## Archive for September, 2022

### Topological Calculus: away with those nasty epsilons and deltas

Continuous functions[figure from Olver (2022a).

A new approach to calculus has recently been developed by Peter Olver of the University of Minnesota. He calls it “Continuous Calculus” but indicates that the name “Topological Calculus” is also appropriate. He has provided an extensive set of notes, which are available online (Olver, 2022a)].

### The 3-sphere: Extrinsic and Intrinsic Forms

Figure 1. An extract from Einstein’s 1917 paper on cosmology.

The circle in two dimensions and the sphere in three are just two members of an infinite family of hyper-surfaces. By analogy with the circle ${\mathbb{S}^1}$ in the plane ${\mathbb{R}^2}$ and the sphere ${\mathbb{S}^2}$ in three-space ${\mathbb{R}^3}$, we can consider hyper-spheres in higher dimensional spaces. In particular, we will consider the 3-sphere which can be embedded in ${\mathbb{R}^4}$ but can also be envisaged as a non-Euclidean manifold in ${\mathbb{R}^3}$.

### Making Sound Pictures to Identify Bird Songs

Top: Audio signal with three chirps. Bottom: Time-Frequency spectrogram of signal.

A trained musician can look at a musical score and imagine the sound of an entire orchestra. The score is a visual representation of the sounds. In an analogous way, we can represent birdsong by an image, and analysis of the image can tell us the species of bird singing. This is what happens with Merlin Bird ID. In a recent episode of Mooney Goes Wild, Niall Hatch of Birdwatch Ireland interviewed Drew Weber of the Cornell Lab of Ornithology, a developer of Merlin Bird ID. This phone app enables a large number of birds to be identified [TM237 or search for “thatsmaths” at irishtimes.com].

### Dynamic Equations for Weather and Climate

“I could have done it in a much more complicated way”,
said the Red Queen, immensely proud. — Lewis Carroll.

Books on dynamic meteorology and oceanography usually have a full chapter devoted to the basic dynamical equations. Since the Earth’s fluid envelop is approximately a thin spherical shell, spherical coordinates ${(\lambda,\varphi, r)}$ are convenient. Here ${\lambda}$ is the longitude and ${\varphi}$ the latitude. In Figure 1 we show the momentum equations as presented in the monograph of Lorenz (1967):

Fig 1. The momentum equations, as in Lorenz (1967). The metric terms are boxed.

### Curl Curl Curl

Many of us have struggled with the vector differential operators, grad, div and curl. There are several ways to represent vectors and several expressions for these operators, not always easy to remember. We take another look at some of their properties here.