Fall 2010 UBC

Archive for November, 2010

Public Lecture

There’s a math conference (Canadian Mathematical Society) going on this Saturday, Sunday and Monday in Vancouver.  I’ll be speaking there, but you probably aren’t interested in a research talk.  What you might be interested in, though, is the Public Lecture.  The Public Lecture is a long tradition in math conferences (and other places): it’s aimed at a non-specialist lay-person audience.  It’s usually entertaining and very down to earth.  This year’s speaker is reported to be excellent.

Saturday, 20:00 – 21:00: Saturday, December 4th, 2010
Coast Plaza Hotel & Suites – 1763 Comox Street, Vancouver


RON GRAHAM, University of California at San Diego
Searching for the Shortest Network 

Suppose you are given some set of cities and you would like to connect them all together with a network having the shortest possible total length. How hard is it to find such a shortest network? This classical problem has challenged mathematicians for nearly two centuries, and today has great relevance in such diverse areas as telecommunication networks, the design of VLSI chips and molecular phylogenetics. In this talk, I will summarize past accomplishments, present activities and future challenges for this fascinating topic.

Here’s the flyer:  Public Lecture



Come to Movie Night and find out how spheres intersecting the plane upset impressionable young polygons in “Flatland: The Movie”, and then enjoy the classic “Good Will Hunting” study break!

December 3rd, MATH 100

FLATLAND, 5:40 pm (trailer)

GOOD WILL HUNTING, 6:20 pm (trailer)

Doors open at 5:30 pm.


Nearing the End!

Just two more classes, can you believe it?  Is there life after vector calculus?

Important things to know as you head toward the unknown:

0) On Wed (Dec 1st) we will have homework hand-in instead of a quiz, no dice roll.  This is because there’s so little precious lecture time left!

1) Office Hours this week:  Wed, Dec 1st, 11-12, Math 126 / Fri, Dec 3rd, 11-12, Sowk 122.

2) Office Hours during the exam period will be announced on this blog.

3) Movie Night is the afternoon of Dec 3rd, and will be advertised in a separate post on this blog.

4) There will be a final Divergence Theorem homework assigned on Wednesday but it is not to hand in (no credit, no due date, but solutions will be posted).

5) Review problems for the final will be posted on a continuing basis, under “Homework” on the website.  In other words, I’ll post a package of review problems shortly, and then as I think of even more review problems, I may post more.

6) Review-Question Game Show!  The last class, Fri, Dec 3rd we’ll have a team competition for prizes.  Info in a separate post.

7) I will be putting up an online system whereby you can check your own grades against my grading sheet to ensure no errors.  Info in a separate post when that happens.

Numerical Answers, Homework #12

By popular demand, here are the numerical answers to the book problems, to help you tell if you’re on the right track.  Important: if your minus sign disagrees, figure out why, don’t just change it and hope!  I will grade your explanation for the sign of your answer carefully, so an unexplained but correct sign won’t get full marks.  Explain yourself!!

17.7 #20: \frac{\pi^3}{6}

#22 \frac{\pi}{3}

#28: 4\pi

#42 0 kg/s

#44 24 \varepsilon_0

17.8 #2 -18\pi

#6 0

#8 2e-4

Hwk #12, Problem 17.7 #42

Q.  For our current homework I was wondering whether, for #42, the density matters at all and if so how would we use it? For my answer I got zero, but if it were to be a non-zero answer how would we include the density in our calculations?

A. Excellent question.   Stewart explains the relationship between flux and fluid flow at the beginning of the section “Surface Integrals of Vector Fields” in 17.7.  You can read that and compare to what I’m about to say.

A useful strategy for physics problems is to think about units.  What does “fluid flow through a surface” mean and how would we measure it?  We would measure it in mass per time. (Aside: We could measure it in volume/time, but if it’s a compressible fluid, the same amount of fluid might occupy a different volume at different times, so that’s not as good a conventional choice.  So mass per time it is.)

