Month: March 2010

A Super Specific Multivariable Calculus Question

Hi all,

I have a question about multivariable calculus, that I need some help with. My kids and I are both slightly stumped about this.

The question we are asked — in a section thrillingly titled, replete with semicolon, “Parametric Surfaces; Surface Area” — is to find the surface area of “The portion of the sphere x^2+y^2+z^2=16 between the planes z=1 and z=2.”

In class, the formula we derived for surface area for any parametric surface \vec{p}(u,v) is

S=\underset{R}{\int\int} \left\Vert \frac {\partial\vec{p}}{\partial u}\times\frac{\partial \vec{p}}{\partial v}\right\Vert dA.

We solved this by converting the (top part) of the sphere to a parametric surface:

x=r\cos(\theta)
y=r\sin(\theta)
z=(16-r^2)^{1/2}

Then we defined \vec{p}=<r\cos(\theta),r\sin(\theta),(16-r^2)^{1/2}> (where \theta ranged between 0 and 2\pi and r ranged between \sqrt{12} and \sqrt{15}. (Those limits for r come from the fact that we want the surface area of the sphere between z=1 and z=2 — which correspond to r=\sqrt{15} and r=\sqrt{12} respectively.) [1]

So I calculate \frac{\partial \vec{p}}{\partial r}=<\cos \theta, \sin \theta, -r(16-r^2)^{-1/2}> and \frac{\partial \vec{p}}{\partial \theta}=<-r\sin \theta, r\cos \theta, 0>.

So to use our surface area formula above, we need to find \left\Vert \frac {\partial\vec{p}}{\partial r}\times\frac{\partial \vec{p}}{\partial \theta}\right\Vert. Calculating that out, we get it to equal \frac{4r}{\sqrt{16-r^2}}. Phew, now we have something we can plug into the surface area formula for that “norm of the cross product” thingie.

Here’s where the question comes in. We know

S=\underset{R}{\int\int} \left\Vert \frac {\partial\vec{p}}{\partial r}\times\frac{\partial \vec{p}}{\partial \theta}\right\Vert dA=\underset{R}{\int\int} \frac{4r}{\sqrt{16-r^2}} dA.

Why is it that when we finally evaluate this beast, dA is not equal to our normal area element for polar, namely r dr d\theta? For the answer to come out right, we need to let dA equal to simply d r d\theta.

WHY? Why don’t we plug in the normal polar area element?

Here’s my thinking. Even though we usually use dA to represent an area element, in this particular surface area formula, it doesn’t represent anything more than du dv (for whatever parametrization gets made). The reason I think this? When I look at the derivation of the formula, it defines du dv to be dA. Simple as that.

I used to think that dA had a fixed meaning: the area element in a particular coordinate system. However, I’m now thinking that it might mean different things in different equations? Either that or our book is being sloppy.

If anyone can follow what I’ve written here and has any help to proffer, I would be much obliged. It’s a small point — one that won’t really matter in the long run for this course — but both my kids and I would like to have this resolved once and for all.

[1] If you don’t see that, imagine you have this sphere and you make a slice at z=1 and another slice at z=2. You want the surface area of that little curved “ring” — and if you find the shadow of that ring on the x-y plane, you’ll get two concentric circles with radius \sqrt{12} and \sqrt{15}. That’s the region R that you will be integrating over.

Surprise ’em with what they don’t know

Sometimes it isn’t that we are bad teachers. And it isn’t that we aren’t giving students the lessons they need. It is that students aren’t willing to shore up their knowledge each night to make sure they know what they know, and figure out how to learn what they don’t know.

So I try to aperiodically remind them of that fact.

Yesterday, for example, I hinted to my students that they might have a pop quiz. We’ve been working on quadratics, and have seen questions like:

Solve 2x^2+5x+7=0

and

Graph x^2+10x-8=y

and the latest feather in our caps

Solve x^2+10 \leq 0

It’s a lot. And quadratic inequalities killed my kids last year. So I told my students to spend the night just reviewing the material and making sure that they can organize the information in their heads. They come to class today and I give them a two question pop quiz, both questions on quadratic inequalities. 6 minutes. Most are frantic. Clearly many didn’t shore up their knowledge.

I then tell them to stop and put their pencils down. I tell them it wasn’t for a grade. I tell them I’m not collecting it. They breathe a sign of relief. We then had a conversation.

What was hard about the pop quiz?
Did you think you knew the material?
Did taking this quiz demonstrate that? Or did it tell you something else?

It was a nice and short conversation and I think it really drove home the point: you think you know, but you have no idea.

