Friar Bellows

Not content to confine my "top three" threads to the Media and Popular Culture forum, here I am contaminating the Science forum with the following: list your three favourite equations from science and/or mathematics. To start the ball rolling:

1. G = T

WTF, has the Friar gone mad? Actually, this is the Einstein field equations, stripped of unimportant constants, and written in a coordinate-free manner. It says the geometry of spacetime as represented by the "Einstein tensor", G, is generated by a quantity called the "stress-energy tensor", T, which represents the energy density, momentum density and stress present at a particular point in spacetime. John Wheeler put it better, but in looser terminology: "Space acts on matter, telling it how to move. In turn, matter reacts back on space, telling it how to curve." That's gravitation for you in a nutshell.

2. E = h f

In 1900, Planck derived the law of "blackbody" (or thermal) radiation, which had been a tough problem to crack in the late 19th century. His simple but powerful model used a collection of linearly oscillating monochromatic resonators enclosed in a cavity with reflecting walls, and he assumed that the energy, E, of each resonator was directly proportional to its frequency, f, with the constant of proportionality, h, being a fundamental constant of nature, now known as Planck's constant. That was the start of one of the biggest ever revolutions in physics: quantum mechanics.

3. 1 + 1/4 + 1/9 + 1/16 + 1/25 + . . . = pi^2 / 6

where "pi" is the familiar ratio of a circle's circumference to its diameter. In words, the above equation can be read as: "The infinite sum of the reciprocals of the squares of the whole numbers is equal to pi squared divided by 6." The first person to get any sort of hold on what the sum of this infinite series might be was Jakob Bernoulli who was born in Switzerland, over 300 years ago. Jakob showed that the series did converge and that it was less than two. But the exact number failed him, as it did such leading mathematical lights as de Moivre, Leibniz and Stirling. The problem was finally solved by another Swiss, Leonhard Euler, who was also a student of Jakob's brother and bitter rival, Johann. So the next time a character from an Orson Welles film tries to tell you that the only contribution of the Swiss to civilisation was the cuckoo clock, you now know better.

[ September 25, 2002: Message edited by: Friar Bellows ]</p>

simian

1. F=ma
2. Conservation of mass/energy (yes, I am too lazy to type it out)
3. (-1)^0.5 = i

Simian

edited to change # 3 - I came up with a good one.

[ September 25, 2002: Message edited by: simian ]</p>

Abacus

I've always liked Euler's equation:

e^(j*theta)=cos(theta) + j*sin(theta)

Ba'alzamon

e^(i*pi)=-1

Intuitively, it just doesn't make any sense, but when you go through the calculus, it turns out to make a surprising amount of sense.

Starboy

1) Maxwell's equations
2) Schroedinger's equation
3) Schmidt Orthogonalization Procedure

Starboy

[ September 25, 2002: Message edited by: Starboy ]</p>

Shadowy Man

You beat me to it. I was going to suggest this one too.

Undercurrent

I'm personally quite fond of equation (2.16) in "An asymptotic expansion for the hypergeometric function _2F_1(a,b;c;x)" Thorsley & Chidichimo, J. Math. Phys. vol 42, no. 4, pp1921-30. But maybe I'm just biased.

As for cool equations, I'd go for:

1) The Cauchy-Riemann equations.

2) Stokes' Theorem.

3) The Euler-Lagrage equations in the calculus of variations.

It's not really an equation, but I'm also absurdly fond of Banach's fixed point theorem. Banach rules!

m.

elwoodblues

Godel's Incompleteness Theorem (not really suited for a T-shirt equation, though )

Mandelbrot's Set:
Z(sub 0) = C, Z(sub n + 1) = Z^2(sub n) + C
(forgive me, I don't know how to code subscripts)

Gravitic equation:
F = Gm1m2/d^2

[ September 25, 2002: Message edited by: elwoodblues ]</p>

wade-w

Its too bad that there still isn't a good way to encode superscripts/subscripts and various calculus symbols, so many of the above equations don't look as elegant as they do in math or physics texts. I suppose I could dig up that old standby, latex2html, and then post the image, but I'm too lazy for that right now.

And while Godel's Incompleteness Theorem is beautiful, it doesn't really qualify as an equation. Same for the Axiom of Choice, etc.

1. d(e^x)/dx = e^x
2. e = mc^2
3. The Lorentz Transformations

[ September 25, 2002: Message edited by: wade-w ]

[ September 25, 2002: Message edited by: wade-w ]</p>

sir drinks-a-lot

I'd have to go with:

1+1=2 The equation that started it all!!

1=0.9999... Where "0.9999..." is a repeating decimal.