Schedule

We study the distance formula and spheres in 3-space.

Here are notes (printer-friendly) and some practice problems.

We also synchronously discussed the syllabus and structure of the course on Zoom.

We study vectors and their operations as well as the properties of vector operations, magnitude and direction, and the standard basis in 3-space. We'll also discuss examples in three parts: Act 1, Act 2, and Act 3.

Here are notes (printer-friendly) and some practice problems.

We study the algebraic and geometric properties of the dot product, together with our first examples; we resume with a discussion of orthogonal projection and see some more examples. We'll wrap things up with a short note on direction angles in 3-space.

Here are notes (printer-friendly) and some practice problems.

We study a derivation of the cross product, and to remember its formula with matrix notation. After a brief aside for our first examples, we will analyze the algebraic properties and geometric Properties of the cross product. We'll finish up with some more examples.

Here are notes (printer-friendly) and some practice problems.

I provided extra office hours 8am - 9:30am and 4:40pm - 6:10pm on Zoom.

We study lines (with a pause for some first examples) and planes (accompanied by more examples and still more examples) in 3-space.

Here are notes (printer-friendly) and some practice problems.

I gave a synchronous quiz administered during scheduled class times. I also set Homework 0: Using Gradescope due today (only accessible on Gradescope via email invitation).

We study cylinders and quadratic surfaces in 3-space with visualization software.

I made interactive GeoGebra demos for quadratic surfaces for you to play with! Drag the sliders around to change the shape. Play with:

the ellipsoid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\),

the elliptic paraboloid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z}{c} = 0\),

the hyperbolic paraboloid \(\frac{x^2}{a^2} - \frac{y^2}{b^2} - \frac{z}{c} = 0\),

the one-sheet hyperboloid \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 1\),

the cone \(\frac{x^2}{a^2} + \frac{y^2}{b^2} - \frac{z^2}{c^2} = 0\), and

the two-sheet hyperboloid \(- \frac{x^2}{a^2} - \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\).

Also learn how to visualize cross-sections of quadratic surfaces, and check out this demo deforming hyperboloids through a cone which explains the similarity of their corresponding equations.

Here are notes (printer-friendly) and some practice problems.

We make a gentle introduction with many pictures, make some computations, followed by an introduction to limits, and finally properties of limits and continuity.

Here are notes (printer-friendly) and some practice problems. See my GeoGebra implementations of the helix, the moment curve, the trefoil knot, a family of toroidal curves, the \((p, q)\)-torus knots, and whatever this is…

We begin by defining the derivative and proceed to study properties of the derivative; finally we naively define integrals. As a bonus, we discuss frames in \(\mathbb{R}^3\) (you only need to know about the unit tangent and unit normal vectors in this video).

Here are notes (printer-friendly) and some practice problems.

A synchronous quiz was administered during the scheduled class time.

First we derive a formula for arc length, followed by a computation of several examples; we then discuss how to reparametrize a curve with arc length, followed by several examples.

Here are notes (printer-friendly) and some practice problems.

Office hours 8am - 9:30pm and 4:40pm - 6:10pm on Zoom. Think of this as a review session driven entirely by your questions.

This exam covered material from Textbook Sections 12.1 - 12.6 and 13.1 - 13.3.

First we discuss some physical interpretations of notions involving space curves; we then compute examples including a projectile motion problem, an example minimizing speed of a particle, and a harder projectile motion problem.

Here are notes (printer-friendly) and some practice problems.

We begin with a brief introduction, followed by two methods of visualization: graphs and level curves. Note that there is a (somewhat artificial) focus on functions of two variables.

Here are notes (printer-friendly) and some practice problems. See my GeoGebra implementations of a function grapher and a level curve generator for building intuition!

We first introduce limits, and then give some remarks on the curves criterion and one more example for good measure (a proof of the Curves Criterion is provided in the notes linked below). We then turn our attention to continuity of multivariate functions, and finally use this to compute more limits.

Here are notes (printer-friendly) and some practice problems.

Regrade requests for Exam 1 were due today! During office hours, we solved Exam 1 together.

We begin with an introduction to the directional derivative (and the special case of partial derivatives), and some examples; our investigation naturally leads us to discover Clairaut's Theorem, after which we make some remarks and computions.

Here are notes (printer-friendly) and some practice problems.

We introduce the multivariate chain rule, followed by an initial example and then some more examples. Next we use the chain rule to derive the Implicit Function Theorem. Finally we introduce the gradient and investigate properties of the gradient with some final examples computing directional derivatives via the gradient.

Here are notes (printer-friendly) and some practice problems.

First we introduce tangent planes and compute examples, and then recast this notion via differentials (there is an application in the notes, but the video file was corrupted).

Here are notes (printer-friendly) and some practice problems.

A synchronous quiz was administered during the scheduled class time.

I gave additional office hours on Zoom between 8am - 9:30pm and 4:40pm - 6:10pm. I did this because I'm dangerously kind.

Office hours 8am - 9:30pm and 4:40pm - 6:10pm on Zoom. Think of this as a review session driven by your questions.

This exam covers material from Textbook Sections 13.4 and 14.1 - 14.6.

