Schedule
Here is a complete list of topics organized by week (you can skip to this week).
Each topic has some associated content for you to study and take notes on; sometimes I link a YouTube playlist of videos I made for a previous semester (like and subscribe, y'all). For more information about the course, see our class homepage (Sections 1 and 2).
Week 01 (08-24 to 08-28)
In the first week I briefly discussed the syllabus and answered some questions about the course.
Geometry in Three Dimensions (§12.1)
Coordinates, distance, and spheres in \(\mathbb{R}^3\). Derived the distance formula for \(\mathbb{R}^3\).
Introduction to Vectors (§12.2)
Vectors as directed line segments (up to equivalence). Vector operations and their properties. The standard basis in \(\mathbb{R}^3\).
Week 02 (08-29 to 09-04)
Homework 00 due 11:59pm on 3 September via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson"). Deadline extended to 8 September; see this video I made on Gradescope submissions.
Dot Product (§12.3)
Introduction to the dot product. Geometric interpretation of the dot product. Orthogonal projection. Direction angles (in 3-space).
Cross Product (§12.4)
Introduction to and derivation of the cross product. Algebraic and geometric properties of the cross product. The magnitude of the cross product and the right hand rule. Using the cross product to determine a plane in \(\mathbb{R}^3\).
I wrote a short document explaining the formula for the determinant of a \(2 \times 2\) matrix.
Week 03 (09-05 to 09-11)
This week there are no classes on Monday (Labour Day) and Wednesday (Rosh Hashanah).
Homework 01 due 11:59pm on 10 September via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Lines and Planes in 3-Space (§12.5)
Lines in 3-space. Planes in 3-space. Computations involving lines and planes. Practice! My YouTube lectures.
Week 04 (09-12 to 09-18)
I gave office hours this week despite the holiday on Thursday (Yom Kippur).
Quadratic Surfaces in 3-Space (§12.6)
Drag the sliders around to change the shape for the following quadratic surfaces.
| Equation | Name | \(x=k\) | \(y=k\) | \(z=k\) |
|---|---|---|---|---|
| \(\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) | Ellipsoid | Ellipse | Ellipse | Ellipse |
| \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z}{c}=0\) | Elliptic Paraboloid | Parabola | Parabola | Ellipse |
| \(\frac{x^2}{a^2}-\frac{y^2}{b^2}-\frac{z}{c}=0\) | Hyperbolic Paraboloid | Parabola | Parabola | Hyperbola |
| \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1\) | One-sheet Hyperboloid | Hyperbola | Hyperbola | Ellipse |
| \(\frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=0\) | Cone | Hyperbola | Hyperbola | Ellipse |
| \(-\frac{x^2}{a^2}-\frac{y^2}{b^2}+\frac{z^2}{c^2}=1\) | Two-sheet Hyperboloid | Hyperbola | Hyperbola | Ellipse |
Also visualize cross-sections of quadratic surfaces, and see how to deform hyperboloids through a cone.
Space Curves and Calculus (§13.1 and §13.2)
Space curves as vector-valued functions. Limits, continuity, derivatives, and integrals of space curves.
Here are some nice examples of space curves I made in GeoGebra.
| helix | moment curve | some toroidal curves |
| this thing | trefoil knot | \((p, q)\)-torus knots |
Calculus on Space Curves (§13.2)
More limits, continuity, derivatives, and integrals of space curves.
Arc Length (§13.3)
Computing arc length of a curve in \( n \)-space (with reminders about some integration techniques).
Week 05 (09-19 to 09-25)
Homework 02 due 11:59pm on 20 September via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Wednesday was a student-driven review session for Exam 1.
Motion in Space (§13.4)
Mostly this is stuff you've already seen, now asked with words (GASP!).
Exam 1 (Friday, 24 September)
This exam covers material from Textbook Sections 12.1 - 12.6 and 13.1 - 13.4. This video may help you before the exam.
Update
Exam 1 has been graded. You should review my comments on Gradescope. Regrade requests are open from 2 October to 4 October.
