Calculating Square Roots By Hand

Before the days when students carried personal calculators that gave instantaneous square root calculations, that is, pre-1970's, most were taught a mysterious method that was accepted without understanding the logic behind it and therefore often forgotten. In the paper Algorithm to Compute the Square Root the technique is re-visited and examples and a justification are provided.

Phase Space Trajectories and Conservation of Energy

The motivation for the derivation of the Schroedinger equation arising from the Law of Conservation of Energy as seen in the blog entry of December 13, 2022, is made clearer when one looks at the phase space trajectories in familiar classical settings. In the paper, Phase Space Curves and Conservation of Energy, we go through a few examples of how Newton's 2nd Law of motion and the conservation of energy are closely related and how the phase space trajectory gives insight into the equation that defines motion in quantum settings.

Geometry Studies

Like all realms of Mathematics, Geometry is an endless landscape where new roads and new interconnections between those roads and older roads are perpetually being introduced by some very clever people who seem to relish such activities. Their work, to most, is beyond the scope of interest. But to those who are interested, it can be absorbing. It is easy to understand why many do not give Geometry any attention. It is definitely not something that will pay the bills unless you happen to be a bit exceptional in the mathematical realm. As deep and extensive as the realm is, it is not widely known and it is not easily accessible for the less than determined. The originalists in the field are often remarkably clever and intellectually determined. For those who enjoy bending their brains a bit in the pursuit of understanding the properties and interrelationships of abstract objects (abstracted from real world objects to which the properties then apply), Geometry in rich in possibility. Almost everyone is introduced to the fundamentals of Geometry in school. Even those who are not particularly drawn to mathematics often find Geometry interesting because it does relate to familiar objects. Unfortuantely, for most, that taste of geometry they get in school is all they get. Even for those who go on to study more math after high school for engineering or science or computer science, etc, further studies of Geometry is not always part of their curriculum. So it is only a subset of even the small set of people who do study mathematics that get to experience the joys of this realm. The paper Geometry Studies takes the reader on a short trip into this realm. It is meant as another taste that many never got. It follows the flow of Chapter 2 in Howard Eves' "A Survey of Geometry, Volume 1". By working the exercises of that chapter in detail, the paper offers some insights for those who might get get their foot caught in some brambles on the side of the road.

The Schrödinger Equation Solution in 1-Dimension

In the paper Schrödinger Equation we describe where the equation comes from in the realm of Quantum Mechanics and how it is solved in the one dimensional case. The solution serves as a good example of solving a separable partial differential equation that satisfies the Sturm-Liouville conditions (reviewed in the Appendix). It also provides a real-world application of Fourier analysis. The paper is meant as an introductory study guide for someone who has heard about Quantum Mechanics and the importance of the Schrödinger equation but never had the chance to study it formally. The author always appreciates hearing about any errors or typos in the paper.

Center of Mass

The center of mass (centroid) has many interesting physical and mathematical applications. For any physical or mathematical entity the center of mass is a point which defines the weighted average position of the total mass. This paper, Center of Mass, provides a rigorous definition, establishes some basic properties, and works many examples of calculating the center of mass in various dimensions and applying it in the proofs of more general results. One such result is the Pappus Centroid Theorem. We show an example where this theorem significantly reduces the work in determining the centroid under investigation. We also show the usefulness of the center of mass in gravitational calculations by providing a detailed proof that under constant gravitational acceleration a body acts on an outside body as if its total mass were concentrated at the center of mass. The Table of Contents lists the worked examples and applications included in the paper. Comments and reporting of errors or typos are always appreciated.

Mathematics of Waves, The Wave Equation, Electromagnetic Waves

The goal of the paper Waves is to show how electromagnetic waves (EM) follow as a consequence of the experimentally established Maxwell equations which were discussed in the previous blog entry. We first review the physical characteristics and properties of waves including the mathematical description given in the linear wave equation in one (spatial) dimension. We discuss how this naturally extends to higher dimensions. Using Maxwell's equations, we show how the properties of EM waves follow as a consequence of those equations where the electric and magnetic fields associated with the EM waves are solutions of the three (spatial) dimensional wave equation. Included in the paper are many illustrative worked examples and a solution of the one dimensional linear wave equation is included as an appendix. In particular, we show that an EM wave in a vacuum propagates at the speed of light and carries a magnetic field and an electric field where the two field vectors are always (that is, at all points in space and time) orthogonal to each other and the (propagation) direction of the wave.

Vector Calculus and Maxwell's Equations

In the paper Vector Calculus and Maxwell's Equations we review in detail the mathematical tools used to express Maxwell's Equations which govern the classical laws of electric and magnetic fields and thereby all EM phenomena. The paper could serve as supplemental reading to a BC level AP calculus class where the student gets an appreciation for real world usefulness of the tools that no doubt arose from such studies.