Teaching
Fall 2019 - Math 233 Section 10 Homepage
Read this first:
The course chair, Paul Gunnells, has created a website with the course-wide information, including the syllabus and schedule of textbook sections to be covered:http://people.math.umass.edu/~gunnells/m233/m233.html.
You are responsible for reading and understanding the information he has provided at this page and its various links!
Please check this page frequently for schedule updates, announcements, and links to resources which are specific to our section, section 10.
About section 10
What: Multivariable calculus, math 233, now featuring discussion sections, and with more vector calculus!
Who: You, one of the students, your instructor (me) Andrew Havens, and your TA Yasin Karacan.
When: Mondays and Wednesdays, 2:30 pm - 3:45 pm for lectures,
Friday 12:20 -1:10 pm for discussion 233.10AA, and
Friday 1:25 -2:15 pm for discussion 233.10AB.
Where: LGRT 121 for lectures, LGRT 141 for discussion 233.10AA, and LGRT 145 fordiscussion 233.10AB.
Why: For the joy of mathematics (me), but mostly because somebody said it was a requirement (but you'll learn to enjoy it anyway I hope.)
Office hours in LGRT 1316 tentatively Mondays from 11:15 am - 12:15 pm and 3:45-4:45 pm, Wednesdays from 12:00 - 1:00 pm, and Thursdays 9:00 - 11:00 am, or by appointment. Appointments must be confirmed by email, preferably a day or so prior to meeting. I'll also be in the CTC (LGRT 140) from 11 am to noon on Tuesdays and Thursdays. Note that you can get help in the CTC from any of the TAs or instructors there, not just your own. The CTC is open Monday through Thursday, from 10:00 am to 3:00 pm, starting the second week of class.
Announcements
Exam 1 details:
The first exam is Tuesday, October 15, 7-9pm. Section 10 of math 233 will be in the Integrated Learning Center North, 151.
Please read the information provided at the course-wide exam info page. Practice materials for review can be found there.
There will be two course-wide review sessions given by two of the teaching assistants.
Saturday, October 12, 10–12am. Marcus 0131.
Sunday, October 13, 2–4pm. Marcus 013
Topics covered: The first exam covers the following sections:
12.1, 12.2, 12.3, 12.4, 12.5, 12.6,
13.1, 13.2, 13.3, 13.4,
14.1, 14.1, 14.3, 14.4, 14.5, 14.6, 14.7
The corresponding topics list is:
3 dimensional rectangular coordinates;
vectors, dot and cross products;
equations of lines and planes;
distances (e.g., point to plane, line to plane, point to line, etc...);
traces, cylinders, and quadric surfaces;
vector valued functions and their limits, derivatives, and integrals;
velocity, acceleration, arc-length, and motion in space;
multivariable functions, their level curves, their graphs, limits, and partial derivatives;
directional derivatives and the gradient vector;
tangent planes and linear approximation, differentials;
local and global extrema, critical points, the second derivative test, and optimization.
Exam 2 details:
The second exam is Tuesday, November 19, 7-9pm. Section 10 of math 233 will be in the Integrated Learning Center North, 151.
Please read the information provided at the course-wide exam info page. Practice materials for review can be found there.There will be two course-wide review sessions given by two of the teaching assistants, TBA.
Topics covered: The second exam covers the following sections:
14.8 - Lagrange Multipliers
15.1 - 15.3: Double integrals over rectangles and general regions including polar coordinates
15.4: Applications of Double Integrals (mass, density, centers of mass, moments, moments of inertia)
15.5: Surface area of bivariate graphs via double integrals
15.6: Triple integrals in rectangular coordinates
15.7: Triple integrals in cylindrical coordinates
15.8: Triple integrals in spherical coordinates
15.9: Change of variables in multiple integrals
Final Exam Details:
The final exam will take place Tuesday, 17 December, 8–10am, in Boyden Gym.
Please read the information provided at the course-wide exam information page. The final exam is cumulative. You must bring your student ID to the exam room! Please arrive early.
Resources
Check back soon! I will post solutions to quizzes, old quiz problems for practice, notes, perhaps a diatribe about determinants... You just have to check!
