Category Archives: Lecture

Wave equation Let $M$ be a surface and $f : \mathbb{R} \times M \rightarrow \mathbb{R}$ a function. The wave equation is given by: \[\ddot{f} = \Delta f.\] After the space discretization we have $f:\mathbb{R} \times V \rightarrow \mathbb{R},$ $p:= \left(\begin{matrix} … Continue reading →

Let $\Omega \subset \mathbb{R}^n$ be a domain, we will consider $\Omega$ to be the “space” and functions: \begin{align} f : \mathbb{R} \times \Omega \rightarrow \mathbb{R}, \\ (t,x) \mapsto f(t,x),\end{align} will be view as time dependent functions \begin{align} & f_t : … Continue reading →

Let $M= \left(V, E, F \right)$ be an oriented triangulated surface without boundary and $p:V \rightarrow \mathbb{R}^3$ a realization. In earlier lectures we considered the space of piecewise linear functions on $M$: \[W_{PL}:=\left\{ \tilde{f} :M\rightarrow \mathbb{R} \, \big \vert \,\left. … Continue reading →

Let $\Sigma = (V,E,F)$ be a triangulated surface (with boundary). A realization of the surface in $\mathbb{R}^3$ is given by a map $p:V \rightarrow \mathbb{R}^3$ such that $p_i,p_j,p_k$ form a non degenerated triangle in $\mathbb{R}^3$ for all $\{i,j,k\} \in \Sigma$, … Continue reading →

Let $M = (V,E,F)$  be a triangulated domain in the plane where $V$ denotes the set of vertices, $E$ the set of edges and $F$ the set of triangles.  We consider the set of functions on the vertices  : \begin{align*} … Continue reading →

Let $M$ be a triangulated domain with the functionspace: \[W_{PL}:=\bigl\{f:M\rightarrow\mathbb{R}\,\bigl\vert\bigr.\,\,\,\left. f\right|_{T_{\sigma}} \mbox{ is affine for all } \sigma \in \Sigma_2 \bigr\}.\] On the interior of each triangle $T_{\sigma}$ in $M$ the gradient of a function $g \in W$ is well … Continue reading →

We want to derive a discrete version of the laplace operator defined on triangulated surfaces. At first we will define what a triangulated surface with and without  boundary is,  and consider triangulated domains of $\mathbb{R}^2$ as an important example. Let … Continue reading →

Let $M \subset \mathbb{R}^2$ be a domain with smooth boundary $\partial M$ and outpointing normal vector field $N$. For a smooth function $f \in C^{\infty}(M,\mathbb{R})$ the gradient vector field $\mbox{grad} \, f :M \rightarrow \mathbb{R}^2$ is defined as : \[ … Continue reading →

What is a typical function? The answer of this question certainly depends on the branch of mathematics you are into – while functions in differential geometry are usually smooth, the wiggling graphs appearing at the stock market are far from … Continue reading →

So far we have seen geometries in 2D and 3D, which are the dimensions we are familiar with.  But mathematician have found interesting geometries in higher dimensions and it would be great if we could visualize them.  In particular a huge … Continue reading →