Normal distribution: The normal probability distribution, which has the familiar bell curve graph to the left, is ubiquitous in statistics. The normal curve is used in physics, biology, and the social sciences to model various properties. One of the reasons the normal curve shows up so often is that it describes the behavior of large groups of independent processes. 8) Wave Equation: This is a differential equation, or an equation that describes how a property is changing through time in terms of that propertys derivative, as above. The wave equation describes the behavior of waves - a vibrating guitar string, ripples in a pond after a stone is thrown, or light coming out of an incandescent bulb. The wave equation was an early differential equation, and the techniques developed to solve the equation opened the door to understanding other differential equations as well. 9) Fourier Transform: The Fourier transform is essential to understanding more complex wave structures, like human speech. Given a complicated, messy wave function like a recording of a person talking, the Fourier transform allows us to break the messy function into a combination of a number of simple waves, greatly simplifying analysis. The Fourier transform is at the heart of modern signal processing and analysis, and data compression. 10) Navier-Stokes Equations: Like the wave equation, this is a differential equation. The Navier-Stokes equations describes the behavior of flowing fluids - water moving through a pipe, air flow over an airplane wing, or smoke rising from a cigarette. While we have approximate solutions of the Navier-Stokes equations that allow computers to simulate fluid motion fairly well, it is still an open question (with a million dollar prize) whether it is possible to construct mathematically exact solutions to the equations. 11) Maxwells Equations: This set of four differential equations describes the behavior of and relationship between electricity (E) and magnetism (H). Maxwells equations are to classical electromagnetism as Newtons laws of motion and law of universal gravitation are to classical mechanics - they are the foundation of our explanation of how electromagnetism works on a day to day scale. As we will see, however, modern physics relies on a quantum mechanical explanation of electromagnetism, and it is now clear that these elegant equations are just an approximation that works well on human scales. 12) Second Law of Thermodynamics: This states that, in a closed system, entropy (S) is always steady or increasing. Thermodynamic entropy is, roughly speaking, a measure of how disordered a system is. A system that starts out in an ordered, uneven state - say, a hot region next to a cold region - will always tend to even out, with heat flowing from the hot area to the cold area until evenly distributed. The second law of thermodynamics is one of the few cases in physics where time matters in this way. Most physical processes are reversible - we can run the equations backwards without messing things up. The second law, however, only runs in this direction. If we put an ice cube in a cup of hot coffee, we always see the ice cube melt, and never see the coffee freeze.
Posted on: Sat, 15 Mar 2014 10:59:53 +0000
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