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.

Analysis of the Curl for Rotations about General Axis of Rotation

In the paper Curl for Rotation about General Axis we present an analysis of the curl of a vector field rotating about an arbitrary axis in space. The analysis includes the definition of the vector field, the computation of the curl, and the breakdown of the curl into components relative to the coordinate planes and arbitrary planes. The paper serves as a practical introduction to how the curl is related to the angular velocity of rotation and how rotations in space can be broken down into two-dimensional components.

Exercises on Surface Curvature from "A Survey of Geometry, Vol.2, 14.6" by Howard Eves

In the paper Exercises on Surface Curvature Section 14.6, we present the solutions to exercises 1-15 from section 14.6 of Howard Eves' "A Survey of Geometry, Vol.2". Exercises 16-20 are addressed in a reference. Eves' approach follows the classic approach of C.E. Weatherburn whose work is also referenced in the paper including a link to an on-line version of his textbook. The problems are stated in the Contents section of the paper and an Appendix is included to cover some background material for some problems that may be useful to the reader. The solutions offer some examples of basic arguments in introductory differential geometry, including Euler's Theorem and Dupin's Indicatrix.