Week 1, January 15–17: sections 11.1–11.4
- Tuesday
- We did introductions and looked at some problems from Section 11.1. For Wednesday, you should read Sections 11.2 and 11.3 and try the exercises.
- Wednesday
- We discussed dot products and cross products of vectors. For Thursday, you should read Section 11.4 about lines and planes, and you should be prepared for a quiz on dot products and cross products.
- Thursday
- We discussed the scalar triple product, took a quiz on dot
products and cross products (solutions are available), and discussed lines and planes in three dimensions.
- You should be working the recommended exercises for each section. The first homework exercises to hand in—think of these as a take-home quiz—are numbers 32, 36, 38, and 46 in the Review Exercises on pages 715–716. These four exercises are due at the beginning of next class (Tuesday, January 22).
Week 2, January 22–24: sections 11.5–11.8
- Tuesday
- We first worked some problems about lines and then looked at more general equations of curves in three dimensions and the notion of the tangent vector. You should work on the suggested exercises from Section 11.6. Tomorrow we will discuss Section 11.5 and then start Section 11.7.
- Wednesday
- We discussed quadric surfaces and how to understand them by looking at two-dimensional slices.
We did not get to Section 11.7, so we will work on that section tomorrow.
Be prepared for a quiz tomorrow on Sections 11.1–11.4 and 11.6.
- Thursday
- Here is the assignment to hand in next class (Tuesday, January 29):
- Section 11.5, page 688, exercise 44.
- Section 11.6, page 698, exercise 60.
- Section 11.7, page 705, exercises 4 and 14.
In class, we took a quiz (solutions are available) and discussed the orthogonal frame of tangent, normal, and binormal vectors as well as the notion of curvature. We skipped over Section 11.8. Next week we will start Chapter 12.
Week 3, January 29–31: sections 12.1–12.4
- Tuesday
- We discussed two graphical ways to understand functions of two variables: level curves (contour plots) in the plane and planar slices of the three-dimensional graph. Also we discussed partial derivatives. (Section 12.2 is held over to the next day.)
- Wednesday
- We discussed limits and continuity of multivariable functions (Section 12.2) and second partial derivatives. Be prepared for a quiz tomorrow on Sections 12.1–12.3.
- Thursday
- In class, we took a quiz (solutions are available) and discussed tangent planes.
- Here is the assignment to hand in next class (Tuesday, February 5):
- Chapter 12 review (page 790), exercises 34 and 42.
- Chapter 11 review (page 716), exercises 68 and 78.
Week 4, February 5–7: sections 12.5–12.7
- Tuesday
- We discussed differentials and the chain rule for functions of more than one variable. You should work on the exercises for Section 12.5. Tomorrow we will work on Section 12.6 (about directional derivatives and the gradient).
- Wednesday
- We discussed directional derivatives, the gradient of a function, and the application to finding tangent planes. Be prepared for a quiz tomorrow on Sections 12.4–12.6.
- Thursday
- In class, we took a quiz (solutions are available) and discussed local extrema of functions of two variables (critical points and the second-derivative test).
- Here is the assignment to hand in next class (Tuesday, February 12):
- Chapter 12 review (page 791), exercises 53 and 61.
- Chapter 11 review (page 716), exercises 51 and 53.
Week 5, February 12–14: section 12.8, review, Exam 1
- Tuesday
- We discussed methods for finding extreme values of a function on the boundary of a region—in particular, the method of Lagrange multipliers.
- Wednesday
- The topic for class is reviewing for the exam that takes place on Thursday.
- Thursday
- The first examination was given, and solutions are available.
Week 6, February 19–21: sections 13.1–13.3
- Tuesday
-
We discussed two-dimensional integrals and computed some examples as
iterated integrals. Tomorrow we will discuss section 13.3.
- Wednesday
- We started section 13.3 about double integrals over regions that
are more general than rectangles. Expect a quiz Thursday over
section 13.2 and the first part of section 13.3 (the
material covered in Exercises 1–18 on page 812 of the textbook).
- Thursday
- We took a quiz (solutions are available) and worked on further exercises from section 13.3.
- Here is the assignment to hand in next class (Tuesday, February 26).
- Exercise 30 on page 805 in section 13.2.
- Exercise 24 on page 812 in section 13.3. (Some of you worked on this exercise in class today.)
- Exercise 44 on page 813 in section 13.3.
- Exercise 32 on page 463 in section 8.1. (This exercise is intended to refresh your memory about integration by parts and substitution.)
Week 7, February 26–28: section 13.4–13.7
- Tuesday
- We discussed polar coordinates (section 13.4) and worked some initial examples of integrals in polar coordinates (section 13.5). We will do more examples on Wednesday.
- Wednesday
- We continued working on section 13.5. Be prepared for a quiz
Thursday on sections 13.3, 13.4, and 13.5.
- Thursday
- We took a quiz (solutions are available)
and worked on some computations of surface area.
- Here is the assignment to hand in next class (Tuesday, March 5).