So here’s something for you to consider doing, if you’re cruel like me: a very occasional fake pop quizzes can be a nice conversation starter about studying and nightly responsibility.

UPDATE: So in this case, the faux pop-quiz was only moderately successful. Last year so many kids didn’t know what to do on the 1D quadratic inequalities question on the final unit assessment.  This year they were less were confused. But still there were enough students who didn’t know how to solve it to give me pause. I realize now that we learned so many different types of linear/quadratic things that students kept confusing “what’s the question asking?” and “how do I solve that kind of problem?” So I need to come up with a way to emphasize at each point of the unit these two fundamental questions. And maybe designing a short activity where students are forced to answer those questions.

Favorite Tweets #2

I did this a while ago — posted my favorite tweet conversations from days past. Favorite, for me at the moment, doesn’t really mean “advice on teaching” but just random convos. The best kind.

…

k8nowak @samjshah aw. You like us! You really like us! *hugs*

samjshah @k8nowak naaah, *like* is maybe too strong a word. you guys just keep me slightly amused. paint drying < y’all < reality tv

calcdave @samjshah You do know “Glee” is not reality tv, right?

Fouss @samjshah With that bit of information, you’re kicked out of the club. Don’t let the door hit you on your way out.

…

dcox21 @samjshah thanks Sam. Re-mixed them to suit my needs, but borrowed the heck out of your stuff. I’ll send you the remix…for approval.;-)

dcox21 Thanks to @samjshah… http://img111.yfrog.com/i/vtkl.jpg/

dcox21 @calcdave Yeah, I wrote it as neatly as I could and then placed the kids in just the right place just before taking the pic. ;-)

…

mctownsley The Teacher Salary Project – teaching is easy…right? http://www.youtube.com/watch?v=czPRKh2ooOY&feature=player_embedded via @scsdmedia

…

SweenWSweens Not gonna lie: pretty sweet artwork, but I really wish you were paying attention to the related rates lesson. http://brizzly.com/pic/1EJC

…

k8nowak @JackieB hey for pi day we are planning a ‘family feud’ assembly, and i put on the survey ‘name the teacher that gives the best hugs’ and..

k8nowak @JackieB the frontrunner as well as the second and third highest vote-getters…English. no joke.

…

k8nowak Just gave a girl a hard time about dropping out of college while buying beer from her. TEACHER OF THE YEAR.

…

busynessgirl #needaredstamp “It will take me more ink to explain what’s wrong than you used to do this problem in the first place.”

…

samjshah cool! how to find things that nearly rhyme… if i were musical this would rock… http://www.b-rhymes.com/

calcdave @samjshah My name is Sam which almost rhymes with stamp. These words I amass to teach you all math.

samjshah @calcdave My name is Dave which rhymes with concave. Up in the schoolz, my kidz be amazed. With my crazy calc skillz, ya heard? Word.

…

sumidiot anybody know a source for this quote about lectures: info from notes of prof to notes of student without passing through mind of either?

…

samjshah um, how do i feel that justin bieber and luda did a song together? someone tell me because i just can’t figure it out. http://www.youtube.com/watch?v=kffacxfA7G4

sig225 @samjshah On behalf of all Canadians, I apologize for Justin Bieber. Please, take our Olympic Men’s Hockey gold medal …

…

jbrtva Guy I know asked me out for coffee…he wanted to tell me that we’re just friends. #didimisssomething?

k8nowak @jbrtva now you invite him for coffee, and inform him you’re not just friends, but mortal enemies. Advise him to sleep with one eye open.

samjshah @k8nowak @jbrtva invite him for coffee, and inform him that you’re not just friends, but in a serious, committed relationship. then propose.

k8nowak @samjshah @jbrtva invite him for coffee and inform you’re not just friends, but actually brother & sister. Expose family’s deepest secret.

samjshah @k8nowak @jbrtva invite him for coffee and inform him you are tyra banks in disguise. scold him for sending mixed signals. KISS MY FAT A**!

…

samjshah just bought my spring break tickets. SAN FRAN HERE I COME!!! and i am taking virgin atlantic one way there… TV!

dcox21 @samjshah stop by Porterville and I’ll buy you dinner. Best Mexican food north of the border.

samjshah @dcox21 that’s 4+ hours away? NO WAY! wait, did you say you’re buying…

samjshah @dcox21 @k8nowak what are all y’all talking about? i missed some bandwagon. there were probably cupcakes on it, i bet.

k8nowak @dcox21 SSHHHH. Do NOT tell him about the cupcakes!

dcox21 @k8nowak Hey if he’s not willing to take me up on a free dinner, I’m not saying nothin’ about the cupcakes.