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.

War and Peas - Relevant

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 Chapmanβ€”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.


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