1 Line Integrals | WeBWorK: LineIntegrals 13,15,16,20,22 |
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| 37 | Apr 23 | 16.2 Vector Fields, Work, Circulation, and Flux | WeBWorK: VectorFields 14,17,22,23,37 |
MMA (vector fields) | MMA (flux and flow) MMA (flux and flow animations) |
|
| 38 | Apr 25 | 16.3 Path Independence, Potential Functions, and Conservative Fields | Turn in Apr 30: 28,31,38 11,20,23,26 |
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| 39 | Apr 28 | 16.4 Green`s Theorem in the Plane | WeBWorK: GreensTheorem 7,19,29,35 |
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| 40 | Apr 30 | 16.5 Surface Area and Surface Integrals & 16.6 Parameterized Surfaces | Turn in 16.3 WeBWorK: SurfaceIntegrals 16.5: 3,6,9,25 |
MMA (parameterization of surfaces) | |
| 41 | May 2 | 16.5 Surface Area and Surface Integrals & 16.6 Parameterized Surfaces | WeBWorK: ParameterizedSurfaces 16.6: 7,13,15,17,31,41 |
MMA (surface integrals) | |
| 42 | May 5 | 16.7 Stokes` Theorem | WeBWorK: Stokes 3,5 |
||
| 43 | May 7 | 16.8 The Divergence Theorem and a Unified Theory | WeBWorK: Divergence | ||
| 44 | May 9 | Review of Chapter 16 | Notes | practice questions | ||
| May 13 | FINAL EXAM on Chapter 16 1:30-3:30pm in class 2008 Final Grade Distribution |
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