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

Abstract:

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

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MOVIE NIGHT IS CONFIRMED!

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.

Kate.

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

http://www.kleinbottle.com/

(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

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