Schedule

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Purpose of this Page

Here is a complete list of topics organized by week. As the semester progresses I add additional content, study suggestions, and descriptions. If you are in the course, you should bookmark this page and check back often.

For more information about the course, see our class homepage (section 3).

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).

A word of caution: sometimes (being human) I make mistakes—when that happens in a YouTube video, I write a note about it in the description of the video (on YouTube).

Week 01 (01-23 to 01-29)

On Wednesday I briefly discussed the syllabus and answer some questions about the course.

Optional assignment 00 is due by <2022-02-11 Fri 23:59> on Gradescope. This should get you acquainted with my expectations for Gradescope submissions; see also this video explaining Gradescope submissions.

Geometry in Three Dimensions (§12.1)

Coordinates, distance, and spheres in \(\mathbb{R}^3\).
Inequalities 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: addition, subtraction, scalar multiplication, and magnitude.
The standard basis in \(\mathbb{R}^3\).

Week 02 (01-30 to 02-05)

Dot Product (§12.3)

Introduction to the dot product.
Algebraic properties of the dot product.
Geometric interpretation of the dot product.
Orthogonal projection.
Direction angles (in 3-space).

Cross Product (§12.4)

Introduction the cross product and determinants.
Algebraic properties of the cross product.
The magnitude of the cross product geometrically.
Other geometric properties of the cross product.
The right hand rule.

I wrote a short document explaining the formula for the determinant of a \(2 \times 2\) matrix. This is not required, but might be interesting to some of you.

Here is a brief video related to the right hand rule.

Week 03 (02-06 to 02-12)

Reminder: Optional assignment 00 is due by <2022-02-11 Fri 23:59> on Gradescope.

Lines and Planes in 3-Space (§12.5)

Lines in \( n \)-space.
Different expressions for lines: vector, parametric, and symmetric equations.
Planes in 3-space.

Quadratic Surfaces in 3-Space (§12.6: Self-study)

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.

Practice problems for 12.6. My YouTube lectures.

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

Week 04 (02-13 to 02-19)

Exam 1 has moved to Friday, 25 February.

Arc Length (§13.3)

Arc length of curves in \( 3 \)-space.
The arc length function.
Unit-speed reparameterization (by arc length!).

Motion in Space (§13.4)

Position, velocity, speed, and acceleration.

Multivariate Functions (§14.1)

Functions of several variables.
Domain and evaluation.
Level curves and contour maps.

See my GeoGebra implementations of a function grapher and a level curve generator for help with intuition.

Multivariate Limits and Continuity (§14.2)

Formal definition of the limit.
The Curves Criterion.
Algebraic manipulation and evaluation.
Limits via coordinate change (e.g. to polar coordinates).
Continuity for functions of several variables.

Week 05 (02-20 to 02-26)

Wednesday's lecture is a review session for the exam on Friday. Exam 1 has moved to Wednesday, 2 March due to inclement weather.

Derivatives of Multivariate Functions (§14.6 and §14.3)

Directional derivative: definition and examples.
Partial derivatives: definition and computational aspects.
Clairaut's Theorem.
Gradient: definition and basic properties.

Week 06 (02-27 to 03-05)

There is an exam on Wednesday…

Exam 1 (Wednesday, 2 March)

This exam covers Chapters 12 and 13.

This video may help you before the exam.

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.

Update: Exams graded.

Review my comments on Gradescope. You will have from 16 March until 23 March to make a regrade request through Gradescope. I will actually REGRADE the problem; this means you could lose points.

Tangent Planes (§14.4)

Introduction to the gradient.
Tangent planes.
The (total) differential.

We discussed this topic in two separate lectures this week, filling in some Calculus I gaps in the process. I gave a geometric argument for the tangent plane formula, and described the tangent plane's relationship with linear approximation. We finally had a brief discussion of the total differential as a formal structure and its use for linear approximation.

Week 07 (03-06 to 03-12)

This is the last week before spring break!

Office hours this week were moved to Friday.

