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.

1. An introduction to these articles. Introduction

2. Triangles and regular polygons are considered as an introduction to the PT. Modern Pythagorean Theorem, Part 1

3. 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). Modern Pythagorean Theorem, Part 2

4. If the summation of a series grows faster than some rate then it diverges and grows without bound. Up until the 17th century, no one knew whether the Harmonic series diverged, or converged, as it grows very slowly but faster than the convergent Geometric Series. Series and their Limits

5. Roots are exponents with a fraction as power. Here, rules are developed for the evaluation of any positive Real radicand. Zero Exponent Rule

6. Trigonometry is the study of the sine and cosine cofunctions, their domains and ranges, along with the rational compositions of the two—all belonging to the geometry of vectors on the Unit Circle. Cofunctions and the Unit Circle

7. The logarithm is presented as an 18th century revelation. The Natural Base, e, is compared to other exponential bases. The Logarithm and Exponential

8. Decomposing functions into a spectrum of sine and cosine waves. Fourier Analysis

9. Tangents in general are considered, particularly those of curves which come to the minds of early modern natural philosophers. Derivatives and Tangents

10. The inverse operation of calculating the tangent of a curve, or derivative, is the area under the curve, or integral. Integrals 1: The Fundamental Theorem of Calculus

11. The Geometric Series is approximated using calculus. Integrals 2: Continuous Geometric Series

12. Integration requires tools for our simple function as integrand, namely integration by parts and the point transformation. Integrals 3: Our Simple Function

13. The representation of a curve with a differential series describes its local approximation. The Taylor Series

14. The Taylor Series of the cofunctions are derived and analyzed for error. Cofunction Series

15. Here, the Trigonometric Taylor Series' are juxtaposed with the Exponential Series of an imaginary argument to derive, Euler's Formula. Euler's Formula

16. A theorem of Mertzbacher is presented, using Fourier analysis to outline a direct route from wave-particle duality to their complex existence. Mertzbacher's Theorem

17. Bibliographical references. References