Posts Tagged ‘differential equations’
Friday, November 20th, 2009
Overview of Isoperimetric Inequalities
Description
A detailed tutorial on isoperimetric inequalities. Step by step tutorial including several examples of isoperimetric inequalities for reference.
Overview
An isoperimetric inequality is actually a geometric inquality. It deals with the square of a circumference of a closed curve in a plane and the area of the region it encloses. Isoperimetric means to have the same perimeter. The isoperimetric problem is used in conjunction the isoperimetric inequality to determine the measure of the plane figure.
Tags: area, circumeference, closed, curve, differential equations, figure, geometric, inequalities, inequality, isoperimetric, meausre, perimeter, plane, problem, region, square
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Tuesday, November 17th, 2009
Overview of Green’s Function
Description
A detailed tutorial on Green’s function. Step by step tutorial including several examples of Green’s function for reference.
Overview
Green’s function is used in deifferential equations as a method of solving certain types of equations that are subject to boundary conditions. There equations are called inhomogeneous differential equations. Unlike many other functions, Green’s function is not studied by how to use it to solve equations, but it is studied by the fundamental solutions.
Tags: boundary, conditions, differential, differential equations, equations, function, fundamental, George Green, Green's function, inhomogeneous, solutions
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Thursday, November 5th, 2009
How to Use Parametrization
Description
A detailed tutorial on how to use parametrization. Step by step tutorial including several examples of how to use parametrization for reference.
Overview
Parametrization can be used in many different branches of math, including algebra and calculus. Parametrization involves setting up parameters necessary for the complete or relevent specification of a geometric object. This means it is only used when calculating a shape or part of a shape, because that is what a geometric object is. Sometimes, this is nothing more than identifying the parameters. Other times it becomes an involved mathematical process that is used to find out what the parameters are.
Tags: Calculus, complete, decide, deciding, define, defining, differential equations, geometric, identify, identifying, parameter, parametrization, relevent, set, setting, shape, specification, vector
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Thursday, November 5th, 2009
Overview of Mass-Energy Equivalence
Description
A detailed tutorial on mass-energy equivalence. Step by step tutorial including several examples of mass-energy equivalence for reference.
Overview
Mass-energy equivalence is the concept that the mass of a body is the measure of its energy content. This is often expressed by a formula written by Einstein, who is also the one that proposed the idea of mass-energy equivalence. This formula is 
, where E is energy, m is the mass, and c is the speed of light in a vacuum.
Tags: Albert, body, c, content, differential equations, E, Einstein, energy, equivalence, equivalent, formula, idea, light, m, mass, measure, speed, vacuum
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Thursday, October 8th, 2009
Definition of a Mandelbrot Set
Description
A detailed tutorial on Mandelbrot sets and identifying Mandelbrot sets. Step by step tutorial including a several visual examples of a Mandelbrot set for reference.
Overview
A Mandelbrot set is defined as a set of points in the complex frame, the boundary of which forms a fractal. This can be mathematically defined as the set of complex values c for which the orbit of zero under iteration of a complex quadratic polynomial remains bounded.
Tags: boundary, complex, differential equations, fractal, iteration, Mandelbrot, Math, point, polynomial, quadratic, set, value
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Thursday, October 8th, 2009
Definition of a Julia Set
Description
A detailed tutorial on Julia sets and identifying Julia sets. Step by step tutorial including a several visual examples of a Julia set for reference.
Overview
A Julia set is a complimentary set defined from a function. A Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. The behavior of the Julia set is classified as “chaotic”.
Tags: chaotic, complimentary, differential equations, function, iterated, Julia, Math, perturbation, set, value
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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
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Tuesday, October 6th, 2009
How to Solve a Dirichlet Problem
Description
A detailed tutorial of solving Dirichlet problems. Step by step tutorial including several examples of how to solve Dirichlet problems for reference.
Overview
A Dirichlet problem is a problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of the region. It was originally supposed to be used for Laplace’s equation, although other equations can use it as well. The Dirichlet problem can be stated as: given a function f that has values everywhere on the boundary of a region in R^n, is there a unique continuous function u twice continuously differentiable in the interior and continuous on the boundary, such that u is harmonic in the interior and u = f on the boundary? A mathematical solution can be expressed as:
Tags: bounded, continuous, differential equations, Dirichlet, equation, harmonic, interior, Laplace, Math, partial differential equation, problem, region, solution, value
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Tuesday, October 6th, 2009
The Application of Bessel Functions
Description
A detailed tutorial on the application of Bessel functions. Step by step tutorial including several examples of the application of Bessel functions for reference.
Overview
Bessel functions are the solution to Bessel’s equation – although they were created by the same person, the question and the answer are seperated into two different things. This is because Bessel functions are also used to solve other things, such as Laplace’s equation and the Helmholtz equation. It can also be used with spherical and cylindrical coordinates. There are many different definitions of Bessel functions, but the most common one involves a Taylor series expansion around x = 0. Mathematically this is expressed as:
Tags: Bessel, cylindrical, differential equations, equation, expansion, function, functions, helmholtz equation, Laplace's Equation, Math, series, spherical, Taylor
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Tuesday, October 6th, 2009
Definition of a Cornu Spiral
Description
A detailed tutorial on Cornu spirals. Step by step tutorial including a visual example of a Cornu spiral for reference.
Overview
The Cornu spiral is also known as a Euler spiral and clothoid. It is generated as a straight line that branches out, and then turns up on one end and down on the other, both spiraling into tight curls. It is formed by a parametric plot of S(t) against C(t). They are very closely linking to Fresnal integrals and have been sometimes thought of as a solution.
Tags: Calculus, clothoid, Cornu spiral, curl, curve, differential equations, Euler spiral, Fresnal integral, Geometry, line, Math, parametric, plot, solution, spiral
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