Course website for 18.089, Summer 2010

Instructor information and contact

Name: Joel Lewis
This page is the archived version of a class taught in summer 2010. For information about current or future versions of this class, please contact the appropriate office in the math department.

Course meeting place/time

June 1 to June 4: 9 AM - 12:30 PM in room 12-142
June 7 to July 9: MTWR 12:30 - 2:30 PM and F 11:30 - 1:30, in room 2-132

Note that class meets every weekday except Monday, July 5.

Basic course data

Overview: This course is a review of single- and multi-variable calculus for students who already have some familiarity with calculus. The first week will be a fast-paced review of single-variable calculus (18.01). The remaining five weeks will cover multi-variable calculus (18.02).
Text: George F. Simmons, Calculus with Analytic Geometry, 2nd edition.
Other texts: Arthur Mattuck's course notes for 18.02 are available from CopyTech, 11-004, or free on OCW. I will primarily rely on Simmons' text, though I may also make use of James Stewart, Multivariable Calculus.

Assignments

Homework 1: Due Wednesday, June 9. Part 1: Choose five questions from the suggested questions for Lecture 5, including at least one question from each of the four sections (length & direction; dot products; determinants; cross products). Part 2: Do 1F-1b, 1F-4, 1E-1a, 1E-3a, and compute the inverse of the matrix in 1G-1 by two different methods.
Homework 2: Due Monday, June 14. Questions (.pdf), final version. (Note that there is no question 9, I'm just too lazy to correct the final PDF.)
Homework 3: Due Tuesday, June 22. Questions (.pdf), final version. Question 1 makes reference to this .jpg file. (Source: Stewart's Multivariable Calculus: Concepts and Contexts, 2nd edition, p. 759, and doubtless several other versions of Stewart's books, as well; distributed to students under the fair-use provision.) For the optional version of question 9, you may use the fact that the given ellipsoid has volume 4pi/3 * abc.
Homework 4: Due in class Friday, June 25.
1. Find a simple (or at least simpler) form for the surface area of the portion of the surface given by the equation z = f(x, y) that lies above the region R in the xy-plane. (Your answer will still involve the function f and/or some of its derivatives, the region R, and a double integral.)
2. Find the area of the paraboloid z = x² + y² that lies below the plane z = a². (Hint: use the previous problem to write down a double integral, then evaluate it by switching to polar coordinates.)
3. Find the volume of the solid lying under the plane z = 2x + 5y + 1 and above the rectangle {-1 ≤ x ≤ 0 and 1 ≤ y &le 4}.
4. Find the volume of the solid in the first octant bounded by the surface z = 9 - y² and the plane x = 2.
5. Let R be the region defined by the inequalities {0 ≤ z ≤ 6 - x - y, 0 ≤ y ≤ 6 - x, 0 ≤ x ≤ 6}. Write the triple integral over R of the function f(x,y,z) as an iterated integral in the order dx dy dz (so integrating first with respect to x, then y, then z).
6. A cube has vertices (0,0,0), (1,0,0), (0,1,0), ..., (0,1,1), (1,1,1). Its density is proportional to the square of the distance from one corner. Compute the mass of the cube (in terms of the constant of proportionality).
7. In class, we discussed the solid bounded below by the xy-plane and above by the surface z = 1 - x^2 - y^2. Find the centroid (= center of mass, assuming constant density δ = 1) of this solid by integrating in cylindrical coordinates. (Feel free to make use of symmetry to decrease the number of integrals you must compute.)
Homework 5: Due Friday, July 2. Questions (.pdf), final version.
Homework 6: Due Friday, July 9. Questions (.pdf), question 8 corrected July 8, 10:45 AM, final version.

Lecture summaries and notes

If possible, I will keep this updated with a brief summary of topics covered in each lecture. Suggested problems come from the 18.02 course notes (linked above).

1 and 2. Taught by Professor Mattuck. Differentiation. Trig and exponential functions. Antidifferentiation by substitution. Fundamental Theorem of Calculus. Basic definite integration. Basic numerical integration.
Simmons chapters 1-4, 5.1-3, 6, 8, 9.

Lecture notes 1 and 2 (.pdf)
3. Applications of integration: area between curves, volumes of solids of rotation, hydrostatic force. Partial fractions. Integration by parts. Differential equations.
Simmons 5.4, 7.1-4, 7.8, 10.

Lecture notes 3 part 1 (.pdf), Lecture notes 3 part 2 (.pdf), Lecture notes 3 part 3 (.pdf)
4. L'Hopital's rule. Improper integration. Numerical series. Taylor series.
Simmons 12.2-4, 13.1-7, 14.

