Ubuntu on Lion under VirtualBox (1)

I posted a few times previously about running Ubuntu in a virtual machine in OS X (here), but I had some difficulty in retracing my steps six months later.

So I thought I would cover the ground again, and expand it. This time I will talk mostly about things that do work, not about my trials and tribulations on the way.

I don't have the luxury of a separate computer to run Linux. So it's very nice that there is a free virtual machine available for me to play with called VirtualBox (wikipedia). Thanks, guys. (It's an Oracle product these days).

I downloaded and installed VirtualBox 4.1.10 for OS X.

For Ubuntu, there is a choice of 32-bit or 64-bit and Desktop or Server. For example:

ubuntu-11.10-desktop-i386.iso
ubuntu-11.10-server-amd64.iso

These designations stand for Intel 80386 32-bit, and AMD x86-64, 64-bit. Either one will work. I chose standard Linux Ubuntu version 11.10 ("Oneiric Ocelot") 32-bit this time. It does not matter that my machine has an Intel chip.

I want to do server-like stuff, but I do find it handy sometimes to have a GUI (e.g. for Firefox and Dropbox), and also I thought it might be instructive to add the relevant software one piece at a time, starting with Apache2. In the Desktop, the 32-bit version is recommended.

The .iso file type is described here. It is basically an archive file format for an optical disc.

After intalling VirtualBox I ran the Oracle VM VirtualBox Manager, which first lead me through setup of a new virtual machine, where I accepted all defaults except that I boosted the RAM to 2048 MB. (I don't plan to do much work on the OS X side while the server is running). Starting the VM gives a "First Start Wizard" which has a dialog where you should navigate to the stored .iso file. I am always too impatient to actually read the instructions, but in this case it would be good to read chapter one of the VirtualBox docs or at least this part of it: Starting VirtualBox.

The install doesn't take long, 10 minutes or so. I had trouble (on several installs) where I had wandered away to the OS X side, and when the install finished and wanted to restart Ubuntu it did bad things to OS X, forcing a hard shutdown with the power switch. So, make yourself a sandwich or something.

At the end, bring up the Terminal. How to achieve this was puzzling at first. A shortcut I found on the web is to do CTL-OPT-t. Another simple method is to search for Terminal after clicking the "dash" icon at the top of the "launcher"---a dock lookalike.

There are methods to add the Terminal to the launcher, but these involve right-clicking, which seems problematic with this setup. I'll have to investigate and get back to this later. I got it to work on my laptop by the two-finger-tap method, but it's not working now on the mini.

Now, the very first thing is to do:

sudo apt-get update

A GUI Update Manager will come up when this finishes, with many items to update (≈ 350). I did this too, though I suspect it's unnecessary for us and it took a long time.

In order to get cut-and-paste to work, we will need what are called the "Guest Additions" from VirtualBox. First, type in this command:

sudo apt-get install dkms build-essential linux-headers-generic

When it's done, restart Ubuntu. Then, from the VirtualBox menu above the Ubuntu window, from Devices choose mount Guest Additions. This "mounts" a "CD" in Ubuntu, when you click on that it will ask you whether you want to run the software, which you should do. When it's done, reboot Ubuntu again, and then "eject" the disk.

Now, cut-and-paste should work between OS X and Ubuntu. The only trick is that you need to click in the Ubuntu/Terminal window twice, once to shift focus to Ubuntu, and then to shift focus to Terminal. For Terminal, we do CTL-SHIFT-v and -c. (If you're in the Text Editor, just do CTL without the SHIFT).

Now, we'll grab two more packages:

sudo apt-get install apache2
sudo apt-get install openssh-server

And at this point, I saved a snapshot of the VM from VirtualBox (under Machine > Take Snapshot). To shut down (in OS X), just click on the red close button (top left of window) and then choose "Save State."

go run go

Go version 1 is released today.

