- #1
gikiian
- 98
- 0
Whyyyyyy??! Whhhhhy?!?
LCKurtz said:There is a real reason. In triple integration, if you use the standard volume element:
[tex]dV = \rho^2\sin(\phi)d\rho d\phi d\theta[/tex]
you want to let θ to from 0 to 2π and φ go from 0 to π, otherwise the sin(φ) factor can be negative. If you don't do that you need absolute values around the sine factor, generally causing twice the work, or worse, incorrect calculation by being unaware of that.
Phi is a mathematical constant equal to approximately 1.6180339. It is often referred to as the "golden ratio" and has been studied extensively in art, architecture, and nature. In spherical coordinate systems, the range of Phi is limited to 0-Pi (or 0-180 degrees) because it represents the angle between the positive z-axis and the projection of a point onto the x-y plane. This range ensures that Phi remains a positive value.
The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding numbers. This sequence is closely related to Phi, as the ratio of any two consecutive numbers in the Fibonacci sequence approaches Phi as the sequence continues. This connection has led to further exploration of Phi and its presence in nature.
Phi has been observed in various aspects of nature, including the growth patterns of plants, the proportions of animal bodies, and the arrangement of branches on trees. It has also been used in art and architecture to create visually appealing designs. In addition, Phi has been studied in economics and finance, as it has been found to have some predictive power in financial markets.
Yes, Phi is a transcendental number, which means it is a real number that is not the root of any non-zero polynomial equation with integer coefficients. This makes it an irrational number, meaning it cannot be expressed as a simple fraction. Phi is also a unique number in that its decimal expansion is non-repeating and non-terminating.
The study of Phi has led to further insights and discoveries in fields such as geometry, number theory, and physics. Its presence in nature and its relationship to other mathematical concepts make it a valuable tool for understanding and solving a wide range of problems. In addition, the exploration of Phi has sparked curiosity and creativity in many scientists and mathematicians, leading to new and innovative ideas and discoveries.