differential operator

In mathematics, and particularly in analysis, an ordinary differential equation (or ODE) is an equation that involves the derivatives of an unknown function of one variable. A simple example of an ordinary differential equation is , where is an unknown function, and is its derivative. See differential calculus and integral calculus for basic calculus background. Important theorems in the field of ODEs include broad existence and uniqueness theorems and for ODEs in the plane, the Poincaré-Ben ...

Including:

• Ordinary differential equation - Definition
• Ordinary differential equation - General application
• Ordinary differential equation - Existence and nature of solutions
• Ordinary differential equation - Types of differential equations with some history
• Ordinary differential equation - Linear ODEs with constant coefficients
• Ordinary differential equation - Linear ODEs with variable coefficient
• Ordinary differential equation - General solution method for first-order linear ODEs
• Ordinary differential equation - Linear PDEs
• Ordinary differential equation - First-order PDEs
• Ordinary differential equation - Singular solutions
• Ordinary differential equation - Reduction to quadratures
• Ordinary differential equation - The Fuchsian theory
• Ordinary differential equation - Lie's theory
• Ordinary differential equation - Bibliography

Read more here: » Ordinary differential equation: Encyclopedia - Ordinary differential equation

In mathematics, a boundary value problem consists of a differential equation to be satisfied at all points in the interior of an interval or a region and a set of boundary conditions specifying the values of the solution or some of its derivatives everywhere on the boundary of the interval or region. Boundary value problems may be posed for ordinary differential equations as well as partial differential equations. Boundary value problems arise in several branches of physics. Problems involving the wave equation, such as the determination ...

Including:

• Boundary value problem - Example
• Boundary value problem - Example ordinary
• Boundary value problem - Example partial
• Boundary value problem - Bibliography

Read more here: » Boundary value problem: Encyclopedia - Boundary value problem

The letter D is the fourth letter of the Latin alphabet. Its name in English is dee. D - History. The Semitic letter Dâlet probably developed from the logogram for a fish or a door. In Semitic, Ancient Greek (Modern Greek /ð/) and Latin the letter was pronounced /d/, in the Etruscan alphabet the letter was superfluous but still maintained (see letter B). Greek letter: Δ (capital) or δ (small) (Delta). The minuscule (lower-case) form of D, consisting of a loop and a tall vertical stroke, de ...

Including:

• D - History
• D - Usage
• D - Alternative representations
• D - Computing
• D - Meanings for D

Read more here: » D: Encyclopedia - D

In calculus, the indefinite integral of a given function (i.e. the set of all antiderivatives of the function) is always written with a constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function f is defined on an interval and F is an antiderivative of f, then the set of all antiderivatives of f is given by the funct ...

Including:

• Arbitrary constant of integration - Where does the constant come from?
• Arbitrary constant of integration - Why is the constant necessary?
• Arbitrary constant of integration - Why is a constant the only difference between two antiderivatives?

Read more here: » Arbitrary constant of integration: Encyclopedia - Arbitrary constant of integration

In vector calculus, curl is a vector operator that shows a vector field's rate of rotation: the direction of the axis of rotation and the magnitude of the rotation. It can also be described as the circulation density. "Rotation" and "circulation" are used here for properties of a vector function of position; they are not about changes with time. A vector field which has zero curl everywhere is ...

Including:

• Curl - Examples

Read more here: » Curl: Encyclopedia - Curl

In mathematics, variation of parameters is a technique used in solving certain second order linear inhomogeneous ordinary differential equations. Variation of parameters is not commonly used in pure mathematics, but is a useful tool in engineering applications. Method of variation of parameters - Technique. We have a differential equation of the form and we define the linear operator where D represents the differential operator. We there ...

Including:

• Method of variation of parameters - Technique
• Method of variation of parameters - Example usage

Read more here: » Method of variation of parameters: Encyclopedia - Method of variation of parameters

In mathematics, de Rham cohomology is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. It is in different, definite senses dual both to singular homology, and to Alexander-Spanier cohomology. De Rham cohomology - Defini ...

