A companion matrix is a matrix with a prescribed characteristic polynomial. I would like to show them from a broader perspective: companion matrices are the matrix version of a shift operator.

My goal is to derive Maxwell’s equations of electromagnetism with almost no effort at all. As often in mathematics, things look simpler when there is less structure. Here, as in mechanics, we do not assume any prior metric, so the geometry of the space at hand is very simple.

The variational equation is a fundamental tool in engineering. It is the description of the sensitivity with respect to the initial conditions. A remarkable fact, often presented only in \(\mathbf{R}^n\), is that this is in fact the solution of another differential equation, sometimes called the variational equation. I would like to show that this is valid in general manifolds.

A little JavaScript application to demonstrate two ideas:

• Equivariance

• Modified vector field