Mathematics E-21a - Multivariable Calculus - Fall 2023
Instructor: Robert Winters. We will also have two Teaching Assistants (Renée Chipman, Kris Lokere, and Jeremy Marcq) who will conduct weekly problem sessions and grade homework assignments.
Recommended Prerequisites: One full year of single variable calculus. Though it may be preferable to have seen some infinite series and differential equations, these topics are not necessary for successful completion of this course.
About the Course: We will cover at least the following topics: Vectors in R2 and R3 and vector methods, functions of several variables, differentiation and integration of functions of several variables, linear and quadratic approximation, optimization, parameterized curves and surfaces, vector fields, line and surface integrals, Green’s Theorem, the Divergence Theorem and Stokes’ Theorem.
Note: The “Graduate” credit option is primarily for students enrolled in certain Extension School graduate programs such as the “Math for Teaching” program. All other students (including high school students) should register for the “Undergraduate” option or the Noncredit option (if you will not be submitting homework or taking exams). Students registered for “Graduate” credit will be asked to complete additional work on supplemental topics.
Textbook: Multivariable Calculus: Concepts and Contexts by James Stewart - either the 4th Edition (published 2010 by Brooks-Cole, ISBN: 0495560545) OR the 3rd Edition (published 2005 by Brooks-Cole, ISBN: 0534410049) OR a comparable text. There are links on the course website to help you find an inexpensive copy. Note: Each homework assignment will be posted as a PDF for those with other editions of the text. We will also rely substantially on Lecture Notes prepared specifically for this course and available on the course website.
Course website: http://math.rwinters.com/E21a and Canvas site
Course meetings: The class meets weekly on Thursdays, 8:00pm to 10:00pm [in person (Harvard Hall 101) or via Zoom] or on-demand in Canvas. Optional problem sessions conducted by our Teaching Assistants will be scheduled after our first class meeting. Additional times for questions may follow as the course proceeds. An optional Q&A session with the instructor may also be scheduled at a day and time to be determined.
Homework: Problem sets will be assigned each week and will be due the following week (typically the Saturday after the following class). You are encouraged to discuss the homework with your fellow students, but you must write up the solutions by yourself without collaboration with others. (This is simply a matter of professional ethics.) Grading policies for the homework will be established after the first class meeting. Homework assignments and solutions will be posted on the course Calendar: http://math.rwinters.com/E21a/calendar.htm. Any unethical behavior on the homework (such as the copying solutions from a solutions manual) may result in only exam scores being used in the calculation of your course grade.
Solutions to the homework problems in PDF format will be made available soon after the due date via a password-protected web page linked from the Math E-21a course website. Selected problems may also be discussed in the problem sessions. Homework submitted after the day of the subsequent class meeting will be accepted only at the discretion of the course assistants.
Exams: There will be two 70 minute midterm exams and a two-hour Final Exam conducted online during a 24-hour window within the Canvas site using the Proctorio browser extension. The dates for the exams are:
Midterm Exam 1: Thurs, Oct 12 - Fri, Oct 13 Midterm Exam 2: Thurs, Nov 30 - Fri, Dec 1
Final Exam: Thurs, Dec 21
Grading: Your final grade will be based on your performance on the homework (20%), the two midterm exams (20% each), and the final exam (40%). These percentages are subject to minor modification.
Computers and calculators: The visualization of surfaces and other geometric objects is an important aspect of this course. Because computerized graphing programs can aid you in developing this ability, you are encouraged to employ them as part of the learning process. Use of mathematical software is nonetheless optional. Computers should be considered solely as an aid to the development of geometric intuition. Calculators will be permitted on the exams, but not computers. Various homework problems will ask you to sketch or otherwise describe various geometric objects. You are strongly advised to struggle with these first without electronic aids, as they may be quite trivial with a graphing program.
Words of Caution and Advice: This course may prove to be more demanding than your previous mathematics courses. The weekly assignments may be somewhat time-consuming, so you should plan now to set aside regular hours to wrestle with them. It is virtually impossible to do well in this course without working the homework assignments in a timely fashion. The course is somewhat fast-paced and new material builds on old, so do not fall behind. If you find yourself falling behind, please contact the instructor immediately so that options for assistance may be discussed.
When you are working your assignments, keep in mind that your success in this course will require more than just memorizing formulas and “plugging in values.” Numerical calculations are still important, but play a smaller role than in single-variable calculus. The key to success is understanding the underlying concepts and working enough problems so that you can employ them in any example thrown at you, especially homework and exam problems which differ significantly from material discussed in class.
Detailed Syllabus. (This may change as the course proceeds.)
| Date | Topics | Text sections |
| Sept 7 | Introduction to R2 and R3. Points vs. vectors. Dot product in R2, R3, and Rn. Scalar and vector projections. Equation(s) of a line. | 9.1 - 9.3 |
| Sept 14 | The cross product in R3. Equations of lines and planes in R3. Intersection of lines and planes. Functions, graphs, and surfaces. (Cylindrical and spherical coordinates.) | 9.4 - 9.6; (9.7) |
| Sept 21 | Vector-valued functions - parameterized curves in R2 and R3. Velocity and acceleration vectors. Arclength. Equations of motion, tangential and normal components of acceleration. | 10.1 - 10.4 |
| Sept 28 | Parametric surfaces. Functions of several variables; limits and continuity; graphs and level curves (contours) of a function of two variables. Partial derivatives. | 10.5; 11.1 - 11.3 |
| Oct 5 | Partial derivatives; differentiability; linear approximation and tangent planes; rate of change of a function along a parameterized curve; the Chain Rule. | 11.3 - 11.4 |
| Oct 12 | Directional derivatives and the gradient vector. Level surfaces of a function of three variables. Midterm Exam I | 11.5 - 11.6 |
| Oct 19 | Optimization; maximum and minimum values of a function. Constrained optimization and the Method of Lagrange Multipliers. | 11.7 - 11.8 |
| Oct 26 | Integration over regions in R2 and R3. Average value of a function. Iterated double integrals and the Fubini Theorem. Double integrals in polar coordinates. | 12.1 - 12.4 |
| Nov 2 | Applications of double integrals: mass, moments, center of mass, moment of inertia, probability. Surface area. Triple integrals in Cartesian coordinates. | 12.5 - 12.7 |
| Nov 9 | Triple integrals in cylindrical coordinates and spherical coordinates; applications of triple integrals. Change of variables in multiple integrals; Jacobian matrices. | 12.8 - 12.9 |
| Nov 16 | Vector fields in R2 and R3. Line integrals and work done by a variable force along a parameterized curve; Fundamental Theorem of Line Integrals, independence of path, conservative vector fields. | 13.1 - 13.2 |
| Nov 30 | Green’s Theorem. Midterm Exam II | 13.3 - 13.4 |
| Dec 7 | Curl and divergence of a vector field; surface integrals; flux of a vector field through a surface. | 13.5 - 13.6 |
| Dec 14 | Stokes’ Theorem; the Divergence Theorem | 13.7 - 13.8 |
| Dec 21-22 | Final Exam |
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