Unix spoken here (5)

A link to a great review of Unix text processing tools.

matplotlib on 10.7.3

I upgraded my home "Desktop" (a Mac mini that had been running OS X Server 10.6) to Lion (10.7.3) yesterday. This post is just to note that the pkgconfig method still works for installing matplotlib. To recap, I used Homebrew to install pkgconfig, which can then be used by the matplotlib build process to find zlib, libpng and libfreetype. It's as simple as:

git clone git://github.com/matplotlib/matplotlib.git cd matplotlib/ python setup.py build sudo python setup.py install

There were a few hiccups, of course. The motivation for changing to 10.7 was that my MobileMe email (and iDisk storage) are going away at then end of June. So I "upgraded" to iCloud, after installing 10.7.3 on my laptop. However, this made Mail extremely sluggish on the machines with 10.6. Mail showed the little triangle (which I think means it's having trouble) for 15 minutes at a time and wouldn't get email from the server.

I had various troubles with the install that I believe relate to having bought Lion using my first Apple ID (which used my work email) rather than my second (which uses my .mac email). It irritated me, but seemed that the simplest solution was to just buy Lion again. So I did that, and everything started working fine.

Another issue was to get Xcode (Developers Tools) from the App Store. This is free, but involves various hoops to jump through, the weirdest of which is that gcc is no longer a part of the default install. You have to start Xcode and go to Preferences > Downloads > Command Line Tools. Why would they do that? Anyway, having gcc

> ls -al /usr/bin/gcc lrwxr-xr-x 1 root wheel 12 Mar 24 23:14 /usr/bin/gcc -> llvm-gcc-4.2

I used Homebrew to get pkgconfig, and then built matplotlib as shown above. Simple scripts work. I used easy_install nose (with sudo) to install nose, which is needed by matplotlib's tests, then did

python tests.py -v -d

from within the matplotlib directory. The result looks good:

Ran 1086 tests in 290.786s OK (KNOWNFAIL=546)

Now I just have to re-install my favorite Safari extensions (forgot to save them), and find the license key for TextMate. I'm making it work, but all these hoops to jump through, and Apple's Big Brother mentality, are making me think seriously about switching to Linux for my hobbyist programming.

iTunes backup playlist

Since they removed the DRM from music downloads at the iTunes store, I've been buying songs from Apple using that approach. It's great for impulse buying. Also, it's nice in combination with an app for the iPhone called Shazam, which can identify a track that is playing wherever you are at the moment. I thought I was fairly modest in my purchases, but now find that I have over 500 songs after 18 months. So the question then is how to archive these purchases so they don't go **kabluie** in the night.

One strategy is to just let Apple do it, but I don't trust them.

The next idea is to manually copy all 500 songs to a backup disk. This would be easy, except that (i) I don't want the whole library, just a playlist, and (ii) iTunes uses nested folders to preserve the artist:album:songtitle information. I need to merge u/v/w with u/v/x to create a directory u/v containing both w and x. You can't do that just copying x.

The playlist info is contained in xml format, file: 'iTunes Music Library.xml' but I decided to export the playlist to disk from within iTunes ('File > Library > Export Playlist...'). Each entry in the exported text file ends with something like this as the last field (tab-separated, one entry per line):

HD:Users:Shared:iTunes Music:Bob Marley:Natty Dread:01 Lively Up Yourself.m4a

The following script finds each song on the playlist and copies it to a directory temp in the directory where the script is run. Simple. The result is 3.77 GB of .m4a files with the desired directory structure.

A detail that is (or should be) embarassing: iTunes (on OS X) uses '\r' (CR) as newline. Talk about the constraints of backward compatibility.

