Spring 2010, Math 236

January and February, 2010

Be sure to check back often, because assignments may change!

All section and page numbers refer to sections from Smith & Minton's Multivariable Calculus, 3rd edition.

Due Friday 1/29 at 9am

suggestions for reading a math text
course policies
syllabus
Fill out the questionnaire on my OnCourse page

• Pay attention to: all of it. Any questions? Please do ask me!

Section 10.1: Vectors in the Plane
Section 10.2: Vectors in Space
Section 10.3: The Dot Product

• To read: All of sections 10.1 and 10.2; up thru the middle of p. 808 in Section 10.3. Don't worry, you won't usually have this much to read!
In this course, the graphs and diagrams are critical, and the authors really put a lot of effort into explaining them. Many people have a hard time plotting points on 3D axes (see section 10.2) -- really try to follow what the authors are describing in their diagrams!

E-mail Subject Line: Math 236 Your Name 1/29

1. Find a unit vector in the direction of <-2,3,-6>.
2. Let a=<2,-1,3>, b=<4,10,-1>, and c=<3,-1,2> be vectors, and let x.y represent the dot product of any vectors x and y.
(a) Find a.b.
(b) Does (a.b).c make sense, and if so, what is it in this case?
(c) Are b and c orthogonal?

Reminders:

(I start out with extensive reminders at the beginning of the semester -- often things I'm afraid I'll forget to mention. I pare them down significantly as the semester goeos by.)

• In these assignments, you should always briefly explain how you arrived at your answers.
• Remember, if you have any questions, please come to my office hours. Office hours will probably be a huge part this class; I expect nearly every one will come see me at least every couple weeks.
• Begin Problem Set 1, listed at the bottom of this course's web page. . The problem sets that are due each Friday (except this first Friday, 1/29) reflect an entire week's worth of work, and you should be working on them throughout the week. PS 1, due 2/5, will actually reflect more than a week's worth of work, and as such will be unusually long, so plan on spending plenty of time on it!

Occasionally check the grades available through OnCourse to see whether I've gotten all your reading assignments: sometimes messages do seem to just disappear. (Note: I do not use the OnCourse grade sheet for anything other than letting you know scores, so don't pay any attention to the totals.)

Due Monday 2/1 at 9am

Multi Problem Sets
Problem Set Guidelines
Section 10.4: The Cross Product

E-mail Subject Line: Math 236 Your Name 2/1

1. Briefly explain what, geometrically, projba and compba are.
2. How is a x b related to a and b geometrically?
3. If a and b are vectors in V2 (that is, in the plane), is a x b defined?
4. If a, b, and c are arbitrary vectors, should we expect that
(a x b) x c=a x (b x c)?

Reminder:

• For the problem sets, it really is a good idea if you work on the practice ("black") problems first -- I choose those to be good problems for helping with understand the ideas and techniques. You'll find that writing up the focus ("red") problems goes much more smoothly if you've already worked through the practice problems.
• PS 1 is an individual assignment. While this of course means that you will each be turning in your own problem set, it does not mean that you can not work with someone else, as long as you do so in a way that helps you all better understand the material. This generally means that you each think about the problems on your own, and then discuss your ideas. It often will also mean working through the problems on your own, and then comparing results. When it comes time to write the problems up, you should each do so on your own. Just to be clear: You absolutely may not divide the problems up between you, nor may you copy somebody else's work, merely changing a word or two (see the section on the Honor Code in the Course Policies).

• Make sure you've read Multi Problem Sets and Problem Set Guidelines.

Due Wednesday 2/3 at 9am

Section 10.4: The Cross Product

• To read: Finish Section 10.4. You will notice throughout this book that there are many applications -- in section 10.4, you encounter torque and magnus force. We won't always be able to spend much time on the applications, but still read them to get a sense of the place of multivariable calculus in the world. Those of you with a related background (in this case, physics) will probably get more out of these applications than others; if the details of such examples prove stressful, focus on the big picture.

E-mail Subject Line: Math 236 Your Name 2/3

1. How can we tell whether two vectors in V3 are parallel? How can we tell whether two vectors are orthogonal?

Reminder:

• Make sure you clear up any difficulties on PS 1 by coming to my office hours or taking advantage of the tutors at the Kollett Center.

