Darwin's Theory: Math Equations That Changed the World

Mathematics is a foundation of technology. Any new cellphone, gadget, or practically any scientific discovery was inspired by Mathematics. Yahoo published a list of the most important Math equations courtesy of Professor Stewart of the University of Warwick. Here they are:

According to Prof. Stewart, the following 17 equations have changed the world:
Pythagoras's Theorem, Logarithms, Calculus, Newton's Law of Gravity, The Square Root of Minus One, Euler's formula for Polyhedra, Normal Distribution, Wave Equation, Fourier Transform, Navier-Stokes Equation, Maxwell's Equations, Second Law of Thermodynamics, Relativity, Schrödinger equation, Information Theory, Chaos Theory and Black Scholes Equation.

Math Equations that changed the world:

1. Pythagoras's Theorem:

$a^2 + b^2 = c^2\!\,$

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

2. Logarithms

$\log_b(xy) = \log_b (x) + \log_b (y). \,$

The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number.

3. Calculus

$\frac{dy}{dx}=f'(x),$

4. Newton's Law of Gravity

Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:
$F = G \frac{m_1 m_2}{r^2}\$

where:

F is the force between the masses,
G is the gravitational constant,
m₁ is the first mass,
m₂ is the second mass, and
r is the distance between the centers of the masses.

5. The Square Root of Minus One

i 2 = -1

The imaginary unit is sometimes written

\sqrt -1

in advanced mathematics contexts (as well as in less advanced popular texts). However, great care needs to be taken when manipulating formulas involving radicals. The notation is reserved either for the principal square root function, which is only defined for real

x \geq 0

, or for the principal branch of the complex square root function

6. Euler's formula for Polyhedra

Realized that Euler has a brilliant, brilliant mind. So any of his formulas could end up on the list. So the list was not talking about this formula:
$e^{ix} = \cos x + i\sin x \$

but rather this formula below.This is also called Euler's Polyhedron formula. Where V refers to vertices, E to Edges and F for Faces. One of the more people-accessible formulas from Euler.

$\chi=V-E+F \,\!$

7. Normal Distribution

$f(x;\mu,\sigma^2) = \frac{1}{\sigma\sqrt{2\pi}} e^{ -\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2 }$

Looks very complex, but this is one of the more practical in the list. Mathematicians call this the Gaussian function or the probability density function. To everyone else,this is the formula for the Bell Curve.

8. Wave Equation

${ \partial^2 u \over \partial t^2 } = c^2 \nabla^2 u$

the wave equation concerns a time variable

t

, one or more spatial variables

x 1, x 2, \dots, x n

, and a scalar function

u = u (x 1, x 2, \dots, x n; t)

, whose values could model the height of a wave

9. Fourier Transform

$\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,dx$

Expresses a mathematical function of time as a function of frequency, known as its frequency spectrum. More common in Physics or Engineering but of course a mathematics construct just like most of them.

10. Navier-Stokes Equation

Navier–Stokes equations (general)

$\rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} \right) = -\nabla p + \nabla \cdot\boldsymbol{\mathsf{T}} + \mathbf{f},$

where v is the flow velocity, ρ is the fluid density, p is the pressure, $\boldsymbol{\mathsf{T}}$ is the (deviatoric) stress tensor, and f represents body forces (per unit volume) acting on the fluid and ∇ is the del operator. This is a statement of the conservation of momentum in a fluid and it is an application of Newton's second law to a continuum; in fact this equation is applicable to any non-relativistic continuum and is known as the Cauchy momentum equation.

11. Maxwell's Equations

could refer to several ground-breaking equations, as Maxwell is considered to be the godfather of Electromagnetism. The more popular one would be a derivation of Maxwell's concepts, more popularly known as Gauss's Law:

$\oiint$ ${\scriptstyle\partial \Omega }$ $\mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{Q(V)}{\varepsilon_0}$

12. Second Law of Thermodynamics

A change in the entropy (

dS

) of a system is the infinitesimal transfer of heat (

δ Q

) to a closed system driving a reversible process, divided by the equilibrium temperature (

T

) of the system.^[1]

$dS = \frac{\delta Q}{T} \!$

The first law of thermodynamics provides the basic definition of thermodynamic energy, also called internal energy, associated with all thermodynamic systems, but unknown in mechanics, and states the rule of conservation of energy in nature.

However, the concept of energy in the first law does not account for the observation that natural processes have a preferred direction of progress

13. Relativity

E = mc²

The two postulates of special relativity predict the equivalence of mass andenergy, as expressed in the mass–energy equivalence formula E = mc², where c is the speed of light in vacuum

14. Schrödinger equation

Time-dependent Schrödinger equation (general)

$i \hbar \frac{\partial}{\partial t}\Psi = \hat H \Psi$

where Ψ is the wave function of the quantum system, i is the imaginary unit, ħ is the reduced Planck constant, and $\hat{H}$ is the Hamiltonian operator, which characterizes the total energy of any given wavefunction and takes different forms depending on the situation.

15. Information Theory

There are so many formulas from the Information Theory, but the most common would be this:

$W = K \log m$

16. Chaos Theory

$| \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |\$

where λ is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

17. Black Scholes Equation

$\frac{dS}{S} = \mu \,dt+\sigma \,dW\,$

where W is Brownian motion. Note that W, and consequently its infinitesimal increment dW, represents the only source of uncertainty in the price history of the stock

The full yahoo article is in the link below:
http://ph.news.yahoo.com/17-maths-equations-that-changed-the-world.html;_ylt=AjJmzT_E5Dr.ZPoASLokzN71V8d_;_ylu=X3oDMTFqaTNjbzlmBG1pdANBcnRpY2xlIEJvZHkEcG9zAzMEc2VjA01lZGlhQXJ0aWNsZUJvZHlBc3NlbWJseQ--;_ylg=X3oDMTJyazB2ZzUyBGludGwDcGgEbGFuZwNlbi1waARwc3RhaWQDMDA5OWIxMzItNGUyZi0zOGIxLWI2YTctZThmMDQwNmFkZjQ4BHBzdGNhdANidXNpbmVzcwRwdANzdG9yeXBhZ2U-;_ylv=3?page=all

Sunday, July 29, 2012

Math Equations That Changed the World

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