One last thing: please, please, please back-up and test before using this. YMMV. Caveat lector. No warranty express or implied. Don't blame me if your library vanishes.

import os, subprocess name = 'playlist' FH = open(name + '.txt', 'r') data = FH.read().strip() FH.close() # iTunes uses '\r' (CR) as newline! data = data.split('\r') # data[0] is metadata (column names) data.pop(0) # file path is the last value L = [item.split('\t')[-1] for item in data] # ':' is path separator L = [item.replace(':','/') for item in L] for item in L: # remove HD name from file path item = item.split('/', 1)[1] # file path has spaces # must be quoted for shell command below src = '"/' + item + '"' artist, album, songfile = item.split('/')[-3:] # construct directory tree if it doesn't exist path = '/'.join(('temp', artist, album)) try: os.stat(path) except OSError: os.makedirs(path) dst = '"' + '/'.join((path, songfile)) + '"' cmd = ' '.join(('cp', src, dst)) obj = subprocess.call(cmd,shell=True) if obj != 0: print 'e', else: print '*', print dst

Airport Express

I saw a blurb (it must be several years ago) about using an Apple wireless device called Airport Express to play music files from my computer over a WiFi network on remote speakers. I bought an AE recently, plus a miniature amplifier from Amphony and some bookshelf speakers from Polk Audio through Newegg.

I was surprised, and quite disappointed, to find that in order to use this device in the normal way one must be a member of the "Apple club." That is to say, in order to send audio from my computer to Airport Express and have it come out through speakers, I need to have an Apple ID so that I can do "Home Sharing." I think this is outrageous. However, I do think it was pretty cool, in the end, to control this setup with my iPhone running an App called Remote.

I've never owned a non-Apple computer. That's a lot of boxes, starting from 1984. And I love OS X and its integration of the pretty Mac side with Unix underneath. However, it's clear that the future is iOS, i.e. not folks like me. So I was pleased to find, in a quick Google search, a somewhat dated article which explains how to stream audio in this way with Linux hardware.

About the amp, it is fun to produce music with so little hardware on site, but there is no ON/OFF switch for output, though there is a dial, which when dialed down still sends a nasty signal through the speakers when powering down. And I do like the sound a lot. Nice speakers!

Go Snail Go

I came across a post which solves what it calls the "snail" problem in Go. We've seen some very pretty examples of the same thing in Python here.

This is the output of my version in Go for S = 9.

1  2  3  4  5  6  7  8  9 32 33 34 35 36 37 38 39 10 31 56 57 58 59 60 61 40 11 30 55 72 73 74 75 62 41 12 29 54 71 80 81 76 63 42 13 28 53 70 79 78 77 64 43 14 27 52 69 68 67 66 65 44 15 26 51 50 49 48 47 46 45 16 25 24 23 22 21 20 19 18 17

You can see why he called it the snail. Anyway, I noticed a chance to use a goroutine for this problem. In any cycle starting with [left] followed by [down] we go n steps in each direction, then n+1 steps in the [right,up] directions. In the code below, we obtain these step values from a Go channel.

A couple of other Go-like things about this code. We stash the 2D array (as a 1D array with a shape parameter) in a struct, and then attach a func to that struct to pretty print it. The details of the pprint function could doubtless be improved---I'm not too swift with formatting. The other Goish thing is to modify both the row and column indices at once, returning what we'd call in Python a tuple value from the step function.

Fun.

package main import ( "fmt" "os" "strings" ) func dist(ch chan int) { var a int = 1 for { ch <- a ch <-a a++ } } var m = map[string]string{"L":"D","D":"R","R":"U","U":"L"} func step(r, c int, dir string)(rn, cn int) { switch dir { case "L": { c-- } case "D": { r++ } case "R": { c++ } case "U": { r-- } } return r,c } type A struct { arr []int; SZ int } func (a *A) pprint() { d := a.SZ for i:= 0; i < len(a.arr); i += d { S := []string{} for _,f := range a.arr[i:i+d] { S = append(S,fmt.Sprintf("%2d", f)) } fmt.Println(strings.Join(S," ")) } } func main() { S := 9 if S%2 != 1 { os.Exit(1) } N := S*S a := A{make([]int,N), S} ch := make(chan int) go dist(ch) dir := "L" var r, c int; r = S/2 + 1; c = r n := <- ch for { for i := 0; i < n; i++ { //fmt.Println(r, c, N) a.arr[(r-1)*S + (c-1)] = N N-- if N == 0 { break } r,c = step(r, c, dir) } n = <- ch dir = m[dir] if N == 0 { break } } a.pprint() }