Due Friday 2/5 at 9am

Section 9.1: Plane Curves and Parametric Equations
Section 10.5: Lines and Planes in Space

• To read: Read all of Section 9.1 (available through my OnCourse page); through middle of page 832 in Section 10.5.

E-mail Subject Line: Math 236 Your Name 2/5

1. What information about a line L do you need to determine an equation for the line?
2. Find parametric equations for the line through the point (1,2,3) and parallel to the vector <3,0,-1>.
3. Find parametric equations for the line through the points (1,2,3) and (7,-1,0).
4. What information about a plane P do you need to determine an equation for the plane?
5. Find an equation for the plane containing the point (1,2,3) with normal vector <3,0,-1>.

Reminder:

• PS 1 is due at the beginning of class Friday.
• Quick reminder: the problem sets that are due each Friday reflect the whole week's worth of work, and should be worked on throughout the week, so plan on starting PS 2 this week-end. As always, start with the practice problems. It's a group problem set, so find a partner to work with now, to work with after you've given some thought to the problems. (You must work in a group. Groups of 2 are best; 3 is less than ideal, but sometimes unavoidable. I do not want you to turn in a hw you've done on your own, neither do I want you to work in a group of 4.) As always, don't split the problems up between you.

Due Monday 2/8 at 9am

Section 10.5: Lines and Planes in Space
Section 11.1: Vector-Valued Functions

• To read: Finish reading Section 10.5; In Section 11.1, read everything but Example 1.8. Really think about Example 1.5 -- before you ever turn the page, can you make some progress on figuring out which vector-valued functions do (or do not) go with which graphs?

By the way, when the book refers to past sections that you haven't studied, don't panic -- just figure out as much as you can about it, and move on.

E-mail Subject Line: Math 236 Your Name 2/8

1. The vector-valued function r(t)=cos(t)i+sin(t)j lies in the plane.
(a) Using Example 1.2 as a guide, what will the graph of this function look like?
(b) Is it possible to rewrite r(t) as a function y=f(x)?
2. What are some advantages to using vector-valued functions?

Reminders:

Due Wednesday 2/10 at 9am

Section 11.2: The Calculus of Vector-Valued Functions

E-mail Subject Line: Math 236 Your Name 2/10

1. If r(t)=tcos(t)i+exp(t^2)j+ln(t)k, what is r'(t)?
Remember exp(t^2) is Maple notation for et2.
2. If r(t) is a vector-valued function, what geometric/graphical information does r'(a) give you?
3. Let r(t)=cos(t)i+sin(t)j and s(t)=sin(5t)i+cos(5t)j.
(a) What do the graphs of r(t) and s(t) look like?
(b) If the graphs of two vector-valued functions r(t) and s(t) are the same, must r'(0)=s'(0)? (Is this a new result, or was it also true for functions f(x) and g(x)?)

Reminders:

• For the group problem sets, each group will turn in one joint problem set. You should work together to solve every problem, then one of you should take responsibility for writing up the entire problem set. That person is the "primary author". On the finished problem set, put a star next to the primary author's name.
• These groups are not permanent -- you'll be switching partners from week to week. You should end up being primary author 1/3 - 1/2 the time (depending on whether you're working in groups of 2 or 3).

Due Friday 2/12 at 9am

Problem Set Guidelines
Section 11.3: Motion in Space

• To read: Review the Problem Set Guidelines again before turning in PS 2. Read all of Section 11.3. Pay particular attention to Examples 3.4 and 3.7. We probably won't spend much time on rotational motion, but still skim through those examples (3.3, 3.5, and 3.6).

E-mail Subject Line: Math 236 Your Name 2/12

1. If r(t) is a vector-valued function representing the motion of an object at time t, what physical information does r'(t) give you? How about ||r'(t)||?
2. Find the velocity and acceleration vectors, if the position of an object moving in space is given by
r(t)=(5/sqrt(t))i+ln(t3)j-tan(3t)k.

Reminders:

• Make sure each member of your group has a photocopy of the problem set (for studying purposes).
• (From Wednesday 2/10:) I had hoped to incorporate a little bit of work with Maple, on laptops Friday. As I adjust my plan for Friday to take into account where we are, it looks like there may be a short opportunity to use it to help with some in-class work at the end of class. It's completely up to you whether you think it's worth it to bring it to class for something we might or might not get to! If not enough people bring them, I'll just post a file that shows useful commands, as I often do.

