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The classical inverse function theorem reads as follows:

Theorem 1 ($latex {C^1}&fg=000000$ inverse function theorem) Let $latex {\Omega \subset {\bf R}^n}&fg=000000$ be an open set, and let $latex {f: \Omega \rightarrow {\bf R}^n}&fg=000000$ be an continuously differentiable function, such that for every $latex {x_0 \in \Omega}&fg=000000$, the derivative map $latex {Df(x_0): {\bf R}^n \rightarrow {\bf R}^n}&fg=000000$ is invertible. Then $latex {f}&fg=000000$ is a local homeomorphism; thus, for every $latex {x_0 \in \Omega}&fg=000000$, there exists an open neighbourhood $latex {U}&fg=000000$ of $latex {x_0}&fg=000000$ and an open neighbourhood $latex {V}&fg=000000$ of $latex {f(x_0)}&fg=000000$ such that $latex {f}&fg=000000$ is a homeomorphism from $latex {U}&fg=000000$ to $latex {V}&fg=000000$.

It is also not difficult to show by inverting the Taylor expansion

$latex \displaystyle f(x) = f(x_0) + Df(x_0)(x-x_0) + o(\|x-x_0\|)&fg=000000$

that at each $latex {x_0}&fg=000000$, the local inverses $latex {f^{-1}: V \rightarrow U}&fg=000000$ are also differentiable at $latex {f(x_0)}&fg=000000$ with derivative

$latex \displaystyle Df^{-1}(f(x_0)) =…

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Team Python

Okay, so you have “registered an interest” or pre-ordered or even (lucky few) ordered an RPi. It’s time to get together the stuff you’ll need.

1. A screen. The most obvious choice is a TV with an HDMI input.

2. A usb mouse. If you don’t have a spare around, why not treat yourself to a new one – you’re worth it!

3. A usb keyboard. Again, you may have a spare. I’ve ordered a small one, for portability.

4. An ethernet cable. For the web!

5. An operating system. I’m still not quite sure if the Pi is going to ship with one at this early stage. There are lots of places where you can find instructions for setting up your SD card. So far Debian and Arch are available to download via the RPi site. Which brings me to:

6. An SD card. The OS images are…

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Team Python

How to use RacyPy or WaryPy.

1. Download the .iso.

2. Download and install DVD Decrypter (or use you favourite .iso burning software).

3. Right click on the .iso file and choose “Burn with DVD Decrypter.

4. Reboot your PC. It should boot from the CD, if it doesn’t do this by default, look for a message telling you what to press for boot options (often F2 or F9).

5. That’s it.


To run WaryPy in Virtual Box.

1. Download WaryPy.vdi.

2. Download and install VirtualBox.

3. Start VirtualBox. Choose “new”.

4. Follow the instructions on screen and choose “Linux” and Linux 2.6.

5. Choose “use existing hard disk” and find the .vdi file.

6. Start your new virtual machine.


Since Liz posted on the RPi homepage about this LiveCD, I’ve had a lot of hits on the blog. Quite a few people have asked about…

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