Go, again

This is a brief, updated report on my exploration of the Go language. (First post here). For an idea about what Go can do I encourage you to check out another video with Rob Pike (and Russ Cox). Not to disrespect Russ, but as a great example, check out the segment starting about 21:25. It's amazing.

The source of that program is here.

In order to play with this stuff, you will have to download the compiler source and build it. I assume you can do that, if not, drop me a line.

I've spent a total of about 40 hours on Go over the last week, and I can say that I believe this is all quite correct:

The Go programming language is an open source project to make programmers more productive. Go is expressive, concise, clean, and efficient. Its concurrency mechanisms make it easy to write programs that get the most out of multicore and networked machines, while its novel type system enables flexible and modular program construction. Go compiles quickly to machine code yet has the convenience of garbage collection and the power of run-time reflection. It’s a fast, statically typed, compiled language that feels like a dynamically typed, interpreted language.

The more I explore, the better I like it. Go does "feel like a dynamically typed, interpreted language." Exactly. And using TextMate I can just do CMD-R and the build, linking and execution happen painlessly.

I wrote about 30 or so short programs to explore simple tasks in Go. I also wrote two different versions of the PSSM code discussed here, and in subsequent posts. And I played with Bruce Eckel's code from here. A zipped folder of all this stuff is on Dropbox. It's not very well documented but only the intrepid will follow this lead anyway.

I'm hooked. I have a lot of work to do figuring out the interface and concurrency stuff. And it is great fun!

Go fib

I've been having fun with a new programming language called Go, developed by Rob Pike and friends at Google (video). Somehow I missed hearing about it until now. I'm not leaving my true love, Python, but there's a lot to like about Go as a replacement for C and C++.

Go has

garbage collection, so there are no worries about managing memory

no classes, but simply attaches methods to types

a simple syntax compared to C---eliminating most semicolons

much less complexity than C++

goroutines to launch parallel processes that communicate using channels

maps and string support, and slices as a kind of resizable array

multiple return values from a function

combined var declaration and assignment with type of rhs

pointers but no pointer arithmetic

the ability to construct and return a local variable from a function.

great docs, is compiled, and fast.

Go has interfaces, which I don't understand very well, but they can be used to give polymorphic behavior.

Here is a simple, familiar Go program using a goroutine and a channel. Notice that the type declaration comes after the variable name, and the arrow symbol for flow into and out of the channel, c.

package main import "fmt" func fib(c chan int) { var a, b int = 1, 0 for { c <- a a, b = a + b, a } } func main() { c := make(chan int) go fib(c) for i := 1; i < 12; i++ { f := <- c fmt.Println(f) } }

fib() will generate values as long as we want, similar to a Python generator. Output:

1 1 2 3 5 8 13 21 34 55 89

I found a Go "bundle" for TextMate here.

Fee, fi, foe, fum

I found myself wondering about the correct pronunciation of the Greek letter φ (phi). Somehow, I acquired the habit of saying "fee." But it turns out that mathematicians say "fi" as in "fly."

It seems that in Greek, the first pronunciation is correct. However, if we were to follow that route, we'd have to pronounce π as "pee," and that just feels wrong.

A more common question is about "data." See howjsay. I pronounce data like "date-uh." I have a memory of this being drummed into me in Latin class. Perhaps we should just "say it like the Romans did," but see the above for one problem with that approach. Anyone know an authoritative source?

One year, one world

Yesterday was the one-year anniversary of the flag counter widget. We have 144 countries, including a new one: Cote D'Ivoire. It makes me very happy to see all these ≈25K (unique) visitors, even if I don't know that much about them.

