Quotient rule with determinants?

I’ve always found the quotient rule for differentiation hard to remember:
$$\dod{}{x}\frac{f}{g} = \frac{f’g – g’f}{g^2}$$

I was looking on the Wikipedia page for the finite difference operator and noticed that they use the determinant in describing the quotient rule for the difference operator. The determinant can also be used to describe the quotient rule for the derivative:
$$\dod{}{x}\frac{f}{g} = \frac{1}{g^2}\begin{vmatrix}f’ & g’ \\f & g\end{vmatrix}$$
I wonder why this is? Is it just a coincidence?

Update: It occurred to me that it can also be written this way:
$$\dod{}{x}\frac{f}{g} = \det\left(\frac{1}{g}\begin{bmatrix}f’ & g’ \\f & g\end{bmatrix}\right)$$

Since the determinant represents the area of a parallelogram, the derivative at $x_0$ is the area of parallelogram formed by $ \langle f'(x_0),\ g'(x_0)\rangle $ and $ \langle f(x_0),\ g(x_0)\rangle $ where $ g(x_0) $ is scaled to 1. Still, this interpretation is kind of forced. What does it mean?

March 25, 2012, 11:34pm by Casey
Categories: Math | Tags: | Permalink | 2 comments

Forwarding all email to a domain with sendmail

I just bought a new domain name (bitsofpancake.com) — I’m actually quite proud of that name, to be honest! I decided that I would forward all the email addressed to it (whatever@bitsofpancake.com) to my personal email. I had previously done that with a convoluted combination of Google Apps and its catch-all email feature, but this should be much less roundabout. Or so I thought: figuring out how to set it up was a pain.

First, I needed to allow sendmail to function as an external mail server.
I’m on Ubuntu Natty, and the sendmail config files are located in /etc/mail/. It turns out that the sendmail.conf is so complicated that there’s a separate file, sendmail.mc, that is designed to be edited and then “compiled” into the real config file.

I had to change this line in sendmail.mc:

DAEMON_OPTIONS(`Family=inet, Name=MTA-v4, Port=smtp, Addr=127.0.0.1')dnl

into:

DAEMON_OPTIONS(`Family=inet, Name=MTA-v4, Port=smtp')dnl

This change tells sendmail to accept external email (source).

Next, I had to enable the virtual user table. I added this line to the same file (sendmail.mc) (source):

FEATURE(`virtusertable',`hash -o /etc/mail/virtusertable.db')dnl

I made a file in /etc/mail/ called virtusertable with this one line:

@bitsofpancake.com	forwardtothisemail@gmail.com

This catches all emails to @bitsofpancake.com and forwards it to my Gmail account. You can add more lines to this file to specify different forwarding rules (source).

I compiled the virtusertable using:

/etc/mail$ sudo makemap hash virtusertable < virtusertable

Next, I added bitsofpancake.com to /etc/mail/local-host-names — this file tells sendmail which domains to consider "local." (Otherwise, it would keep on trying to send to itself in an infinite loop!)

Then, I reloaded and "compiled" sendmail's config with:

/etc/mail$ sudo sendmailconfig

Finally, I added port 25 (the port for SMTP) to my iptables firewall and updated my MX DNS records to point to the server — it works!

March 25, 2012, 8:27pm by Casey
Categories: Server Administration | Tags: | Permalink | 2 comments

Hello!

Hi! I’m Casey, and this is my new blog. I’ll be throwing stuff that’s interesting to me and hopefully interesting to you, too! (That’ll probably mean lots of math and computer programming, just to warn you.) A lot of the time, I’ll have some thoughts on some math theorem or an idea for a web site but leave them once I’ve thought about them enough. I figured: why not have a place to put everything so I can look back at them or so others can be inspired? (Yeah, let me dream a little. :])

Since I have a lot of time over the next two weeks, I’ll also try to dig up some interesting ideas I’ve had in the past and make them into blog posts. I’ll see how this goes. Wish me luck!

March 25, 2012, 1:14pm by Casey
Categories: Meta | Permalink | 2 comments

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