by Adam Levine, GSAS

1. (From freshman year multivariable calculus) If psi = (2x+y cos(xy))dx+x cos(xy)dy, show that psi is exact by finding a function f such that df = psi, and compute the integral of psi around the boundary of T.

f is x to the power of 2

Plus the sine of its product with y.

We can see the equality’s true

’Twixt df and our given form psi.

We can then say that psi is exact

So it’s closed – cause for joy, not despair

For this gives us the wonderful fact

That d psi equals nil everywhere.

Now to integrate psi on the bound

Of T, use the theorem of Stokes:

Integrating d psi all around

Gives the answer. I mean it! No jokes!

So I say in this couplet of heros

That the integral comes out to zero.

2. (From senior year algebraic topology) Let X be a finite, connected CW complex of dimension n, and let Y be a space such that pi_i(Y) is finite for i<=n. Show that [X, Y] is finite.

Let X be a CW complex

That’s finite. Take another space, called Y .

In each dimension less than that of X

Assume Y has a finite pi sub i.

Now take the set of classes of functions

Equivalent up to homotopy.

Inducting on our X’s dimension

We show that finite this has got to be.

The largest cells in number are but few

We may restrict our function to each one.

And then we also may restrict it to

A map into the largest skeleton.

The classes to a finite set inject

Which proves that our theorem is correct.

Winner

First Runner-Up

Second Runner-Up

Dishonorable Mention

Dishonorable Mention

Dishonorable Mention

Dishonorable Mention