From Google's stats, it looks like most people are interested in maybe 25-30 pages. I think there is more good stuff buried here, so perhaps I'll make a post about my favorites one of these days. What I'd love to see, is data that showed a person arriving and then poking around for a while. Sometimes you see a hint of this, but the stats really don't tell that much.

Anyway, thanks for reading! And don't forget to check out the Python book.

And, I'm still waiting for Tonga.

R.W. Hamming

I received a new book in the mail today. I'm a huge fan of Richard Hamming (e.g. this), and the book is everything I hoped for.

I thought I would just post a quote (from him, not one of the many interesting quotes he includes):

The taste to work on the right problem at the right time and in the right way is the secret of doing significant things.

To focus on just one aspect: what you are trying to do must be doable. Timing is (almost) everything.

SVD

I found a nice tutorial on SVD (Singular Value Decomposition). I love the intro:

Most tutorials on complex topics are apparently written by very smart people whose goal is to use as little space as possible and who assume that their readers already know almost as much as the author does. This tutorial’s not like that. It’s more a manifestivus for the rest of us. It’s about the mechanics of singular value decomposition, especially as it relates to some techniques in natural language processing. It’s written by someone who knew zilch about singular value decomposition or any of the underlying math before he started writing it, and knows barely more than that now. Accordingly, it’s a bit long on the background part, and a bit short on the truly explanatory part, but hopefully it contains all the information necessary for someone who’s never heard of singular value decomposition before to be able to do it.

An excellent place to start. And check out the animation at wikipedia.

MathDoctorBob and Fibonacci

I ran into an interesting page which purported to be about various ways of looking at (or deriving) the trigonometric identity

sin² + cos² = 1

but the site has lots of stuff including the trail I'm following here, which involves the Fibonacci numbers. Strang derives the Binet formula for the n^th Fibonacci number, F_n, using linear algebra:

F_n = 1/√5 [φ₁ⁿ - φ₂ⁿ]

where φ₁ is the golden mean or golden ratio, and

φ₁ + φ₂ = 1

The approach is very nice, because it uses the eigenvalues and eigenvectors of the matrix

[ 1 1 ]
[ 1 0 ]

which involves solving λ² - λ -1 = 0, whose solutions are φ₁ and φ₂. That is, φ₁ and φ₂ are eigenvalues of that matrix.

However, I got lost in the middle of Strang's version (in the book), and cast about for another explanation. I found a great one here.

This video turns out to be one of a large number by MathDoctorBob. I am very impressed with the quality of these videos, based on the first half-dozen or so that I've looked at.

Finally, back at the first site, we use two limits (as n => ∞):

φ₁ⁿ => ∞
φ₂ⁿ = (1 - φ₁)ⁿ => 0

and the Binet formula to get a limit for the ratio with large n:

F_n+1 / F_n = φ

MathJax

Let's follow up on a few previous posts (a calculation of d/dt of the total energy in a gravitational field, a post about typesetting math, and an introductory exploration of LaTeX). I'm grateful for a comment from a reader, with a link to this page, which makes it pretty clear that, yes, you can use their server to get the script in your pages.

So that's what we'll do. I had to modify the LaTeX commands a bit for this but it's mostly the same as before.