Due Monday 2/15 at 9am

11.3: Motion in Space

Reminders:

• Just a reminder that you are responsible for all of the practice problems, and that you really will find the focus problems easier if you work through the practice problems first.

Due Wednesday 2/17 at 9am

Anton, Bivens, and Davis Section 13.3: Change of Parameter; Arclength
Section 11.4: Curvature

• To read: Read all of Section 13.3 from the Multivariable Calculus book by Anton, Bivens, and Davis, available through my OnCourse page. In Section 11.4, read through Example 4.2, top of page 889.

E-mail Subject Line: Math 236 Your Name 2/17

1. Consider the circle r(t)=cos(t)i+sin(t)j.
• (a) Find a change of parameter t=h(τ) such that the circle is traced counter-clockwise as τ increases over the interval [0,1/2].
• (b) Use the chain rule for vector-valued functions to differentiate r(τ).
• (c) Does the change of parameter you found preserve orientation or reverse it?
2. Let r(t)=t3i+sin2(t)j.
• (a) The graph of r(t) is displayed below, with arrows super-imposed attempting to show the tangent vectors (the lenths are all wrong, but the direction is all that matters here anyway). Based on the graph, does r(t) have a continuously turning tangent vector?

Reminders:

1. Once again, remember that while it's okay to consult with others on the big ideas behind problems, you get alot more out of it if you think through the problems as much as you can before and after such consultations. Also, remember you must write up the problems on your own, in your own words. If you're having any difficulties at all, please do come to my office hours and/or go to the tutors at the Kollett Center.

Due Friday 2/19 at 9am

Section 11.4: Curvature

E-mail Subject Line: Math 236 Your Name 2/19

1. Explain the idea of curvature in your own words.
2. If the helix in Example 4.5 were changed to r(t)=< 2sin(t), 2cos(t), 4t2>, will the curvature still be constant? Don't actually do the calculation, but give an intuitive justification - think about how changing the z-coordinate to 4t2 from 4t will affect the graph.

Reminders:

• PS 3 is due at the beginning of class.

Due Monday 2/22 at 9am

Section 11.5: Tangent and Normal Vectors

E-mail Subject Line: Math 236 Your Name 2/22

1. Suppose you are skiing down a hill along a path that curves left. Describe the direction of the unit tangent and principal unit normal vectors to the curve that describes your motion.

Reminders:

• The problem set due Friday is another group problem set. Try working with a different partner this time! As usual, don't divide the problems up between you. If possible, the primary author this time around should not have been the primary author last time.

Due Wednesday 2/24 at 9am

Section 10.6: Surfaces in Space

• To read: All. If possible, read while sitting by a computer with Maple, and try doing their examples in Maple, using the Maple "cheat" sheet I handed out in class. If it's not working for you, though, don't waste hours on it -- instead come talk to me soon about getting comfortable with Maple!

E-mail Subject Line: Math 236 Your Name 2/24

Consider the surface x=4y2+4z2.

1. Which coordinate plane does the equation z=0 define?
2. What does the trace of this surface in the xz-plane look like?
3. What do the traces of this surface in the planes x=k look like?
4. What is this quadric surface called?

Reminders:

• Those of you who have a laptop - although it will only be for a few minutes, I think having some laptops in class to allow people to practice the various ways to enter the commands for graphing surfaces could be handy
• The in-class portion of Exam 1 will be Wednesday 3/3. You will receive the take-home portion that day, and return it Monday 3/8.

Due Friday 2/26 at 9am

Section 10.6: Surfaces in Space

• To read: Re-read Section 10.6 Again, use Maple while you're reading and really get a feel for the different surfaces. Pay particular attention to the table at the end of the section.

Reminders:

• Laptops will again be helpful in class Friday.
• Make sure each member of your group has a copy of the problem set. Put a star next to the primary author's name.

Here ends the reading for January and February
Go to the reading assignments for March!

Janice Sklensky
Wheaton College
Department of Mathematics and Computer Science
Science Center, Room 101A
Norton, Massachusetts 02766-0930
TEL (508) 286-3973
FAX (508) 285-8278
jsklensk@wheatonma.edu

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