Notes
| Mon | Matthew Mauersberg (s01) | Quinn O'Brien (s01) | Nicholas Frank (s02) | Depo Oyerinde (s02) |
I wrote a short reminder of integration techniques because I am dangerously kind. This is only meant to get you thinking again, and it is not meant to be comprehensive. You should consult the textbook (or me) if you would like help with any of the techniques.
Week 06 (09-26 to 10-02)
Multivariate Functions (§14.1)
Functions of several variables.
See my GeoGebra implementations of a function grapher and a level curve generator for help with intuition.
Multivariate Limits and Continuity (§14.2)
Limits and continuity for functions of several variables. Too many examples.
Derivatives of Multivariate Functions (§14.6 and §14.3)
Derivatives of functions of several variables. Directional derivative. Partial derivatives.
Week 07 (10-03 to 10-09)
Tangent Planes (§14.4)
Tangent planes. The differential.
Multivariate Chain Rule (§14.5 and §14.6)
The chain rule for functions of several variables. The Implicit Function Theorem.
Multivariate Optimization (§14.6 and §14.7)
Introduction to the gradient. Rephrasing past results with the gradient (directional derivatives and the chain rule). Brief review of optimization ideas from Calculus I.
Week 08 (10-10 to 10-16)
There are no office hours on Thursday and no class on Friday (Fall Break). You could use the day off to get a jump on the homework due Monday (18 October) and to study for the next exam…
More Optimization (§14.7)
Fermat's Extremum Theorem. Extreme Value Theorem. Local optimization with the Second Derivative test.
Lagrange Multipliers (§14.8)
An overview of Lagrange Multipliers with examples.
I made GeoGebra sheets to illustrate level curves and the gradient and to help you visualize the method of Lagrange multipliers.
Week 09 (10-17 to 10-23)
Homework 03 due 11:59pm on 18 October via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Homework 04 due 11:59pm on 22 October via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Make sure you start studying for the next exam…
Double Integrals (§15.1, §15.2, and §15.3)
Definite integrals of functions of two variables over a rectangle. Integrating over more general domains. Coordinate changes to polar coordinates for integration.
I made a GeoGebra sheet for approximating regions by rectangles (in case you forgot from Calculus I).
Practice! My YouTube lectures. More Practice! Another of My YouTube lectures.
Note that we did not cover section 15.9 on general coordinate transformations yet (I'm running slightly behind my plan, but right on time with this schedule).
Week 10 (10-24 to 10-30)
Monday is a review session for Exam 2, driven entirely by your questions.
Office hours are moved to Tuesday this week (the day before the exam). So I will be in WH 332 from 08:00 – 11:00 on Tuesday, no appointment needed. I will NOT hold office hours on Thursday.
Exam 2 (Wednesday, 27 October)
This exam covers material from Textbook Sections 14.1 - 14.8 and 15.1 - 15.3.
Do get a good night's sleep before the exam; that's more important than any last-minute studying. This video may be useful while you study; if that doesn't help, try this video instead.
Update
Exam 2 has been graded. You should review my comments on Gradescope. Regrade requests are open from 29 October to 2 November.
Introduction to Triple Integrals (§15.6)
Triple integrals over general regions. Discussion of reparameterization. Examples and student questions.
Notes
In Friday's lecture I made a mistake regarding a weirdly parameterized region. See these notes for a discussion of my mistake and corrected computations (always modulo arithmetic).
| Fri | Cody Jacobs (s01) | Adin Keiter (s01) | Colin Schepis (s02) | Hannah Brodsky (s02) |
Week 11 (10-31 to 11-06)
This week I will do some travelling to give a maths talk, so there will not be office hours on Thursday; these office hours are made up in week 13.
Coordinate Changes for Triple Integrals (§15.7, §15.8, and §15.9)
Cylindrical coordinates (i.e. one plane in polar coordinates). Spherical coordinates (i.e. the 3D version of polar coordinates). General coordinate changes in integration.
Vector Fields (§16.1)
Vector fields. Conservative vector fields. Computing potential functions.