[ UPDATE: I do see a problem, now. The script applies to all LaTeX on the page, which if you go to the main page for the blog, includes the previous post... Just click on the post title, to see the original formatting code. ]



$\mathbf{F} = m\mathbf{a}= m\ddot{\mathbf{r}}
= m\frac{d^2}{dt^2}\mathbf{r}$

$\mathbf{F} = -$$\nabla$$V(\mathbf{r})$

$E = \frac{1}{2}m|\dot{\mathbf{r}}|^2+ V$

$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m
|\dot{\mathbf{r}}|^2 + V )$ $= ?$


$|\dot{\mathbf{r}}|^2 = |\dot{\mathbf{r}}|
|\dot{\mathbf{r}}| = \dot{\mathbf{r}}
\cdot\dot{\mathbf{r}}$

$\frac{d}{dt}\frac{1}{2}m|\dot{\mathbf{r}}|^2
= \frac{1}{2}m\frac{d}{dt}(\dot{\mathbf{r}}
\cdot\dot{\mathbf{r}})$$ = m\dot{\mathbf{r}}\cdot\ddot{\mathbf{r}}$$ = \dot{\mathbf{r}}\cdot-($$\nabla$$V$$)$


$\nabla$$V$ $= < \frac{\partial{V}} {\partial{x}}, \frac{\partial{V}} {\partial{y}},\frac{\partial{V}} {\partial{z}} >$

$\dot{\mathbf{r}}$ = $<$ $\frac{dx}{dt}$, $\frac{dy}{dt}$,$\frac{dz}{dt}$$ >$

$\nabla$$V$ $\cdot$$\dot{\mathbf{r}}$ $= < \frac{\partial{V}} {\partial{x}} \frac{dx}{dt}, \frac{\partial{V}} {\partial{y}} \frac{dy}{dt}, \frac{\partial{V}} {\partial{z}}\frac{dz}{dt}>$$=\frac{d}{dt}V$


$\frac{d}{dt} E = \frac{d}{dt} ( \frac{1}{2}m
|\dot{\mathbf{r}}|^2 + V )$ $= ?$

$\frac{d}{dt}$E $= \dot{\mathbf{r}}$$\cdot$(-$\nabla$V) + $\nabla$V $\cdot$ $\dot{\mathbf{r}}$ = 0

Typesetting, again

Here are the equations from the last post, in LaTeX. It took about two hours, fairly painstaking work, but I think it looks pretty good..

UPDATE:

Here is a nice simple reference for LaTeX. I started with "LaTeX Bootcamp" (pdf).

$\mathbf{F} = m\mathbf{a}= m\ddot{\mathbf{r}}
= m\frac{d^2}{dt^2}\mathbf{r}$

$\mathbf{F} = -$$\nabla$$V(\mathbf{r})$

$E = \frac{1}{2}m\norm{\dot{\mathbf{r}}}^2+ V$

$\frac{d}{dt} E = \frac{d}{dt} (\frac{1}{2}m
\norm{\dot{\mathbf{r}}}^2 + V)$

\hspace*{2em}$ = ?$\\

$\norm{\dot{\mathbf{r}}}^2 = \norm{\dot{\mathbf{r}}}
\norm{\dot{\mathbf{r}}} = \dot{\mathbf{r}}
\cdot\dot{\mathbf{r}}$

$\frac{d}{dt}\frac{1}{2}m\norm{\dot{\mathbf{r}}}^2
= \frac{1}{2}m\frac{d}{dt}(\dot{\mathbf{r}}
\cdot\dot{\mathbf{r}})$

\hspace*{2em}$ = m\dot{\mathbf{r}}\cdot
\ddot{\mathbf{r}}$

\hspace*{2em}$ = \dot{\mathbf{r}}\cdot-($
$\nabla$$V$$)$\\

$\nabla$$V$$ = <\frac{\partial{V}} {\partial{x}}, \frac{\partial{V}} {\partial{y}},\frac{\partial{V}} {\partial{z}}>$

$\dot{\mathbf{r}}$ = $<$$\frac{dx}{dt}$, $\frac{dy}{dt}$,$\frac{dz}{dt}$$>$

$\nabla$$V$$\cdot$$\dot{\mathbf{r}}$$=< \frac{\partial{V}} {\partial{x}} \frac{dx}{dt}, \frac{\partial{V}} {\partial{y}} \frac{dy}{dt}, \frac{\partial{V}} {\partial{z}}\frac{dz}{dt}>$

\hspace*{2em}$=\frac{d}{dt}V$\\

$\frac{d}{dt}$E = $\dot{\mathbf{r}}$$\cdot$(-
$\nabla$V ) + $\nabla$V $\cdot$ $\dot{\mathbf{r}}$ = 0

Vector fun

In Marsden & Tromba's Vector Calculus, I found the following problem. It involves time-derivatives of the position vector r(t), for which we're using the dot notation of the physicists (and Newton).

