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Posts Tagged ‘Science’

Fourier Transforms

Tuesday, October 6th, 2009

Fourier Transforms Explained

Description

A detailed tutorial on Fourier transforms. Step by step tutorial including several examples of Fourier transforms for reference.

Overview

A Fourier transform is an operation that transforms one complex-valued function of a real variable into another. The domain of the original function is typically referred to as the time domain, because it is a representation of time. The domain of the new function represetns frequency. The Fourier transform itself is often called the frequency domain representation of the original function because of this.

Tags: complex, differential equations, domain, Fourier, frequency, function, Math, Physics, real, Science, time, tranform, value, variable
Posted in Differential Equations | No Comments »

Superposition Principle

Tuesday, September 22nd, 2009

Superposition Principle Explained

Description

A detailed tutorial on the solving of the Superposition Principle. Step by step tutorial including several examples of how to solve the Superposition Principle for reference.

Overview

The Superposition Principle, or the Superposition Property (also know mathematically as additivity) states that the net response at a given place and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually. Mathematically, this is saying that for all linear systems F(x) = y, where x is some sort of stimulus or input and y is some sort of response or output, the superposition or sum of the stimuli yields a superposition of the respective reponses:

$F(x_1+x_2+\cdots)=F(x_1)+F(x_2)+\cdots$

Tags: additivity, input, linear systems, net response, output, Physics, responses, Science, stimuli, stimulus, sum, superposition, superposition principle, superposition property
Posted in Differential Equations | No Comments »

Maxwell’s Equations

Tuesday, September 22nd, 2009

How to Solve Maxwell’s Equations

Description

A detailed tutorial on the solving of Maxwell’s Equations. Step by step tutorial including several examples of how to solve Maxwell’s Equations for reference.

Overview

Maxwell’s equations are a set of four partial differential equations that describe the properties of electric and magnetic fields and relate them to their sources, charge density and current density. The result of these equations is that they show light is an electromagnetic wave. The four different equations and the way to express them is as follows:

Gauss’s Law: relates electric charge contained within a closed surface to the surrounding electrical field.

Differentiation: $\nabla \cdot \mathbf{D} = \rho_f$

Integration: $\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf D\;\cdot\mathrm{d}\mathbf A = Q_{f}(V)$

Gauss’s Law for Magnetism: states that the total magnetic flux through a closed surface is zero.

Differentiation: $\nabla \cdot \mathbf{B} = 0$

Integration: $\iint_{\partial V}\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\;\;\;\subset\!\supset \mathbf B\;\cdot\mathrm{d}\mathbf A = 0$

Maxwell-Faraday Equation: describes how a changing magnetic field can create an electric field.

Differentiation: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$

Integration: $\oint_{\partial S} \mathbf{E} \cdot \mathrm{d}\mathbf{l}� = - \frac {\partial \Phi_{B,S}}{\partial t}$

Ampere’s Circuital Law: states that magnetic fields can be generated by electrical current and changing electric fields.

Differentiation: $\nabla \times \mathbf{H} = \mathbf{J}_f + \frac{\partial \mathbf{D}} {\partial t}$

Integration: $\oint_{\partial S} \mathbf{H} \cdot \mathrm{d}\mathbf{l} = I_{f,S} + \frac {\partial \Phi_{D,S}}{\partial t}$

Tags: Ampere's Circuital Law, change density, closed surface, current density, density, electric charge, electric fields, electrical current, electromagnetic wave, Gauss's Law, Gauss's Law for Magnetism, Guassian surface, light, magnetic field, magnetic flux, Maxwell's equations, Maxwell-Faraday Equation, Physics, Science, zero
Posted in Differential Equations | No Comments »

The Ritz Method

Tuesday, September 22nd, 2009

How to Use the Ritz Method

Description

A detailed tutorial on the solving of problems using the Ritz Method. Step by step tutorial including several examples of how to solve problems with the Ritz Method for reference.

Overview

The Ritz Method, often called the Rayleigh-Ritz Method, is the finite element method used to find the eigenvalues of a Hamiltonian system. The Ritz Method starts out with a trial wave function, that is expressed as $E_0 \le \int \Psi^* \hat{H} \Psi \, d\tau$

From there, we can substitute a value for the trial wave function, which is $\Psi = \sum_{i=1}^N c_i \Psi_i.$

The final expected value can be written as:

$\varepsilon = \frac{\left\langle \sum_{i=1}^N c_i\Psi_i \right| \hat{H} \left| \sum_{i=1}^Nc_i\Psi_i \right\rangle}{\left\langle \sum_{i=1}^N c_i\Psi_i \Big| \sum_{i=1}^Nc_i\Psi_i \right\rangle} = \frac{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jH_{ij}}{\sum_{i=1}^N\sum_{j=1}^Nc_i^*c_jS_{ij}} \equiv \frac{A}{B}$

Tags: boundary conditions, differential equations, energy, Hamiltonian system, orthogonal, Physics, rayleigh-ritz method, ritz method, Science, trial wave function, Walter Ritz, wave
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Helmholtz Equation

Tuesday, September 22nd, 2009

How to Solve the Helmholtz Equation

Description

A detailed tutorial on the visual representation of the Helmholtz Equation. Step by step tutorial including several examples of the visual representation of the Helmholtz Equation for reference.