Notes on Vector Algebra - Contains a discussion of vector arithmetic and elementary properties of vectors (which are stated sufficiently generally to work as axioms of a vector space), the dot product, cross product, lines and planes. The exercises and problems are purely optional, for your practice, amusement, and enrichment. In the 233 of yesteryear I used them as extra credit problems to allow students to make up for lost quiz points.See me in office hours if you wish to discuss any of these problems.
The Suspended Banana Problem - a written out example computing tensions using Newton's second law.
Notes on Vector Valued Functions and Curves - Contains a discussion of calculus with vector valued functions including arc-length, as well as projectile motion. These notes do not discuss curvature (as curvature is not in the 233 curriculum). The optional notes below cover topics skipped in our coverage of chapter 13, and then some.Notes on Curves, Curvature, and Natural Frames - These optional notes contain additional material on differential geometry of curves, including curvature, torsion, and acceleration expressed in natural frames (such as tangential and normal components). There is also an introduction to cylindrical and spherical coordinates (the spherical coordinates in these notes use a different convention than we will later in the course; the effect is exchanging sin phi and cosine phi). Some desired pictures and figures are missing (until I have time to scan some drawings and update.) The final sections discuss the Keplerian problem of orbital mechanics and include some challenge problems leading up to an interesting proof of Kepler's laws via vector algebra and calculus of curves. These are provided as is, for the benefit of curious math, physics, and astronomy majors, or anyone else with an interest in these topics.
Notes on Partial Derivatives - these notes cover partial derivatives, the multivariable chain rule, tangent planes, linear approximations, differentiability, directional derivatives, and the gradient. They contain examples and exercises throughout.Slides on limits and continuity for multivariate functions - This slideshow covers the definitions of limits and continuity for multivariate functions, and goes through some case studies in limits of two variable functions. It briefly notes how these ideas generalize for functions of 3 or more variables.
Notes on Double integrals - Just scans of lecture notes I prepared. Apologies for the orientation of the last page; thedepartments unix machine lacks the pdf editing tools needed to correct the orientation of a single page that was merged in. Someday I'll figure out a fix, until then, you'll have to rotate the page in your viewer.
Notes on Vector Calculus - these notes review gradients from the perspective of vector fields, and provide an introduction to vector fields in 2 and 3 dimensions, and develop the theory of line integrals, culminating in the fundamental theorem of line integrals and Green's Theorem, flux, surface integrals, Stokes-Kelvin Theorem and the divergence theorem. They contain many examples and difficult problems, and are still under construction. The problems aren't assigned, but are there for you to sharpen your abilities and delve more intensely into the concepts if desired (if you are a mathematics major, then the delving is recommended). The sections on Stokes-Kelvin and the Divergence theorem are under development and will be expanded.
Course description
Multivariate calculus extends the tools of differential and integral calculus to the study of functions of multiple variables. We will generally confine our study to functions of two and three variables, and parametric functions in two and three dimensions, with one or two parameters. The geometric counterparts to these are surfaces in three dimensional space, as well as plane and space curves. Another major part of the modern language of multivariate calculus is vectors, and as such we will study vector calculus in two and three dimensions. This gives us a unified language to describe coordinates, curves, motion, force, and fields. There is a very geometric flavor to this course, both in the collection of objects studied and in the types of applications we will see.
We begin by introducing three dimensional rectangular coordinates. We will then introduce the language of vectors and the dot and cross products. These notions together will allow us to study lines and planes in 3-dimensions, as well as quadric surfaces and cylinders. This is the essential material of chapter 12 of Stewart's textbook.
In the next chapter, we encounter vector valued functions, which allow us to study space curves and their derivatives. This leads us to the notions of velocity and acceleration, which are essential in the physical study of kinematics in 3 dimensions.
Starting in chapter 14, we initiate an earnest study of functions of multiple variables. We define the corresponding notions of limits and continuity for multivariate functions, from which we can give a definition of partial derivatives. We’ll explore the derivative laws as they generalize to partial derivatives. Equipped with a calculus of partial derivatives, we can find tangent planes and linear approximations of multivariate functions and define the gradient and directional derivative operators, which are useful tools in the study of surfaces and multivariate functions. We can then approach optimization problems, and describe the method of Lagrange multipliers.
