## Archive for April, 2009

### Differential Forms

I’m always on the lookout for non-rigorous, ‘pictorial’ explanations of abstract mathematical topics. Because ones that are good, and also mostly correct, are difficult to find. And also I am not particularly fond of learning mathematics by plodding through theorems in a book. I find that only with the appropriate picture does a particularly daunting and abstract topic come into focus. And then plodding through the theorems is actually quite easy.

I found this short, but sweet introduction to differential forms, via the Googles. It must be good, because PageRank puts it on the first page.

### Topology Apology

In lieu of actually delving into this, I’ll just give a short description and let you go look it up on Wikipedia.

There’s a way of classifying or generating seemingly complicated surfaces using a sort of surface (or shape, I guess) that isn’t really that complicated.

The way you do it is this: Take one vector of the polygon, say the top A vector, and glue it head-to-head and tail-to-tail with the other A vector. Now do the same with the B vector. What you literally get is a kind of weird, oriented, triangular surface. In topology-land, of course, you can continuously deform this weird pseudo-prism into something we all know and love, namely a sphere. Or, in the fancy terms, you’re utilizing a homotopy equivalence between the two surfaces. The same thing can be done for arbitrary surfaces, as well, provided they are also closed.

For example, here’s a torus:

This is a neat way to visualize certain surfaces, especially when they cannot be embedded in three-dimensional space without self-intersections (such as the Klein bottle, or the real projective plane).

### The Lawyer’s Expert Opinion

Andrew McCarthy, noted retard, writes about the release today of four torture memos written under the presidency and administration of George W. Bush (also a noted retard):

A terrible decision, pushed for aggressively by AG Eric Holder… Holder’s press release, which suggests that the interrogators are in fact guilty of torture (“‘The President has halted the use of the interrogation techniques described in these opinions, and this administration has made clear from day one that it will not condone torture,’ said Attorney General Eric Holder. ‘We are disclosing these memos consistent with our commitment to the rule of law.'”)…

If only McCarthy had bothered to adjust his ocular pitch about negative two degrees. You know, like when he’s “talking to a member of the opposite sex”. And by “talking”, I totally mean “thinking about tittie banging that loudmouthed broad”. If that wasn’t obvious.

### Oh, Well That’s Gone

To the person who broke into my car and stole my GPS:

I appreciate that you did not steal the pants lying in the back seat, though you clearly rifled through them.

I must remark, however, that the theft of my umbrella was a pretty dick thing to do.

Today it rained, and I did not have a spare.

I hope you die, forever.

### Tensors

See here for a very friendly introduction to tensors (~25 pages).

See here and then here for a very friendly introduction to the Levi-Civita tensor (~3 pages).