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Linear transformations on the two-dimensional plane

A linear transformation in two dimensions can be visualized through its effect on the two orthonormal basis vectors $\hat{\imath}$ and $\hat{\jmath}$. In general, it can be represented by a $2 \times 2$ matrix, $\boldsymbol{T}$, which acts on a vector $v$ to map it from a vector space spanned by one basis onto a different vector space spanned by another basis: $\boldsymbol{v'} = \boldsymbol{T}\boldsymbol{v}$. This change of basis can be visualized by drawing the basis vectors in the two-dimensional plane, along with equally-spaced "grid lines" parallel to each of them. A linear transformation keeps the grid lines evenly spaced, and the origin fixed.

What N Things?

The mildly controversial geocoding system What3words encodes the geographic coordinates of a location (to a resolution of 3 m) on the Earth's surface into three dictionary words, through some proprietary algorithm. The idea is that human's find it easier to remember and communicate these words than the sequence of digits that makes up the corresponding latitude and longitude. For example, the Victoria Memorial in front of Buckingham Palace in London is located at (51.50187, -0.14063) in decimal latitude, longitude coordinates, but simply using.woods.laws in the language of What3words.

Visualizing the Earth's dipolar magnetic field

The magnetic field due to a magnetic dipole moment, $\boldsymbol{m}$ at a point $\boldsymbol{r}$ relative to it may be written $$ \boldsymbol{B}(\boldsymbol{r}) = \frac{\mu_0}{4\pi r^3}[3\boldsymbol{\hat{r}(\boldsymbol{\hat{r}} \cdot \boldsymbol{m}) - \boldsymbol{m}}], $$ where $\mu_0$ is the vacuum permeability. In geomagnetism, it is usual to write the radial and angular components of $\boldsymbol{B}$ as: $$ \begin{align*} B_r & = -2B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\cos\theta, \\ B_\theta & = -B_0\left(\frac{R_\mathrm{E}}{r}\right)^3\sin\theta, \\ B_\phi &= 0, \end{align*} $$ where $\theta$ is polar (colatitude) angle (relative to the magnetic North pole), $\phi$ is the azimuthal angle (longitude), and $R_\mathrm{E}$ is the Earth's radius, about 6370 km. See below for a derivation of these formulae.

Some analysis of the demographics of India

This Jupyter Notebook can be downloaded and viewed at my GitHub repository.

Impact craters on Earth

The Earth Impact Database is a collection of images, publications and abstracts that provides information about confirmed impact structures for the scientific community. It is hosted at the Planetary and Space Science Centre (PASSC) of the University of New Brunswick.