Overview

The Helmholtz Equation is an elliptic partial differential equation, which can be used to calculate waves. It is similar to the wave and heat equations in that manner, but the formula is very different. The formula can be expressed as $\nabla^2 A + k^2 A = 0$ where k is the wavenumber and A is the amplitude. The Helmholtz Equation is often used for problems involving partial differential equations in both space and time.

Tags: amplitude, elliptic, helmholtz, helmholtz equation, partial differential equations, Physics, Science, space, time, wavenumber, waves
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Astronomy: Earth’s Moon

Tuesday, September 22nd, 2009

What Are The Phases Of The Moon

Description

A detailed tutorial explaining the four main phases of the moon.

Overview

The phases of the moon are caused by the relative positions of the earth, sun, and moon. The moon goes around the earth in 27.3 days, or 27 days 7 hours 43 minutes, on average. This measurement is relative to the stars and is called the sidereal period or orbital period. However, because of the earth’s motion around the sun, a complete moon cycle (New Moon to New Moon) appears to earthbound observers to take a couple of days longer: 29.5305882 days to be exact. This number is called the synodic period or “lunation”, and is relative to the sun.

The sun always illuminates the half of the moon facing the sun (except during lunar eclipses, when the moon passes through the earth’s shadow). When the sun and moon are on opposite sides of the earth, the moon appears “full” to us, a bright, round disk. When the moon is between the earth and the sun, it appears dark, a “new” moon. In between, the moon’s illuminated surface appears to grow (wax) to full, then decreases (wanes) to the next new moon. The edge of the shadow (the terminator) is always curved, being an oblique view of a circle, giving the moon its familiar crescent shape.

New Moon

Waxing Crescent

First Quarter

Waxing Gibbous

Full Moon

Waning Gibbous

Last Quarter

Waning Crescent

Tags: Astronomy, earth's moon, moon, moon phases, Science
Posted in Astronomy | No Comments »

Physics: Acceleration

Monday, September 21st, 2009

How To Calculate Acceleration

Description

A detailed tutorial on how to calculate an object’s acceleration. A step by step guide on solving for an object’s acceleration.

Overview

Acceleration is defined as the change in velocity over the change in time.

$a = \frac{\Delta v}{\Delta t}$

Velocity is measured in meters per second while acceleration is measured by meters per second per second.

$\frac{m}{s * s} = \frac{m}{s^2}$

Acceleration is also expressed as the derivative of velocity.

Tags: acceleration, motion, Physics, rectilinear motion, Science
Posted in Calculus, Physics | No Comments »

Telegrapher’s Equations

Friday, September 18th, 2009

How to Solve Telegrapher’s Equations

Description

A detailed tutorial on the solving of Telegrapher’s Equations. Step by step tutorial including several examples of how to solve Telegrapher’s Equations for reference.

Overview

Telegrapher’s Equations, sometimes referred to simply as telegraph equations, are a pair of differential equations which meausre the voltage and current on a transmission line with regard to distance and time. An example would be a telegraph, hence the name. Instead of having an actual set of equations, Telegrapher’s Equations tend to more oftenbe expressed as a schematic, with the equations only being used for things such as loops and transmissions.

Tags: attenuation constant, differential equations, distance, loops, magnetic field, pair, phase constant, Physics, primary line constants, propagation constant, Science, telegraph, telegraph equations, telegrapher's equations, time, transmission
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The Heat Equation

Friday, September 18th, 2009

How to Solve the Heat Equation

Description

A detailed tutorial on the visual representation of the heat formula. Step by step tutorial including a 3D representation of the heat equation for reference.

Overview

The heat equation is a partial differential equation that describes the distribution of heat or the variation in temperature. The heat equation can be expressed as:

$\frac{\partial u}{\partial t} -k\left(\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}\right)=0$

It can also occasionally be expressed as $\frac{\partial u}{\partial t} = k \nabla^2 u$

Tags: distribution, function, heat, heat equation, partial differential equations, Physics, Science, temperature
Posted in Differential Equations | No Comments »

The Wave Equation

Friday, September 18th, 2009

How to Solve the Wave Equation

Description

A detailed tutorial on the solving of the wave equation. Step by step tutorial including several examples of how to solve the wave equation for reference.

Overview

The wave equation is an important equation in partial differential equations. In its basic form, it is expressed as:

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

Every so often, for questions involving dispersion, the variable c must be changed to $v_\mathrm{p} = \frac{\omega}{k}.$

Tags: constant, dispersion, electromagnetics, hyperbolic, light wave, partial differential equations, Physics, Science, sound wave, velocity, water wave, wave equation
Posted in Differential Equations | No Comments »