# Monthly Archives: June 2019

## Tutorial 11: CMC Snails, Tires and Soap Bubbles

In previous tutorials we compute immersed minimal surfaces. These immersions were the critical points of Dirichlet energy. You have seen in the lecture that, if we additionally include a volume constraint, then the critical points have no longer mean curvature … Continue reading

## Symplectic Maps and Flows

Consider a physical system consisting of $k$ massive particles at positions in $\mathbb{R}^3$ which we can assemble in a vector in $\mathbf{q}\in\mathbb{R}^m$ where $m=3k$. Likewise, the velocities of these particles at a given instant of time can be described by … Continue reading

## Tutorial 10: Wave Equation and Verlet Method

Let \(M\) be a Riemannian surface and let \(\Delta\colon C^\infty M \to C^\infty M\) denote the corresponding Laplace operator. The second order linear partial differential equation \[\ddot{u} = \Delta u\]is called the wave equation and—as the name suggests—describes the motion of … Continue reading

## Tutorial 9: Minimal Surfaces and Plateau Problem in \(\mathbb S^3\)

We are back at minimal surfaces in \(\mathbb S^3\). This time the goal is to visualize minimal surfaces in \(\mathbb S^3\) first described by Lawson, which are constructed by reflections and rotations of a fundamental piece which is a minimal … Continue reading

## Tutorial 8: Electric Fields on Surfaces

As described in the lecture a charge distribution \(\rho\colon \mathrm M \to \mathbb R\) in a uniformly conducting surface \(M\) induces an electric field \(E\), which satisfies Gauss’s and Faraday’s law\[\mathrm{div}\,E = \rho, \quad \mathrm{curl}\, E = 0.\]In particular, on a simply … Continue reading