Lecture notes 4 (.pdf)
5. Vectors and vector operations. Matrices and determinants.
Simmons 17.3, 18.1-3.

Lecture notes 5 (.pdf)

Suggested problems: Length and direction: 1A-1, 1A-2, 1A-3, 1A-7. Dot product: 1B-1 through 1B-5, 1B-7. Determinants: 1C-1, 1C-3, 1C-4, 1C-5. Cross products: 1D-1, 1D-2, 1D-5, 1D-6 (hint: you can choose where to orient the cube in space to make the numbers nice to work with).
6. Matrices. Linear transformations. Matrix multiplication. Matrix inverses (two different methods). Equations of lines and planes in 3-space.
Simmons 18.4.

Lecture notes 6 (.pdf)

Suggested problems: Lines and planes: 1E-1, 1E-3, 1E-4, 1E-6 (although in fact the first three go beyond what we did today). Matrix arithmetic: 1F-1, 1F-3, 1F-5, 1F-8a. Inverting matrices: 1G-1, 1G-3, 1G-7.
7. Vector functions and parametric equations: parametric equations, velocity, speed, acceleration, arclength, curvature. The cycloid.
Simmons 17.1-5. A nice discussion of one situation in which the cycloid arises is given in the appendix to chapter 17 (pp. 629-631).

Lecture notes 7 (.pdf)

Suggested problems: Wrap-up of lines and planes: 1E-1 through 1E-6. Parametric equations: 1I-1 through 1I-4, 1J-1, 1J-2, 1J-5, 1J-9, 1J-10. For more of a challenge, try the sequence of proof-based questions 1J-3, 1J-4, 1J-5, 1J-7, and 1J-8. Cycloid: how does the arclength of one arch of the cycloid compare to the radius of the circle?
8. Curvature and the unit normal. The "natural" tangent-normal coordinate frame. Polar coordinates.
Simmons 17.5-6, 16.1

Lecture notes 8 (.pdf)

Suggested problems: Polar coordinates: in the 18.01 readings (OCW link (.pdf)), 4H-1, 4H-2a(i), 4H-2b, 4H-3. See also assigned homework above.
9. Converting polar coordinates to parametric or rectangular coordinates. Arclength, areas and tangent lines in polar coordinates.
Simmons chapter 16

Lecture notes 9 (.pdf)

Suggested problems: In the 18.01 readings (OCW link (.pdf)), 4I-1 through 4I-5.
10. Functions of several variables. Contour plots/level curves. Partial derivatives. Higher partials.
Simmons 19.1-2

Lecture notes 10 (.pdf)

Suggested problems: Graphing and partials: 2A-1 through 2A-5.
11. Tangent planes. Differentials. Linearizations/linear approximations.
Simmons 19.3-4

Lecture notes 11 (.pdf)

Suggested problems: Any question from section 2B or 2C. Those that are closest to what we've discussed in class are 2B-1, 2B-2, 2B-3, 2B-5, 2B-8 (helpful thing to remember or look up: Law of Cosines), 2C-1; questions 2B-4, 2B-6, 2B-7, 2B-8b, 2B-9, 2B-10, 2C-2 and 2C-3 relate to error estimation, which we didn't really discuss in class but which is one possible application of the linearization method.
12. Directional derivatives. Definition of the gradient. Exam 2.
Simmons 19.5

Lecture notes 12 (.pdf)

Suggested problems: 2D-1, 2D-2, 2D-6, 2D-7. Note that the notation "df/ds|_u" for the directional derivative of f is a variation on the awful notation used by Simmons; in class, we wrote this as "D_uf".
13. The gradient and why we love it. Level surfaces. Chain rules for partial derivatives.
Simmons 19.2 (for level surfaces), 19.5-6

Lecture notes 13 (.pdf)

Suggested problems: More on gradients and directional derivatives: 2D-3, 2D-4, 2D-9. For the chain rule, any question from section 2E is appropriate; if you just want plug-and-chug practice, try a few parts from 2E-1, 2E-2 and 2E-4. For a slightly more theoretical flavor, try 2E-3, 2E-6, 2E-7, 2E-8. (Of these, I personally think 2E-7 is most interesting.)
14. Max-min problems. 2nd derivative test. Lagrange multipliers.
Simmons 19.7-8

Lecture notes 14 (.pdf)

