The Fibonacci Numbers have been of great interest since they were first seen as a solution to an algebra word problem published in 1202. The Fibonacci Numbers have many interesting properties which have many applications. For this the focus will be on finding a formula to find the nth Fibonacci number in the sequence.
In Physics of Waves by William C. Elmore and Mark A. Heald the book begins by looking at waves with the derivation of the formula for motion of a Transverse Wave on a string. In exploring wave phenomena, this material is a good starting point for comprehending both the physics and the mathematics of waves.
Have you ever watched an athlete leap into the air and they appear to float for a long time.? We can watch Ballet Dancers and Basketball Players do this; but what is going on?
This is a commented on phenomena that shows the subtlety in analyzing physics equations. It has been said that a good athlete appears to float in mid-air. The equations of motion for free fall hint at this but let us play with the equations to see this more clearly.
James Nearing in the book Mathematical Tools for Physics (Dover Books on Physics) shows a technique to avoid integration by parts. This is done by using Differentiation and Parameters for Integrals. I found this technique to be an interesting method for obtaining results to definite integrals. So here I play with the technique.