In the first panel, we have Newton's second law, then the statement that the force is minus the gradient of the gravitational potential V, followed by a calculation of the total energy in the system. The problem is to compute d/dt of the energy. (hint)

The answer is given in the last panel, below. According to the book, it's a "simple calculation."

The first step is to compute d/dt of the kinetic energy. We use the formula from above, plus a trick to convert the squared term back to the dot product of dr/dt with itself. Then we use the chain rule, and finally, the definition of the force in terms of the gradient of the potential.

Calculation of d/dt of the potential energy puzzled me for quite a while, though it really shouldn't have. My solution was to work backward from the answer, as shown.

We put the two results together, and notice that they cancel. Surprise!

Typesetting math

This post is really just a note to myself about something I need to investigate more. The problem I'd like to solve is how to format mathematical equations for the web. You can see the caveman approach on most of my posts here, like this one.

I just make a table in html

<div style="overflow-x: scroll;"> <table bgcolor="#ffffb0" border="0" padding="4"> <tbody> <tr><td><pre style="font-family: monaco;">

I like the background color, and vary it depending on whether the content is code or output from a program.

It might be nice to have something prettier. So, looking around, I happened across MathJax, and also this post which explains how to use it on Blogger.

Here's a screenshot of the example:

The method used in the post is to load the script from the mathjax server, but I think what I'm probably supposed to do is direct people to resource on my (nonexistent) server. OTOH, they link to the post on the mathjax site.

What I'll probably do is just look into how to use LateX and then post screenshots.

Any thoughts?

Cycloid

While working through the MIT ocw lectures on multi-variable calculus (Prof. Denis Auroux, here), I particularly enjoyed his discussion about the cycloid. Above is a graphic from the wikipedia article (actually the graphic is an animated gif, but I grabbed one of the frames). The red curve is generated by the motion of a point on the edge of a rolling circle.

In addition to the beauty of the curve, it turns out that the length and area under the curve have simple values that are relatively easy to calculate. See wikipedia for the details.

One thing the article doesn't explain is how to get the "parametrization" for the curve. This looks hard, but is made easy by using vectors. It's explained in the second half of Auroux's fifth lecture.

Another thing the article doesn't explain is how to integrate

√(2 - 2 cos t)

Start from the double angle formula:

cos 2s = cos2s - sin2s cos 2s = 1 - 2 sin2s 2 sin2s = (1 - cos 2s) 2 sin2(t/2) = (1 - cos t)

It's straightforward from there.

The Mathworld article is also quite nice, and references a famous challenge in history, the one which led to this quote (in reference to Newton):

"Ah, I know the lion by his paw!"

Pilgrim's progress

Lately I've been working (again) on understanding multi-variable calculus. I always wished I had time to cover this material in college, but I took lots of courses in biochemistry and molecular biology instead. I wanted to put up the best resources I've found for this so far.

First, and no surprise, the ocw videos at MIT by Denis Auroux (here). I am currently at #25, about to jump off into integrals in 3D space. (It's my second time to get this far). This time I think I really have everything under control. The proof of Green's theorem was perfect, a kind of mathematical satori.

I also work through everything in Gilbert Strang's book as well. I need to read carefully there, because he's so concise, but his insight is just incredible. Check out the proof of the flux version of Green's theorem.

Two more are recently discovered resources that are especially helpful because they develop everything slowly but completely:

Paul's lecture notes
A beautiful set of pages from someone at U Minnesota

All highly recommended.