Fall 2010 UBC

About “Detecting Holes”

I made mention in class that

1) We always have curl \nabla f = 0.  In other words, if \mathbf{F} is conservative, then it has zero curl.

2) When you have a field \mathbf{F} which is defined on a simply connected domain, then if curl \mathbf{F} = 0, it must be that \mathbf{F} = \nabla f.  That is, on simply connected domains, all fields with zero curl come from the gradient of a function (i.e. are conservative).

A few notes to ward off common confusions:

A) If \mathbf{F} is defined on a domain with a hole in it, and curl \mathbf{F} = 0, then we don’t know anything about whether it is conservative.  The gravitational field is an example that has a hole in it, but it’s still conservative.

B) If \mathbf{F} is defined on a domain with a hole in it, and curl \mathbf{F} = 0, then we could consider just part of that domain that doesn’t have a hole in it, and on that restricted domain, \mathbf{F} must be conservative.  So being conservative depends on the domain!  A field can be conservative on one part of its domain without being conservative on all of it.  The pdf exam conceptual review packet “Important Example” number one works like this.

Finally [this paragraph is extra-curricular material not necessary for the exam, but for general interest], in class I said that we can detect holes in a domain by using the fact that the implication “zero curl implies conservative” only works on simply connected domains.  More precisely, what I mean is that if the domain has a hole in it, then there will be some field that has zero curl but is not conservative.  If there is no hole, then there will be no such field.  (Of course, it’s not obvious how to check whether some field with a given property exists among all infinitely many possible fields, so I don’t mean this as a practical detection tool, at least not without doing more theoretical work.)

Here are a selection of Students’ Study Tips.  Feel free to post comments and more tips in the comments section.

On saving time integrating

“When taking a double integral of a function f(x,y) dx dy.  If you can write f(x,y) as g(x)h(y), then you can take the integrals separately.  That is, f(x,y) dx dy = g(x) dx h(y) dy.  It can save a lot of time!”  – Huge

[EDIT:  This is a great suggestion, but you have to be careful about the limits.  For example, consider \int_0^1 \int_x^{x^2} xy \; dydx.  According to this suggestion, you can turn it into \int_0^1 \int_x^{x^2} y \; dy \; x \;dx.  But you can’t change it into \int_0^1 x \; dx \int_x^{x^2} y \; dy because the latter won’t simplify to a number! ]

On surface integral formulas

Anonymous was pleased to clear up the following confusion about surface integrals.  The surface integral formula

\iint_S \mathbf{F} \cdot \mathbf{n} dS

becomes

\iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) dA

when you choose your parametrisation.  At first, this seems wrong, because the first formula appears to have a unit normal, while the second does not.  But in fact, this is because dS becomes | \mathbf{r}_u \times \mathbf{r}_v |dA, so we observe some cancellation:

\iint_S \mathbf{F} \cdot \mathbf{n} dS = \iint_D \mathbf{F} \cdot \frac{\mathbf{r}_u \times \mathbf{r}_v}{| \mathbf{r}_u \times \mathbf{r}_v | } | \mathbf{r}_u \times \mathbf{r}_v | dA = \iint_D \mathbf{F} \cdot (\mathbf{r}_u \times \mathbf{r}_v) dA

On the meaning of “closed”

When I say “closed” in this class, I mean “has no boundary,” and this occurs in two situations:

A “closed loop” is a curve with the same endpoint as starting point.

A “closed surface” is a surface which cuts out a finite piece of 3D space, i.e. has a finite inside distinct from its outside.  This also has no boundary or “edge”.

You may also come across the term “closed” in mathematics to mean “complement of an open set.”  (You have probably heard of a “closed interval” in this sense.)  In this course we don’t talk about open sets much, and I don’t think we’ve talked about closed sets at all.  So when I say closed, I am speaking of a curve or a surface, and I mean that it has no boundary.

— Kate (channelling Pippi Longstocking)

On visualising graphs

“Coming from a Math background and not a physics background, I always had trouble visualizing graphs in 3-D. I usually just refused, put it off, or tried to graph it online. Sometimes I would even plug in number after number until I somewhat could visualize what it looked like. It seems extremely trivial now, but I learned to cover up one of the variables and graph only the other 2, and then repeat this process for the other two combinations. So, take F=<P,Q,R>, I would cover up the P value and know that I am looking down the x-axis onto the yz plane (this is what I never understood before; the fact that I was looking down the axis that I was covering up). I would graph this and then repeat by covering up the Q value, which corresponds to looking down the y-axis onto the xz plane, and the R value which corresponds to looking down the z-axis which looks onto the xy plane. Once I had a graph for each plane (xy, xz, and yz), it became much easier to put it all together and see the graph in 3-D. It’s so easy now but has honestly saved me so much time and frustration! It has also helped me get a better understanding of calculus in general as I am more capable of visualizing what happens to surfaces. ” – Anonymous