Chapter 15 concerns multiple integrals, which allow us to compute volumes, surface areas, masses, moments, and centers of mass, as well as moments of inertia and probabilities. The material of this chapter is widely used in probability and statistics, physics, and engineering courses. In many cases, the old techniques of integration suffice, but there is the new puzzle of dealing with boundaries of regions, which are no longer merely a pair of points as was the case for simple 1D integrals. We will rely on substitution, integration by parts, and a few tricks concerning the description of boundaries to turn multiple integrals into iterated integrals, allowing us to integrate with respect to one variable, and then another, exhaustively.
In the final portion of the course we will focus on vector calculus. We define vector fields and line integrals. This enables the study of work of a force field acting on a particle moving along a space curve. We will examine the fundamental theorem of line integrals, and Green’s Theorem, which give generalizations of the fundamental theorem of calculus. We will describe curl and divergence operators, flux from surface integrals, and Stokes’ theorem, further generalizing the fundamental theorem of calculus to study vector fields and their integrals in three dimensions.
Schedule
I will update this as we go, so there is a record of where our section is in the material.
September 2-6: Week 1.
We met only on 9/4, and covered 3 dimensional rectangular coordinates and elementary vector algebra, roughly the material of sections 12.1 and 12.2 of Stewart's textbook.
September 9-13: Week 2.
We covered further vector algebra, including dot products and projections, as well as cross products and the scalar triple product. We then covered parametricdescriptions of lines in 3-space and the equations of planes, using the ideas from vector algebra. We discussed quadric surfaces and cylinders, and the concept of trace curves of a surface. This is roughly the material of sections 12.3, 12.4, 12.5, and 12.6 of Stewart's textbook.
September 16-20: Week 3.
We studied vector valued functions and space curves, including calculus of such functions. We capped off this discussion of vector valued functions of a single parameter by examining integrals, arc-lengths, and projectile motion. The material this week is roughly that covered in sections 13.1, 13.2,13.3, and 13.4 of Stewart's textbook.
September 23-27: Week 4.
We introduced multivariate functions and discussed limits. We introduced partial derivatives. The corresponding texbook sections are 14.1, 14.2, and 14.3.
September 30 - October 4: Week 5.
We covered tangent planes, linear approximations, differentials and differentiability of bivariate functions. We discussed the chain rule, and used it to prove a calculational formula for the direcitonal derivative that we realized geometrically as a dot product with a very important vector called the gradient. We interpreted the gradient geometrically in terms of optimizing the direcxtional derivative and in terms of producing normals to level sets. This week's material corresponds to sections 14.4, 14.5, and 14.6 of the textbook.
October 7-11: Week 6.
We finished the material of chapter 14 by discussing local and global extrema ,critical points and Fermat's theorem, the second derivative test, the Extreme Value Theorem, and optimization, including constrained optimization via Lagrange multipliers. This is the material of sections 14.7 and 14.8 of the textbook.
October 13-18: Week 7.
We introduced double integrals, first over rectangles, and thenover general regions. We learned to evaluate double integrals via iterated integrals, how to use double integrals to calulate volumes between surfaces, and initiated a study of double integrals expressed in polar coordinates for regions called `polar rectangles'.
October 20-25: Week 8.
We finished discussing integrals in polar coordinates and discussed applications of double integrals, including calculation of surface areas. We introduced triple integrals.
October 27-November 1: Week 9.
We continued to study triple integrals, and learned to compute them in rectangular, cylindrical, and spherical coordinates using iterated integration. We began studying change of variables in multiple integrals.
November 3-8: Week 10.
We finished our discussion of change of variables in multiple integrals and began studying vector fields. We introduced scalar line integrals.
November 10-15: Week 11.
We continued studying line integrals, including the fundamental theorem of line integrals, and the process of finding potentials for a conservative vector field.
November 17-22: Week 12.
We studied Green's Theorem, and the differential operators of Curl and Divergence, first in the plane (motivated by understanding Green's for work and for planar flux), and then in 3 dimensions.
November 24-29 : Thanksgiving Break
December 1 - 6: Week 13.
We studied surface integrals, Stokes-Kelvin theorem, and the Divergence theorem. A snow day compactified this exploration to just one hour and 15 minutes, but we resumed with examples in week 14.
December 8-13: Week 14.
In the last week of classes, we wrapped up our discussion of the big theorems (FTOLI, Green's/Stokes-Kelvin, Divergence), and then explored their applications, e.g., in Maxwell's equations of classical electromagnetic theory.