Fall 2019
Multivariable Calculus
Math 18.02, Concourse - MIT
Fall 2019

Lectures and Recitations by:
Robert Winters
rwinters@mit.edu
Office: 16-137
Phone: x3-2050
(but e-mail is better)

Concourse 18.02 Stellar site

The text for the course is
Multivariable Calculus, 6th Edition by Edwards & Penney (ISBN 0130339679 for softcover edition)
[Click on the image below for prices.]

• Ch. 12 - Vectors, Curves, and Surfaces in Space

• Ch. 13 - Partial Differentiation

• Ch. 14 - Multiple Integrals

• Ch. 15 - Vector Calculus

Announcements:

Problem Set #11 (due Fri, Dec 6)
References: 15.6 (The Divergence Theorem); Notes V10 (The Divergence Theorem); RW-Orthogonal Curvilinear Coordinates - Div, grad, curl, and Laplacian; 15.7 (Stokes' Theorem); Notes V13 (Stokes' Theorem); Notes V14 (Some Topological Questions); Notes V15 (Relations to Physics); and Lecture Notes (see Calendar).

FYI - The Final Exam is scheduled for Tuesday, December 17, from 9:00am to noon in 16-160.

Practice Exam A (from mainstream 18.02)    Solutions

Practice Exam B (from mainstream 18.02)    Solutions

Exam #3 will take place on Tuesday, Nov 19 from 12:00pm to 1:30pm in 16-160. A Practice Exam and Solutions will be posted. Possible topics include: Integration over two- and three-dimensional regions; double and triple integrals in Cartesian, cylindrical, and spherical coordinates; Fubini Theorem and interchanging order of integration; applications of integration – areas, volumes, mass, averaging, weighted averages, centroids and center of mass, moment of inertia, general change of variables for double and triple integrals, Jacobian determinants; Integration of functions over curves and surfaces (line and surface integrals) with applications to mass, averaging, centroids, flux, etc.; flux integrals in 2- and 3-dimensions; conservative vector fields and potential functions; Fundamental Theorem of Line Integrals, Green’s Theorem.

 Approximate letter grades for Exam #1 Total points on exam was 50. Median score was 43.5. Mean score was 43.8. Standard deviation was 4.2. score grade score grade 45+ A 33+ C+ 43+ A– 30+ C 41+ B+ 27+ C– 39+ B 25+ D 36+ B– 0-24 F Exam #1 Solutions

 Approximate letter grades for Exam #2 Total points on exam was 50. Median score was 44. Mean score was 44.2. Standard deviation was 4.5. score grade score grade 45+ A 33+ C+ 43+ A– 30+ C 41+ B+ 27+ C– 39+ B 25+ D 36+ B– 0-24 F Exam #2 Solutions
 Approximate letter grades for Exam #3 Total points on exam was 50. Median score was 48. Mean score was 45.5. Standard deviation was 5.8. score grade score grade 45+ A 33+ C+ 43+ A– 30+ C 41+ B+ 27+ C– 39+ B 25+ D 36+ B– 0-24 F Exam #3 Practice Problems     Solutions Exam #3 Solutions

Problem Set #10 (due Wed, Nov 27 - but I'll be very understanding due to the Thanksgiving holiday)
References: Notes V3 (Two-dimensional Flux); Notes V4 (Green's Theorem in Normal Form); 14.8 (Surface Area); Notes V7 (Laplace's Equation and Harmonic Functions); 15.5 (Surface Integrals); Notes V9 (Surface Integrals); RW-Supplement on surface integrals; 15.6 (The Divergence Theorem); Notes V10 (The Divergence Theorem); and Lecture Notes (see Calendar).

Problem Set #9 (due Wed, Nov 20)
References: 14.8 (Surface Area); 15.1 (Vector Fields, Divergence and Curl); Notes V1 (Plane Vector Fields); Notes V8 (Vector Fields in Space); 15.2 (Line Integrals); Notes V11 (Line Integrals in Space); 15.3 (Fundamental Theorem and Independence of Path); Notes V2 (Gradient Fields and Exact Differentials); Notes V12 (Gradient Fields in Space); 15.4 (Green's Theorem)

Exam #2 took place on Monday, Oct 28 during class. Topics included: Partial derivatives, linear approximation, differentials, gradient vector, normal vectors to curves and surfaces, directional derivative, the Chain Rule, implicit differentiation, stationary points, second derivative test, optimization, Method of Lagrange Multipliers. Double integrals in Cartesian and polar coordinates and applications; Fubini Theorem and interchanging order of integration.