On remembering the curl formula

“I always had trouble with how to remember which P-partial went with which R-partial, etc., when we’re trying to find 3-D vector field is conservative by equating certain partial derivatives. So, I finally realized there was a trick to remember this:

I first start out by writing, PQR in a column on the top of my exam, since they’re in alphabetical order. In the next column, shift PQR down by one spot, to obtain QRP
P = Q
Q = R
R = P

Then, the variable you’re taking the partial with respect to corresponds with where the corresponding variable in the column to the right is, looking at the vector equation.
For example, thinking of <P,Q,R> as <x,y,z> :
Py  = Qx (since Q is in the y-spot and P is in the x-spot)
Qz = Ry (since R is in the z-spot and Q is in the y-spot)
Rx = Pz (since P is in the x-spot and R is in the z-spot)

Then, to check, you should have the capital letters (field components) in alphabetical order on the left side, and the ‘with respect to’ variables in alphabetical order on the right side. :)” — Anonymous

A useful website (from Anonymous)

http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx

Curl and divergence:  which is which?

“At first, I keep messing up Curl and Divergence. I couldn’t remember which uses cross product and which uses dot product. After a while, I remembered it by thinking that Curl starts with a ‘C’ like ‘C’ross product and Divergence starts with a ‘D’ like ‘D’ot product.It’s that simple. =)” — Anonymous

Orientation of surfaces with several parts

“If you are integrating on a surface that consist of a least two different pieces, then the orientation of each integration has to agree with the others. Pay special attention if the question is asking for outward orientation instead of upward. For example, a hemisphere with its bottom disc will have normal vectors that are pointing in different directions for the two pieces. For the part of the hemisphere, its normal is pointing more upwards, but for the bottom disc, it’s pointing downward. ” — Jimmy Wales’s Personal Appeal

On prerequisites and review

“To do well in this course, I would HIGHLY recommend doing a thorough review of 3D surface/volume parameterizations. While it has been a year since I took MATH 253, I felt as though I remembered enough of it to be able to get by. I soon found out, however, that much of the material that is covered in MATH 317 draws heavily on what was taught in Multivariable Calculus, and it would have been wise for me to ensure that my understanding of the material from that class was solid. Much of my trouble with the material in this course isn’t a result of the new concepts being difficult to grasp, but rather because I am not as familiar with material that was learned in previous courses.”  — Anonymous

On choosing the orientation

“When computing for the normal sometimes it is confusing to set the order of the vector multiplication.  Just cross product the two vectors in whatever order and then later on visualize the actual situation or graph and set the sign accordingly without thinking about it too much during the calculation.”   — Anonymous

FRIDAY GAMESHOW!

On Friday, we’ll talk a bit more about the handout I gave on Wednesday — please bring it again.  Then we’ll have a Math 317 Review Gameshow!  You can choose to be a non-participating observer only, but if you want to play, make yourselves into teams of 3-4 people (I’ll help you find teammates in class if you don’t already have some in mind).  Start thinking now of your awesome mathematical team name!  And bring your lecture notes and text (it’s open book).

Expect snacks and prizes!

So all 88 of you were teleported to your own individual earth-like planets and you were all very curious to measure the mass.  (Back at Thanksgiving, remember?)

You came up with a good half-dozen different ways to measure the mass, only some of them having anything to do with Kepler’s Laws.  One student poked the sun with a giant stick made by gluing together little sticks!  One student made a telescope out of the microscope.  One got eaten by the monkeys (they cooked the unlucky student up over a fire kindled with the calculus textbook), although several others bribed them in various ways and made good use of them.  The monkeys got to do all kinds of epic tasks, like walking over the horizon with a perfectly upright stick, and carrying one student for 30 days.  Some of the monkeys got a nice retirement and were well taken care of.  Most of them got a marshamallow.  One of the students had a really really lucky gnome that was so lucky it knew the circumference of the planet.  More often the gnome was smashed to smithereens over the edge of a cliff. The snails seemed mostly useless except as an amusement.  One student suggested measuring the mass of the planet wasn’t interesting enough and gave a link to something more fun to do with your time.

Thank you to all of you for brightening my day, and my semester.

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

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.

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.

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