Mathematical Physics and Mathematics

This area is divided into Analysis, Maps and Programming

Analysis

These articles began to discuss a few interesting problems, but now that they have multiplied considerably, I notice that they pretty well cover mathematics from arithmetic through differential equations, and beyond, at a university level. However, in the present form they are not a proper textbook, since exercises are lacking, and they are not logically arranged, but just in a heap. Exercises can be found in the References, however, so these pages can make a valuable supplement. Many subjects are included that are not normally found in the usual course. Cultural and historical notes are also frequent.

• The Beautiful Theory: Lagrange's Analytical Mechanics
• Groups for Dummies: another beautiful theory, properties of the irreducible representations of finite groups.
• Euclidean Tensors: more powerful than vector analysis.
• The Cardioid The interesting heart-shaped plane curve.
• Logistic: How to do calculations in all sorts of ways.
• Diangles: Find out what the angle of the diangle is.
• Bessel's Functions: the most useful special functions of all.
• The Zeta Function and Euler's Product.
• The Circumference of the Ellipse How elliptic integrals arise.
• Jacobian Elliptic Functions: What is the mysterious function sn(u)?.
• Legendre Polynomials and Jacobi polynomials, as used in spherical harmonics.
• Hermite Polynomials, and their application to quantum mechanics.
• Laguerre Polynomials: clarifying the confusion due to different definitions.
• Spherical Harmonics: solutions of Laplace's Equation in spherical coordinates.
• Quaternions: Hamilton's hypercomplex numbers, rotations and relativity.
• The Catenary: statics of a hanging chain.
• The Parabola: the world of the second most useful curve.
• The Ellipse: how ellipses are drawn and used.
• The Hyperbola: the neglected conic section has some beauties not apparent to the eye.
• The Hanging Chain, with Rayleigh's Principle, too.
• Problems With Axial Symmetry: using cylindrical coordinates.
• Algebra: a review of the toolbox of mathematics, with how to do many things.
• Trigonometry: a review of a very practical branch of elementary mathematics.
• Calculus: its fundamentals, and the role of the calculus course in scientific education.
• Complex Variables: a synopsis of the theory of analytic functions.
• Heaviside, Laplace and the Inversion Integral: an explanation of the complex inversion integral.
• Differential Equations: a survey of methods of integration.
• Solving The Harmonic Oscillator Equation, using the D operator.
• Nöther's Theorem: conservation laws from symmetry.
• Curves, plane curves, theory and examples.
• Curves and Surfaces in Space, differential geometry of curves and surfaces.
• Partial Derivatives, partial derivatives and Jacobians.
• Logarithms, defining the logarithm as an integral.
• The Heine-Borel Theorem, understanding this important theorem.
• Infinity, the lore of infinite series.
• Travel by Brachistochrone, a modest proposal.
• Square Roots can be found by hand!
• Chinese Numbers: come from calculating rods.
• Math Battles: more bad news from the American school classroom.

Maps

• Mercator's Map and the properties of conformal maps.
• Lambert's Map and applications to analytic functions.

Programming

Recently, I realized I had not used some old programming tools for a long while, and became interested in reviewing these packages and seeing if I could get them to work on Windows 98. All, in fact, did work, and very well indeed. In each case, the preparation of a program to display "Hello, world!" on the screen is included. The difficulty of doing this ranges from FORTH, in which it can be done in seconds, to the preparation of a Windows program with C++, requiring an absurd amount of overhead and complication (it is simpler in C-Windows, where only two routines are necessary). This tedious exercise is not treated here. In Pascal and FORTRAN, as well as in C-DOS, it is on the simple end of the scale. I would like to include ALGOL-60 and COBOL for added variety, but I do not have platforms available. It is a shame that Windows has rendered PC programming inaccessible to so many; DOS is a much better platform for small programs, especially in science and engineering, and for laboratory use. Although the modern practical applications of computer programming are boring in the extreme, and repellent to the intelligent, it is still a fascinating game for us!

• The HP-48G An introduction to programming; programmng the HP-48G.
• Turbo Pascal Using an integrated development environment.
• FORTRAN 77 Command-line compiling and linking for DOS.
• PC/FORTH A unique interactive user interface.
• BASIC IBM QBASIC and Applesoft.
• Apple II Fire up your Apple II and learn machine-language programming.
• The ACIA Asynchronous serial communication, ACIA's, plug-in boards and the Apple.
• Hello, world! How to make a Hello, world! program in machine and in assembly language under Windows.
• 8086 Instruction Set Machine language programming for the 8086 family.
• The Romulator Microcontroller experiments with a ROM emulator and an 8051

The picture shows interference colors in a thin film of oxidized oil on water