Author Archives: pinkall

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 →

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

Let $M=(V,E,F)$ be an oriented triangulated surface without boundary and $p:V\to {\mathbb{R}}^3$ 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 $M$ be a triangulated domain with the functionspace: $$W_{PL}:=\{f:M\to \mathbb{R}|{f|}_{T_{\sigma }}\text{ is affine for all }\sigma \in {\mathrm{\Sigma }}_2\}.$$ On the interior of each triangle $T_{\sigma }$ in $M$ the gradient of a function $g\in W$ is well … 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 $f\in C^{\mathrm{\infty }}(M,\mathbb{R})$ the gradient vector field $\text{grad}f:M\to {\mathbb{R}}^2$ is defined as : \[ … Continue reading →

Let $M\subset {\mathbb{R}}^n$ be an open set and $f:M\to {\mathbb{R}}^k$ a smooth map. Then for each $p\in M$ the Jacobian $f^{\prime }(p)$ is a $k\times n$ matrix. The derivative $d_{p}f$ of $f$ at $p$ then is the linear map given by … Continue reading →