Problem Set #8 (due Wed, Nov 13)
References: 14.6 (Triple Integrals); 14.7 (Integration in Cylindrical and Spherical Coordinates); 14.8 (Surface Area); 14.9 (Change of Variables in Multiple Integrals); SN-I. Limits in Iterated Integrals; SN-CV. Changing Variables in Multiple Integrals; SN-G. Gravitational Attraction; and Lecture Notes (see Calendar).

Problem Set #7 (due Tues, Oct 29)
References: SN-N. Non-independent Variables, 14.1 (Double Integrals); 14.2 (Double Integrals Over More General Regions); 14.3 (Area and Volume by Double Integration); 14.4 (Double Integrals in Polar Coordinates); 14.5 (Applications of Double Integrals); SN-I. Limits in Iterated Integrals; and Lecture Notes (see Calendar).

Problem Set #6 (due Thurs, Oct 17)
References: Read sections 13.5 (Multivariable Optimization Problems), 13.9 (Lagrange Multipliers and Constrained Optimization); and Lecture Notes (see Calendar).

Exam #1 - Mon, Sept 30 - Vectors and vector algebra, dot product, cross product, applications to working with lines and planes, areas, volumes, angles, etc.; parameterized curves, velocity vectors, speed, arclength, unit tangent and normal vectors; partial derivatives, linear approximation, differentials, and directional derivative. You should also be familiar with methods for solving systems of linear equations (such as when finding the intersection of lines or planes) and related matrix ideas.

Topics and Assignments are posted in the Course Calendar.

Lecture times: Mon, Wed 3:00pm-4:30pm in 16-160

Recitation times: Tues, Thurs 12:00-1:00pm in 16-160

Office hours: Tues, Thurs 2:00-3:00pm and at other times to be determined

Mathlet (Java applet) for Curves and Surfaces (may be helpful for P-sets)

Text: Multivariable Calculus, 6th Edition by Edwards & Penney (ISBN 0130339679 for softcover edition).

Supplementary Notes: 18.02 Notes authored by Prof. Arthur Mattuck of the MIT Mathematics Department.

 Condensed Syllabus: (See the Calendar for day-by-day details and assignments, updated as the course proceeds.) Vectors and vector algebra in R2 and R3; dot product, cross product, projection, equations of lines and planes. Matrix methods. (Chap. 12) Parameterized curves and surfaces in R2 and R3; velocity and acceleration vectors; tangent vectors; arclength. (Chap. 12) Functions of several variables - limits, continuity, and differentiabilty; partial derivatives, gradients, linear approximation, directional derivatives, Chain Rule. (Chap. 13) Optimization - unconstrained and constrained. (Chap. 13) Integration over regions in R2 and R3 and their applications, using Cartesian, polar, cylindrical, and spherical coordinates. (Chap. 14) Vector fields and their applications. (Chap. 15) Integration over curves in R2 and R3 by parameterization; work integrals, and applications. (Chap. 15) Integration over surfaces in R3 by parameterization - flux integrals, surface area, and applications. (Chaps. 14,15) Calculus of vector fields; curl and divergence of vector fields; Green’s Theorem, Stokes’ Theorem, and the Divergence Theorem. (Chap. 15)

Text: Multivariable Calculus, 6th Edition by Edwards & Penney (ISBN 0130339679 for softcover edition).

Singular Sensations - Steve Strogatz in the New York Times

Supplementary Notes: 18.02 Notes authored by Prof. Arthur Mattuck of the MIT Mathematics Department.

Homework: Homework will be posted on the course website and will be due approximately weekly. Typical assignments will include some exercises that are to be turned in as well as additional practice problems. Homework may be submitted in class or at my office, but it should be completed by the posted due date. Additional time will only be given if requested before the due date and if appropriate for the circumstances. You should not consult any solutions manual in preparing your assignments. You are encouraged to work with your fellow students on the homework, but your written solutions must be your own. Solutions will be made available (as PDF files) on the course website shortly after they are due.

Topics and Assignments are posted in the Course Calendar.

Mathlet (Java applet) for Curves and Surfaces (may be helpful for P-sets)

If ever the MIT mail servers are not accessible from the outside world and you need to get in touch, you can also contact me at either robert@math.rwinters.com or Robert@rwinters.com.