If our vector field is a velocity field (the question stipulates this, as does the discussion in the chapter that a referenced above), then the units of the velocity field are distance per time (aka velocity).  When we do a surface integral \iint_S \mathbf{F} \cdot d\mathbf{S}, we are adding up little pieces “\mathbf{F} \cdot \mathbf{n}“.  The unit normal is unitless; it’s just a way of taking a component of \mathbf{F}, i.e. \mathbf{F} \cdot \mathbf{n} has units distance/time.  So when we add up lots of distance/time over an area (area is (distance)^2), we get distance^3/time.  In other words, that gives us an answer in volume/time.  If we want an answer in mass/time, we need to multiply by the density, mass/volume.  So the calculation the question is actually asking for is \iint_S \rho \mathbf{F} \cdot d\mathbf{S}.

Unfortunately, the answer is zero, and 0 = \rho 0, so the zero obscures the dependence on \rho.  But the correct answer is actually 0 kg/s.  (I’m glad you asked.  Stupidly, #41 also has answer 0.  Obviously the one who designed the questions didn’t think about this type of possible misunderstanding.)

Numerical Answers, this homework

Hi everyone,

Hope you’re enjoying those parametrisations.  I hope you’ll check in with each other as you do them, to catch errors.  Here are the numerical answers to the book problems, to help you tell if you’re on the right track:

17.6 #38: 9 \sqrt{30} \pi

#42 \frac{1}{24}(26^{3/2} - 10^{3/2})

#46: \frac{\pi}{2}(\sqrt{2} + \ln(1+\sqrt{2}))

#58 16

17.7 #6 \frac{1}{\sqrt{6}}

#10 \frac{4\pi}{3}

#14 (\frac{32}{3} - 6\sqrt{3})\pi

#18 241\pi

#38 108 \sqrt{2} \pi

Klein Bottles!

For all kinds of fun, visit


(To see the guy who sells these, check out this video.  He’s a real character.)

Great ideas for presents for dad or for the fluid dynamics buff in your family.    Just make sure to check out the details.  If you want to make your own, sew up the “hole” in the mobius strip you made (you might find this easier if you knit it out of wool):

Klein Bottle Hat

Möbius Strip

Today in class I demonstrated a Möbius strip.  If you missed it, you’ll regret it forever:  I bet you never thought your teacher would demonstrate a strip, did you?  (Yes, I heard the giggles in class.)

For Friday’s class, please bring a Möbius strip with you (but it’s ok if you forget, your neighbour will share).   You can make it with a long narrow strip of paper and a piece of tape:  here’s how.

For fun, try drawing a line down the middle.  This shows it has one side.  Make a few of them and cut one of them right down the centre.  What do you get?  Or try 1/3 of the way from one edge.  Pretty fun.  Here is a video.

Study Suggestions with an emphasis on the conceptual

Several students have come to me asking for advice on studying the “conceptual” side of calculus.  Many of you have had previous calculus courses that emphasized the computational, and my philosophy is to emphasize the conceptual a little bit more.   The textbook and its homework problems are very computational, and computational questions are a dime-a-dozen:  you can just invent loads of them and do them one after another after another (and that can be fun, but that’s not all of mathematics).  Studying for conceptual understanding is a much more difficult process.  There aren’t an easily infinite supply of questions, although I try to give you as many as I can.  Here are some tips:

1) Read Nykamp’s conceptual readings as well as your textbook.  But don’t just read them, work through them.  Be active in your reading.  Read a sentence, then ask yourself, why is that true?  If it describes something, draw a picture.  If it defines something, come up with an example.  If it gives a logical step of reasoning, tell yourself why that step is valid.  If it references a theorem, state the theorem to yourself.  Can’t remember it?  Try to figure out from context what it might say, then look it up.  If it says “if A then B,” ask yourself, “is it also true that `if B then A’?”  If it says “whenever A is true, then B is true,” find an example where A is false and B is false.  Whenever it talks about anything, come up with your own examples, examples, examples.  The art of doing this — coming up with your own questions and examples — is just that:  an art.  You’ll slowly get better at it as you practice.  But it provides an infinite source of conceptual practice with math, right there in your readings.  Finally, when you are done reading a section, close the book and explain what you just learned to your mother.  I’m serious: call her up and tell her your newfound knowledge.  Or your roommate.  Or your hamster.  Or just your diary.  But explain it (out loud: moving your mouth or your pencil, not just in your inner voice) and you will discover, through explaining, where your hidden misunderstandings lie.  If your listener is animate, they can ask you questions and that’s excellent too.  I know it sounds outlandish but this is a tremendous way to learn!  (Many mathematicians, myself included, will write a book about whatever they’re learning, just to exercise the explaining neurons: the book doesn’t necessarily come out that great, but it’s a log of all the thoughts and ideas that have gone into the head.  The more times stuff passes in and out and swirls around in there, the better.)