- Exercise 34 in the chapter review on page 864, which asks for the volume of the solid bounded by the cylinder \(x^2+y^2=4\) and the planes \(z=0\) and \(y+z=3\).
- Exercise 4 on page 832 in section 13.6 (about center of mass).
We did not do an example like this in class, so here is what you need to know: The mass \(m\) is the double integral (over the region) of the density function \(\rho(x,y)\). (That Greek letter is called rho.) The \(x\) coordinate of the center of mass is \((1/m)\) times the double integral of \(x\rho(x,y)\), and the \(y\) coordinate of the center of mass is \((1/m)\) times the double integral of \(y\rho(x,y)\).
Reality check: for a rectangular region, the center of mass had better end up being a point that is somewhere inside the rectangle!
- Exercise 4 on page 834 in section 13.7 (about surface area).
Hint: The double integral can be done by a routine substitution if you choose the right order for the iterated integral.
Week 8, March 5–7: sections 13.8–13.11
- Tuesday
- We worked on examples of triple integrals.
- Wednesday
- We discussed cylindrical coordinates and spherical coordinates and the corresponding volume elements \(r\,dr\,d\theta\,dz\) and \(\rho^2 \sin(\phi)\,d\rho\,d\theta\,d\phi\). The topics for Thursday's quiz are center of mass, surface area, and triple integrals in rectangular coordinates (sections 13.6–13.8).
- Thursday
- We took a quiz (solutions are available)
and worked on triple integrals in spherical coordinates. We did not get to section 13.11 about changing variables in multiple integrals.
The assignment for over Spring Break is to stay safe.
Spring Break, March 11–15
Week 9, March 19–21: sections 14.1–14.3
- Tuesday
- We discussed change of variables in multiple integrals and the Jacobian (Section 13.11).
- Wednesday
- We looked at examples of using a line integral \(\int_C \vec{F}\cdot d\vec{r}\) to compute work done by a force as a particle moves on a curve \(C\).
- Thursday
- In class, we worked on additional line integrals (work, mass of a wire, center of mass).
- Here is the assignment to hand in next class (Tuesday, March 26).
- In Section 14.2 on page 882, exercise 12.
- In the Chapter 13 review on pages 863–864, exercises 28, 68, and 76.
(These three problems all are triple integrals. In the first one, the main difficulty is figuring out the limits of integration. The second problem involves converting the integral to spherical coordinates. The third problem requires working out a Jacobian to implement a change of variables in a triple integral.)
Week 10, March 26–28: review, Exam 2
- Tuesday
- We discussed some of the homework problems and continued with section 14.3.
- Wednesday
- We reviewed for the second examination.
- Thursday
- The second examination was given, and solutions are available.
Week 11, April 2–4: sections 14.4–14.5
- Tuesday
- We derived Green's theorem and worked on examples.
- Wednesday
- We did some more examples of applying Green's theorem, and we saw the definitions of the gradient, divergence, and curl.
Be prepared for a quiz tomorrow about Green's theorem.
- Thursday
- Here is the assignment to hand in next class (Tuesday, April 9).
- On page 673 in section 11.3, exercise 14 (about finding the area of a parallelogram by using a cross product).
- On page 755 in section 12.4, exercise 4 (about a tangent plane).
- On page 835 in section 13.7, exercise 8 (about surface area).
- On page 905 in section 14.5, exercise 20 (about curl).
- In class, we took a quiz (solutions are available) and worked on the curl and the divergence of a vector field.
Week 12, April 9–11: sections 14.6–14.7
- Tuesday
- We discussed parametric surfaces and surface area.
- Wednesday
- We worked on surface integrals of the form \(\iint f(x,y,z)\,dS\). Be prepared for a quiz tomorrow on sections 14.5 (curl and divergence) and 14.6 (surface area of parametric surfaces).
- Thursday
- Here is the assignment to hand in next class (Tuesday, April 16).
- Exercise 74 on page 716 in the Chapter 11 Review.
- Exercise 46 on page 790 in the Chapter 12 Review.
- Exercise 30 on page 863 in the Chapter 13 Review.
- Exercise 38 on page 940 in the Chapter 14 Review.
- In class, we took a quiz and worked on flux integrals.
Week 13, April 16–18: sections 14.8–14.9
- Tuesday
- I posted solutions to the quiz from last time.
- In class, we discussed Stokes's theorem and worked some examples.
- Wednesday
- We worked more examples of applying Stokes's theorem. Be prepared for a quiz tomorrow on sections 14.7 and 14.8.
- Thursday
- We took a quiz (solutions are available) in groups, worked some examples of the divergence theorem, and discussed the Möbius strip (a nonorientable surface).
- There is no assignment to hand in next week, but you should be working exercises to review for the final examination.
Week 14, April 23–25: section 14.10, review
- Tuesday
- We worked on the true/false review exercises at the ends of the chapters.
- Wednesday
- We compiled lists of the highlights from Chapters 11, 12, 13, and 14.
- Thursday
- This was our final class meeting for the semester.
May 3, Final Examination, 15:00–17:00
Solutions to the final exam are available.