Articles

Physics is motivated by observed phenomena and seeks to explain the empirical data using fundamental laws as first principles. These articles seek to lay out some of the basic definitions and applications we use, focusing on the math to support the laws, in order to get started with this vibrant field.

An introduction to these articles, with some set notation and the field of real numbers.
An end to the mathematical frustration of man comes with analysis of the binomial (1620), and the generalization of the formula to a series solution of the binomials with non-natural number exponent (1665).
If the summation of a series grows faster than some rate, then it diverges. Up until the 17th century, no one knew whether the Harmonic series diverged, or converged, as it grows very slowly, but faster than the Geometric series.
Exponents are a simple algebraic construction and notation; they are used in scientific notation and are implicit in the decimal notation. Roots are an intrinsic part of exponents (the fractional exponents). So, in this section the original exponent value (zero) is analyzed.
The logarithm is presented as an 18th century revelation. The natural exponential is compared to other exponential bases, graphically.
Tangents in general are considered, particularly those of curves which come to the minds of early modern natural philosophers.
The inverse operation of the tangent of a curve, is the summed area under such function.
In the modern frameworks, a ubiquitously appearing tool, is the representation of a any curve in terms of a polynomial with differential coefficients.
Trigonometry is rounded out with the Sine and Cosine functions, and the Taylor Series of the cofunctions are considered.
Here, the Trigonometric Taylor Series' are juxtaposed with the Exponential Series of an imaginary argument to derive, Euler's Formula.
A theorem of Mertzbacher is presented, using Fourier analysis to outline a direct route from wave-particle duality to their complex existence.
Bibliographical references.