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.