# Monthly Archives: June 2018

## Tutorial 6 – Biot-Savart field of polygons

The velocity field of a fluid in \(\mathbb R^3\) (at rest at infinity) can be reconstructed from the vorticity by the Biot-Savart formula. Let us assume a fluid with vorticity concentrated along a filaments: Given a curve \(\gamma\colon \mathbb S^1 … Continue reading

## Tutorial 5 – Solid angle of space curves

Let \(\gamma\colon \mathbb S^1 \to \mathbb R^3\) be a closed space curve and \(m\in\mathbb R^3\setminus \gamma(\mathbb S^1)\). The solid angle \(\Omega_{\gamma}(m)\) of \(\gamma\) subtended to the point \(m\) is defined as the (signed) spherical area enclosed by the projection \(\hat\gamma\) … Continue reading

## Smoothing

Consider the space $$C^0(\mathbb{Z}^n)= \{f:\mathbb{Z}^n \to \mathbb{R} \,\,|\,\, f\text{ is bounded} \}.$$ Suppose we have a probability distribution $p$ on $\mathbb{Z}^n$ with mean zero: \begin{align*}p:\mathbb{Z}^n &\to \mathbb{R} \\\\ p(\mathbf{n})&\geq 0 \quad \text{for all}\quad \mathbf{n}\in \mathbb{Z}^n\\\\ \sum_{\mathbf{n}\in \mathbb{Z}^n} p(\mathbf{n})&=1 \\\\\sum_{\mathbf{n}\in \mathbb{Z}^n} … Continue reading

## Tutorial 4 – Darboux transformations

In class you defined the Darboux transformation \(\eta\colon \mathbb R\to \mathbb R^3\) of a smooth space curve \(\gamma\colon \mathbb R\to \mathbb R^3\). By construction, the distance between corresponding points is constant. Or phrased differently, there is \(S\colon \mathbb R \to … Continue reading