Tag: Mathematics

Lie groupoids and differential equations

In this blog post, I want to explain a way in which I like to motivate Lie groupoids via the theorem on the existence and uniqueness of solutions to ordinary differential equations. The basic idea is that in trying to do away with the choices – the initial time and the initial conditions – we’re naturally led to consider a ‘universal fundamental solution’. This is a matrix function depending on two parameters, the initial and final times, and it must satisfy a homomorphism-type condition. In order to axiomatize this condition, we are automatically led to the definition of a Lie groupoid. Once this has been absorbed, we can then rephrase the existence and uniqueness theorem as a correspondence between ODEs and representations of the pair groupoid of the real line. This is a special case of Lie’s second theorem for Lie groupoids.

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The Volume of an Even Dimensional Ball

What is the volume V(r,d) of a ball of dimension d and radius r ? This ball is the following subset of \mathbb{R}^d :

B(r,d) = \{ (x_{1},..., x_{d}) \in \mathbb{R}^d \ | \ \sum_{i = 1}^{d} x_{i}^2 \leq r^2 \},

and we measure the volume using the standard Euclidean measure.

If you only need the answer for even dimensions, here’s an easy way to remember: define V(r) to be the sum of the volumes of balls of radius r over all of the even dimensions:

V(r) = \sum_{n = 0}^{\infty} V(r, 2n) .

Remarkably, the formula for this function turns out to be V(r) = e^{\pi r^2} . This is easy to remember since it’s the exponential of the volume (i.e. the area) of a disc. Then, you can read off the volumes V(r,2n) as the terms in the Taylor expansion of V(r) = e^{\pi r^2} . Explicitly:

V(r,2n) = \frac{\pi^n r^{2n}}{n!}.

A nice, and immediate, corollary is that the derivative of V(r) , with respect to the radius r , gives you the sum of the volumes of all of the odd-dimensional spheres:

2 \pi r e^{\pi r^2} = \sum_{n = 0}^{\infty} Vol(S^{2n + 1}) r^{2n + 1}.

By looking at the coefficients of the Taylor expansion, we can conclude that the volume of the unit sphere of dimension 2n + 1 is \frac{2 \pi^{n+1}}{n!} .

There’s a lot more to say about the volumes of balls and spheres, and if you’re interested, the Wikipedia page is a good place to start.

Okay, but why is the formula for V(r) true? And what’s special about even dimensions? That’s what I want to explain in this post.

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