We first at the gradient and optimization. We then state and prove Fermat's Extremum Theorem, and look at methods of analyzing critical points (with examples in Act I and Act II). Finally we compute absolute extrema using a method analogous to the "closed interval method" from Calculus I.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly).

We first look introduce Lagrange multipliers, followed by several examples of various difficulties: a simple first example, another simple example, an example with three function variables, and finally a geometrically motivated example with multiple constraints. The notes also contain several other examples worked out completely.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly). See my GeoGebra sheets illustrating level curves and the gradient and a visualization of the method of Lagrange multipliers.

We begin with integrals over rectangular domains and work out several rectangular examples, and compute the average value. We then compute integrals over general domains with several general examples. Finally we explore properties of the integral. As an (optional) application, we compute the volume of the sphere.

Here are notes (printer-friendly) and some practice problems. See my GeoGebra implementation for visualizing approximations of regions by rectangles (in case you forgot from Calculus I).

During office hours, we solved Exam 2 together.

Take the day to shore up your knowledge of double integrals (see Friday's playlist).

First we introduce coordinate transformations by analogy to the substitution method from Calculus I, followed by several examples. Next we study the polar substitution and compute more examples including the area of a rose petal. As (optional) applications we compute the area of a general ellipse and the volume of the sphere (again, but easier).

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly).

This is a required meeting. Bring questions.

Office hours 8am - 9:30pm and 4:40pm - 6:10pm on Zoom. Think of this as a review session driven by your questions.

This exam covered material from textbook sections 14.6 - 14.8, 15.1 - 15.3, and 15.9 (for double integrals).

We begin with an introduction and examples (Act I and Act II), and then compute the volume of a tetrahedron. Finally we discuss the center of mass.

Here are notes (printer-friendly) and some practice problems.

We will solve Exam 3 during office hours!

We begin with an overview. Next we study cylindrical coordinates and compute examples (Act I, Act II, and Act III). Next we study spherical coordinates and compute examples (Part I and Part II). Finally we compute the volume of the ellipsoid.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly).

We introduce vector fields and discuss conservative vector fields, followed by examples (Act I, Act II, and Act III).

Here are notes (printer-friendly). Also see my GeoGebra sheet for vector fields.

We introduce line integrals with respect to arc length and compute another example, and then look at line integrals with respect to other variables (preparing for Green's Theorem). We next compute line integrals of vector fields; finally we introduce the Fundamental Theorem of Line Integrals and work through examples on 2-space and 3-space. Finally we make some (optional) remarks on independence of path.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly).

We introduce Green's Theorem, compute examples (Act I, Act II, and Act III), and compute the area of the ellipse via a line integral.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly).

We study the curl (working several examples) and the divergence of a vector field, and finally study their properties. This is all setup for Stokes's Theorem and the Divergence Theorem.

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly). See my GeoGebra sheets for computing divergence and curl interactively.

PS. I decided not to include the content from the last three pages of my notes, but I left them in the PDF. Let me know if you're interested in seeing the proofs.

Use today as an opportunity to catch up on your work from the previous week.

During office hours there will be a synchronous quiz.

We first introduce parametric surfaces and discuss nonorientable surfaces. We next describe tangent planes to parametric surfaces (with examples), and then introduce surface integrals (with examples); as an application, we compute surface areas (including for the sphere and torus). Finally we discuss flux (part 1 and part 2).

Here are notes (printer-friendly) and some practice problems with a few requested solutions (printer-friendly). See my GeoGebra sheets for plotting parametric surfaces, the torus \((\sqrt{x^2 + y^2} - r)^2 + z^2 = R^2\) with major and minor radii \(R > r > 0\), and surfaces of revolution about the \(z\)-axis, the Moebius band, a construction of the tangent plane, and surface approximations.

Today we will have a synchronous participation-type quiz during the scheduled meeting time.

Take this opportunity to do lots of practice and catch up on videos!

Office hours 8am - 9:30pm and 4:40pm - 6:10pm on Zoom. Think of this as a review session driven entirely by your questions.

I'm giving a review session beginning at 4pm and going until I'm tired or you run out of questions (whichever comes first).

This exam covers material from Textbook Sections 15.6 - 15.9 and 16.1 - 16.7.

The exam is on Zoom from 5:40pm - 7:40pm for all sections.

No class today because \(\sqrt{-1} / 8 \pi\)…

We introduce Stokes's Theorem (with a proof) and compute an example both ways to illustrate; after two more examples (example 1 and example 2), we discuss some simple corollaries.

Here are notes (printer-friendly) and some practice problems.

We solved Exam 4 together during Office Hours!

We introduce the Divergence Theorem and illustrate with an example; we then prove the Divergence Theorem and use this result to prove Green's Theorem. Finally we do more examples (an in-depth example and then example rapid fire).

Here are notes (printer-friendly) and some practice problems.

Take this opportunity to do lots of practice and catch up on videos!

Office hours! Bring questions on Stokes's Theorem and the Divergence Theorem.

Synchronous FINAL QUIZ! Topics: Stokes's Theorem and the Divergence Theorem.