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)

Directional derivatives via the gradient.
The gradient points in the direction of greatest increase!
Brief review of optimization ideas from Calculus I.
Fermat's Extrema Theorem: Extrema occur at critical points.
Extreme Value Theorem.
Local optimization with the Second Derivative Test.

Week 08 (03-20 to 03-26)

Note the week-long gap (for spring break).

Friday's class is a review session for the exam on Monday.

Lagrange Multipliers (§14.8)

Method of Lagrange Multipliers.
Many, many examples.

I made GeoGebra sheets to illustrate level curves and the gradient and to help you visualize the method of Lagrange multipliers.

Week 09 (03-27 to 04-02)

Due to a dangerous kind streak, I am offering a review session in WH 100E on <2022-03-27 Sun> from 16:00–18:00. Bring questions and problems you want to work through!

Exam 2 (Monday, 28 March)

This exam covers Chapter 14.

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: Exams graded.

Review my comments on Gradescope. You will have from 30 March until 6 April to make a regrade request through Gradescope. I will actually REGRADE the problem; this means you could lose points.

Double Integrals (§15.1 and §15.2)

Definite integrals of functions of two variables over a rectangle.
Integrating over more general domains.

I made a GeoGebra sheet for approximating regions by rectangles (in case you forgot from Calculus I).

Here are requested solution to suggested problems 15.1.21 and 15.2.39.

Week 10 (04-03 to 04-09)

More Double Integrals (§15.3 and §15.9)

General coordinate changes and the Jacobian.
Coordinate changes to polar coordinates for integration.

Here is a requested solution to one of the practice problems.

Introduction to Triple Integrals (§15.6)

Triple integrals over general regions.
Discussion of reparameterization.

Here is the solution to the exercise from the end of lecture.

Week 11 (04-10 to 04-16)

There is no class on Friday.

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).

Week 12 (04-17 to 04-23)

There is no class on Monday; Wednesday is a review session for Exam 3.

Exam 3 has moved to Monday, 25 April.

Vector Fields (§16.1: Self-study)

Vector fields.
Conservative vector fields.
Computing potential functions.

I made a GeoGebra sheet to help you visualize vector fields.

Practice problems for 16.1. My YouTube lectures.

Line Integrals (§16.2 and §16.3)

Line integrals of vector fields.
Fundamental Theorem of Line Integrals.

Here is a requested solution to [https://ia601400.us.archive.org/28/items/math323-s22/solution₁₆-2x3-1e-practice.pdf][practice problem 16.2-3.1e]] (note that we did not cover line integrals of scalar functions in lecture, so this will not be tested).

Week 13 (04-24 to 04-30)

Exam 3 (Monday, 25 April)

This exam covers Chapter 15.

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: Exams graded.

Review my comments on Gradescope. Regrade requests will be open from <2022-05-02 Mon 00:00> to <2022-05-06 Fri 23:59>. Remember that you can lose points on regrades!

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.

Parametric Surfaces (§16.6 and §16.7)

Introduction to surfaces in \( \mathbb{R}^3 \). 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
Computing with surfaces in \( \mathbb{R}^3 \).

Week 14 (05-01 to 05-07)

Parametric Surfaces (§16.6 and §16.7)

Tangent planes.
Introduction to surface integrals: Flux.

Stokes' Theorem (§16.8)

Introduction to Stokes' Theorem.
Many examples.

Divergence Theorem (§16.9)

Introduction to the Divergence Theorem.
Many examples.

Week 15 (05-08 to 05-14)

Review for Final Exam.

Monday is a wrap-up lecture and problem session. Wednesday is a full review session.

Here are some promised solutions:

Another example using the Divergence Theorem (in two ways)
Another Stokes's Theorem problem (in three ways)
Yet another Stokes's Theorem problem

Final Exam (Friday, 13 May)

The final exam is scheduled for 13 May at 08:00am 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!

Final Update

Your final grades have been submitted (they should be available on BU Brain as soon as grades roll). I won't release the final exams on Gradescope (finals are confidential). If you would like to know your grade on the exam, please email me to set up a time to meet in my office. I will be available only until Friday, 20 May 2022 for this purpose.

This is my last update of this page. Thank you all for the semester, and best of luck in the future!