Christmas Lecture

For the Christmas lecture of my multivariable calculus module I tried find something entertaining to present. This turned out to be quite difficult, but in the process I across this multivariable calculus themed comic.

In the end I settled on showing how to evaluate

$\displaystyle \zeta(2) = \sum_{n=1}^\infty \frac 1{n^2} = \frac{\pi^2}6\,.$

There are many proofs for this identity—fourteen have been been collected by Robin Chapman—and it is often done as an application of the theory of Fourier series. One of the proofs, however, uses double integrals: using the geometric series one can show that

$\displaystyle \sum_{n=1}^\infty \frac 1{n^2} = \int_0^1 \int_0^1 \frac{1}{1-xy} \,\mathrm{d}x \,\mathrm{d}y\,.$

Evaluating this integral is an instructive exercise because it confounds many of the assumptions about double integrals students may have: the integration domain is as simple as one might hope, yet the right approach is to perform a change of coordinates; the resulting domain can be parametrized as one piece, yet it is better to split the domain in two; non-trivial trigonometric identities are used to simplify the integrands.

Whether or not the students were entertained remains unknown, but as a result I wrote some notes explaining the calculation in great, perhaps excessive, detail.