2) You can also make more out of your computational homework.  When you sit down to do your homework, don’t review first.  Just dive into the problems.  Give yourself only as much help as you need.  If you start a problem and think, “hmm, I know we did an example like this in class, but I don’t remember how it goes” then instead of looking it up, try to figure out how it should go.  Pretend you’re the first person ever to face this question and you have to invent the mathematics to do it.  Forgotten the formula for something?  Don’t look it up!  Forgetting it is a golden opportunity: now you can re-derive it like a newborn, and that’s a very valuable learning experience.  If you find you can’t invent the method for solving the problem or come up with the formula, after giving it a real honest try, then go get a hint, as tiny a hint as you can.  Maybe read the first few sentences of a similar problem in your notes or text.  Then try again with the books and notes closed.  Proceed in this way, giving yourself little hints, little bits of help, just a tiny bit at a time.  Your goal is to solve the problem having used the minimum of help from outside your head (its kind of like titrating hints).  The learning happens in the process of struggling, not in succeeding. Also, a good indicator is if you find yourself saying “ooooohh!  that’s how it has to work!” when you finally figure it out.  If you just look up the method right off the bat, there’s no eureka moment, and less learning.  (If you’ve been struggling with it, your brain starts to want the answer like a drug, and it gets very excited about it when it gets it; then it stores it better.)  And one more thing:  many of the textbook questions seem random, but they aren’t.  After you do one of those “what the heck is the point of that?” kind of questions in Stewart, ponder a bit what the point is.  Sometimes there really is one (not always).  For example, this week 17.5 #32 has some real interest to it that you can learn about in the reading I suggested next to it on the homework list.

3) To sum up:  you can take the amount of struggle as an indicator of how successful your studying has been.  Good studying should feel like good exercise on the treadmill:  if you set the level too easy, you can do the 30 minutes with less benefit.  Similarly, if you use your textbook or your solutions manual as a crutch, you can do the 30 questions with less benefit.  And if you just read the readings, instead of reading them actively, it’s like walking the mile instead of running it.  Same distance, less muscle growth.

4) But finally, have fun with it!  Math really is supposed to be fun.  (So is exercise!) Puzzles are fun — crosswords, sudoku, brainteasers.  Math is supposed to be like that.  Approach it that way and when you get frustrated, take a break.  Study long in advance of the midterm so you can approach it in a relaxed way, doing only as much at one time as you have the interest for.  I know this can be hard in a fast-paced academic schedule, but do the best you can.

Good luck!

Homework Help

Hi all,

Some hints for homework problems that came up in office hour:

– when you find parametric equations in 17.6 19-26, your domain needn’t be just a rectangle.  You could have a domain like u^2 + 2v^2 < 5 or something.

– for 17.6 #26, parametrize it as the graph of a function, not as a plane, and it will be easier

– for Extra Problem 2, don’t forget the geographic longitude and latitude have slightly different definitions than that the angles in spherical coordinates.  Wikipedia is your friend.

– for matching problems (17.6 #13-18) here are some ideas:  look at grid curves, look at intercepts, look for surfaces of revolution, and don’t be afraid to use elimination.  These are tricky!

– for 17.5 #23-28, don’t try to be clever:  these are just tedious.  Just write everything carefully and plug along.  But they’re good practice!  My advice is to move everything over to one side so you have something of the form “stuff = 0” and then prove it by slowly simplifying the “stuff” until you get the “0”.  The method of so called “left-side-right-side proofs” (curse the person who invented these!) do essentially the same thing in a form 1) which is easier to make mistakes in, 2) tempts you (or sometimes even tells you) to write false things, 3) laughs at logical principles and 4) is more confusing to read.  If you want to know more about why I hate left-side-right-side, read this.


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