How a Boat Rotates in a River: Visualizing Rotation in Flow Fields
Wigand Rathmann
Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Department Mathematik, Erlangen, Germany, and
Center of Mobile Learning with Digital Technology, University of Bayreuth, Bayreuth, Germany
Abstract
We all know rivers and can imagine, to sit a boat enjoying sun and clouds at the sky and we are drifting along a river not using any paddel.
Generated using Stable Diffusion
Sounds like vacation and the rotation of the boat brings as the \(\operatorname{curl}\)-Operator. Thus we wake up back having vector fields in mind thinking about a boat became a single point and moving as a particle through the vector field along a curve \(k\) with an angle velocity induced by the fluid.
In engineering maths vector fields and vector analysis are demanding concepts. Vector fields are considered in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) and are (easy) to visualize using JSXGraph. To find a way to visualize \(\operatorname{curl}V(\mathbf{x})=\nabla\times V(\mathbf{x})\) of a vector field \(V:\mathbb{R}^3\to\mathbb{R}^3\) is very demanding.
In this talk I consider 2D vector fields \(V:\mathbb{R}^2\to\mathbb{R}^2\) and will demonstrate a visualization of the rotating angle of particle/point along a curve \(k:\mathbb{R}\to\mathbb{R}^2.\)