I made a GeoGebra sheet to help you visualize vector fields.
Notes
The lectures on Wednesday were delivered by substitute lecturers Uly and Nick.
Week 12 (11-07 to 11-13)
Homework 05 due 11:59pm on 10 November via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Line Integrals (§16.2 and §16.3)
Line integrals of scalar functions. Line integrals of vector fields. Fundamental Theorem of Line Integrals.
Green's Theorem (§16.4)
Green's Theorem for converting line integrals to double integrals.
Curl and Divergence (§16.5)
Curl of a vector field on \( \mathbb{R}^3 \). Divergence of a vector field. Relationship between curl and divergence.
I made GeoGebra sheets to compute divergence and curl interactively; this should help build your intuition.
Notes
I had to leave town on Monday afternoon; the lectures on Wednesday and Friday were online (as were office hours on Thursday).
| Mon | Kali Hayes (s01) | Anonymous (s01) | Lake Hakes (s02) | Jonathan Sarasohn (s02) |
| Wed | Renee Mui (s01) | Bingxin Chen (s01) | Bryce Stracher (s02) | |
| Fri | Anonymous (s01) | Candace Polisi (s02) | Tony Ni (s02) |
Week 13 (11-14 to 11-20)
Homework 06 due 11:59pm on 15 November via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Wednesday is a review session, driven entirely by your questions, for Exam 3.
This week I will hold office hours on both Tuesday (thus making up the office hours of week 11) and on Thursday from 08:00–11:00.
Parametric Surfaces (§16.6)
Introduction to surfaces in \( \mathbb{R}^3 \). Note that this material is not on Exam 3 (but will be on Exam 4).
I made some GeoGebra sheets for interesting surfaces.
- torus \((\sqrt{x^2 + y^2} - r)^2 + z^2 = R^2\) with major and minor radii \(R > r > 0\)
- surfaces of revolution about the \(z\)-axis
- Moebius band
- general parametric surfaces
- tangent planes
- surface approximations via tangent planes
Exam 3 (Friday, 19 November)
This exam covers material from textbook sections 15.5 - 15.8 and 16.1 - 16.5.
Failure to attend the exam will result in an immediate score of 0, so don't skip.
Don't over-stress about the exam; you may find these three videos useful while studying. Also be sure to get a good night's sleep before the exam.
Update
Exam 3 has been graded. You should review my comments on Gradescope. Regrade requests are open from 26 November to 29 November.
Notes
This week lectures are online, as are office hours.
I wrote a solution to homework 5 problem 2 because it was oft-requested. This is handwritten (at least until I find the time to properly write it up in LaTeX).
| Mon | Yumiko Inoue (s01) | Cody Jacobs (s01) | Triana Cano (s02) | Stephanie Beck (s02) |
Week 14 (11-21 to 11-27)
Monday is the only lecture this week, and there are no office hours on Thursday for the holiday. Happy Thanksgiving!
Calculus on Parametric Surfaces (§16.6 and §16.7)
Computing with surfaces in \( \mathbb{R}^3 \). Tangent planes. Surface area. Brief introduction to surface integrals.
Week 15 (11-28 to 12-04)
Surface Integrals (§16.7)
Computing surface integrals directly. Flux.
Stokes' Theorem (§16.8)
Introduction to Stokes' Theorem. Many examples.
Week 16 (12-05 to 12-11)
Homework 07 due 11:59pm on 8 December via Gradescope. Collaboration encouraged, after your own first attempt; you must cite your collaborators (e.g. "I worked with Sally O'Student and Jimbo McPerson").
Friday is a review session, driven entirely by your questions, for the Final Exam.
Divergence Theorem (§16.9)
Introduction to the Divergence Theorem. Many examples.
Final Exam (14 December)
The final exam is scheduled for 14 December at 12:50pm in GW 69EX. The page linked above has an up-to-date schedule in case something changes; that page supersedes this one in case of conflict.
Do email if you have any questions or concerns. Be sure to take a break from studying to do something relaxing. Best of luck!