Description
A detailed tutorial on the Gram-Schmidt process. Step by step tutorial including a visual example of the Gram-Schmidt process for reference.
Overview
The Gram-Schmidt process is a process used for orthogonalizing a set of vectors in an inner product space. What the Gram-Schmidt process does is it takes a finite and linearly independent set and converts it to an orthogonal set that spans the same amount of space.
Description
A detailed tutorial on the solving of Fourier series. Step by step tutorial including several examples of how to solve the Fourier series for reference.
Overview
The Fourier series is very common in differential equations and partial differential equations. What a Fourier series does is decompose a periodic function into the sum of simple oscillating functions, like sine and cosine. Fourier series are part of Fourier analysis and were first introduced by Joseph Fourier to solve the heat equation. So another name for the Fourier series is the heat equation, although they are considered different things. Both the heat equation and the Fourier series are partial differential equations.
Description
A detailed tutorial of uniform convergence. Step by step tutorial including several example problems of uniform convergence for reference.
Overview
Uniform convergence is a very strong type of convergence, even stronger than pointwise convergence. A sequence {fn} of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. This concept is important because several properties of these functions are transferred to the limit f if the convergence is uniform.
Description
A detailed tutorial on the solving of the Dirichlet kernel. Step by step tutorial including several examples of how to solve the Dirichlet kernel for reference.
Overview
The Dirichlet kernel is this collection of functions: 
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:
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:
Integration:
Gauss’s Law for Magnetism: states that the total magnetic flux through a closed surface is zero.
Differentiation:
Integration:
Maxwell-Faraday Equation: describes how a changing magnetic field can create an electric field.
Differentiation:
Integration:
Ampere’s Circuital Law: states that magnetic fields can be generated by electrical current and changing electric fields.
Differentiation:
Integration:
Description
A detailed tutorial on the explanation of the Galerkin Method. Step by step tutorial including several examples of how to solve problems using the Galerkin Method for reference.
Overview
The Galerkin Method is used for converting a continuous operator problem, like a differential equation, to a discrete problem. This makes it a link between Discrete Math and Differential Equations. You can use different approximation techniques with the Galerkin Method, sometimes changing the name to fit the technique you are using. The credit of this method goes to Russian mathematician Boris Galerkin.
Description
A detailed tutorial on the solving of the Euler-Lagrange Equation. Step by step tutorial including several examples of how to solve the Euler-Lagrange Equation for reference.
Overview
The Euler-Lagrange Equation, sometimes just called Lagrange’s Equation, is a differential equation which has a solution that is function for which a given functional is stationary. The Euler-Lagrange Equation is an equation satisfied by a function q of a real argument t which is a stationary point of the functional 
After finding q, the derivative of q, and L, which can all be expressed by seperate equations. the Euler-Lagrange Equation can be written as an ordinary differential equation expressed by
Description
A detailed tutorial on the solving of Parseval’s Theorem. Step by step tutorial including several examples of how to solve Parseval’s Theorem for reference.
Overview
Parseval’s Theorem is sometimes called Rayleigh’s Energy Theorem, or Rayleigh’s Identity. The term is used to describe the unitarity of any Fourier’s transform, but the most general form of this property should be referred to as the Plancherel Theorem.
Parseval’s Theorem states: Suppose that A(X) and B(x) are two Riemann integrable, complex-valued functions on R of period 2pi with Fourier series 
Description
A detailed tutorial and explanation of the Gibbs Phenomenon. Step by step tutorial including several examples of the Gibbs Phenomenon for reference.
Overview
Gibbs Phenomenon refers to the way the Fourier series of a piecewise continuously differentiable periodic function behaves at a jump discontinuity. What happens is the nth partial sum of the series has large oscillations near the jump, which could possibly overshoot the partial sum above the function. However, the overshoot does not die out as the frequency increases – it approaches a finite limit.