Suggested problems: Basic max-min problems: 2F-1, 2F-2, 2F-5. 2nd derivative test: 2H-1, 2H-6. Lagrange multipliers: any of 2I-1 through 2I-4.
15. Absolute max and min on a closed and bounded domain. Double integrals and iterated integrals. Applications of double integrals. Double integrals in polar coordinates. Parametric surfaces.
Simmons 20.1-20.4

Lecture notes 15 (.pdf)

Suggested problems: Max-min: 2H-5 and 2H-4b (for 2H-4b, consider separately the problem of maximizing the function on each of the four boundary segments). Double integration: 3A-2, 3A-3, 3A-5. In polar coordinates: 3B-2, 3B-3, 3B-4. Centers of mass: 3C-2, 3C-4. (In fact, all of the problems in section 3A and 3B are fine, but the trickiest problems relating to double integration tend to be those that relate to finding the correct regions and bounds of integration.)
16. Parametric surfaces. The torus. Surface areas. Cylindrical coordinates.
Simmons 18.7 and 20.8.

Lecture notes 16 (.pdf)

Suggested problems: ???
17. Spherical coordinates. Triple integrals. Triple integrals in cylindrical coordinates.
Simmons 18.7, 20.5, 20.6.

Lecture notes 17 (.pdf)

Suggested problems: 5A-2. Any other question from section 5A of the notes would also be fine; for questions concerning the moment of inertia, see Simmons 11.4 and 20.3, or see Wikipedia, or (my recommendation) don't worry about what the words "moment of inertia" mean and just find the right bounds of integration (without worrying at all about what the integrand should be).
18. Triple integrals in spherical coordinates. The Jacobian and general change of variables.
Simmons 20.7. OCW 18.02 notes section CV.

Lecture notes 18 (.pdf)

Suggested problems: Spherical: 5B-1, 5B-2, 5B-4. (Be especially careful with 5B-1c -- neither of the bounds on ρ are constant. You need to figure out the equation of the plane z = 1 in spherical coordinates to get the lower bound on ρ, and the equation of the sphere centered at (0,0,1) (NOT the origin) with radius 1 in spherical coordinates to get the upper bound.) Change of variables: 3D-1, 3D-4.
19. Line integrals and vector fields.
Simmons 21.1, OCW 18.02 notes V1.

Lecture notes 19 (.pdf)

Suggested problems: 4A-1, 4A-2, 4A-3, 4A-4.
20. Work done by a vector field. Path-independence. Conservation of energy.
Simmons 21.2, OCW 18.02 notes V1.

Lecture notes 20 (.pdf)

Suggested problems: Computational practice: 4B-1, 4C-1, 4C-2. Good questions to test your conceptual understanding: 4B-2, 4B-3, 4C-3.
21. Conservative vector fields. Green's Theorem.
Simmons 21.3.

Lecture notes 21 (.pdf)

Suggested problems: Green's Theorem and conservative fields: 4D-1 through 4D-4, 4D-6.
22. Proof of Green's Theorem. Non-simply-connected regions. Line integrals for flux. Computing potential functions. Three dimensions.
Simmons 21.3, OCW 18.02 notes V2, V5, V8, V11; notes V3 and V4 are also relevant but deal with what I consider more tangential issues; notes V6 and V7 are both interesting but I won't cover the material in them at all.

Lecture notes 22 (.pdf)

Suggested problems: 4E-1, 4G-2, 4G-4, 4G-5, 4G-6, 6A-1 through 6A-4, 6D-1, 6D-2, 6E-5.
23. The differential operator ∇ ("del"). Divergence and curl. div curl and curl grad. Physical interpretations of divergence and curl.
Simmons, beginnings of sections 21.4 and 21.5.

Lecture notes 23 (.pdf)

Suggested problems: 6C-1, 6C-2, 6E-2, 6E-3, 6E-5, 6H-3
24. Surface integrals. The Divergence Theorem.
Simmons 21.4, OCW 18.02 notes V9 and V10

Lecture notes 24 (.pdf)

Suggested problems: 6B-1 through 6B-7, 6C-3 through 6C-7, 6C-11.
25. The Divergence Theorem. Stokes' Theorem.
Simmons 21.5, OCW notes V13

Lecture notes 25 (.pdf)

Suggested problems: Divergence Theorem: 6C-3 through 6C-7, 6C-11, 6F-4b. Stokes' Theorem: 6F-1, 6F-3.
26. Stokes' Theorem.
Simmons 21.5, OCW notes V13

Lecture notes 25 (.pdf)

Suggested problems: Stokes' Theorem: 6F-1, 6F-3, 6F-5. General interest: 6H-9