Including:

• De Rham cohomology - Definition
• De Rham cohomology - de Rham cohomology computed
• De Rham cohomology - de Rham's theorem
• De Rham cohomology - Sheaf-theoretic de Rham isomorphism
• De Rham cohomology - Proof
• De Rham cohomology - Related ideas
• De Rham cohomology - Harmonic forms
• De Rham cohomology - Hodge decomposition

Read more here: » De Rham cohomology: Encyclopedia - De Rham cohomology

Operators are described usually by the number of operands: monadic or unary operators take one argument. dyadic or binary operators take two arguments. triadic or ternary/trinary/tertiary operators take three arguments. The number of operands is also called the arity of the operator. If an operator has an arity given as n-ary (or n-adic), then it takes n arguments. In programming, outside than functional programming, the -ary terms are more often used than the ot ...

See also:

Operator, Operator - Operators and levels of abstraction, Operator - Describing operators, Operator - Notations, Operator - Examples of mathematical operators, Operator - Linear operators, Operator - Operators in probability theory, Operator - Operators in calculus, Operator - Fundamental operators on scalar and vector fields, Operator - Relation to type theory, Operator - Operators in physics

Read more here: » Operator: Encyclopedia II - Operator - Describing operators

Oliver Heaviside - Early years. Heaviside was born in London's Camden Town, He was short and red-headed, and suffered from scarlet fever during his youth, the illness having a lasting impact on him, leaving him partly deaf. Although he was a good scholar (placed fifth out of five hundred students in 1865), he left school at 16 and began learning about Morse code and electromagnetism. Heaviside became a telegraph operator, initially in Denmark and, later, at the Great Northern Telegraph Company. Heaviside c ...

See also:

Oliver Heaviside, Oliver Heaviside - Biography, Oliver Heaviside - Early years, Oliver Heaviside - Middle years, Oliver Heaviside - Later years, Oliver Heaviside - Innovations and discoveries, Oliver Heaviside - Maxwell reformulation and mathematics, Oliver Heaviside - Electromagnetic terms, Oliver Heaviside - Publications

Read more here: » Oliver Heaviside: Encyclopedia II - Oliver Heaviside - Biography

The Laplacian can be extended to functions defined on surfaces, or more generally, on Riemannian and pseudo-Riemannian manifolds. This more general operator goes by the name Laplace-Beltrami operator. One defines it, just as the Laplacian, as the divergence of the gradient. To be able to find a formula for this operator, one will need to first write the divergence and the gradient on a manifold. If g denotes the (pseudo)-metric tensor on the manifold, one finds that the vol ...

See also:

Laplace operator, Laplace operator - Definition, Laplace operator - Coordinate expressions, Laplace operator - Identities, Laplace operator - Laplace-Beltrami operator, Laplace operator - Laplace-de Rham operator, Laplace operator - Properties

Read more here: » Laplace operator: Encyclopedia II - Laplace operator - Laplace-Beltrami operator

Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which it gets hot most quickly. The magnitude of th ...

See also:

Gradient, Gradient - Interpretations of the gradient, Gradient - Formal definition, Gradient - Example, Gradient - The gradient on manifolds

Read more here: » Gradient: Encyclopedia II - Gradient - Interpretations of the gradient

Linear - Linear functions. In mathematics, a linear function f(x) is one which satisfies the following two properties (but see below for a slightly different usage of the term): Additivity property (also called the superposition property): f(x + y) = f(x) + f(y). This says that f is a group homomorphism with respect to addition. Homogeneity property: f(αx) = Î ...

See also:

Linear, Linear - Mathematics, Linear - Linear functions, Linear - Linear polynomials, Linear - Physics, Linear - Electronics, Linear - Military tactical formations, Linear - Music

Read more here: » Linear: Encyclopedia II - Linear - Mathematics

Å - Ångström, A - Ampere, Area, a Blood type, a Spectral type, Vector potential, Work, B - B meson, a Blood type, Boron, Luminance, Magnetic field, a Spectral type, C - Carbon, Degrees Celsius, Set of complex numbers, Coulomb, Molar heat capacity (Cp), a Programming language, Specific heat capacity, D - Deuterium, Differential operator, Electric displacement, E - Electric field, Energy, SI prefix: (exa-), Expected value, F - degrees Fa ...

See also:

List of letters used in mathematics and science, List of letters used in mathematics and science - Latin, List of letters used in mathematics and science - Greek, List of letters used in mathematics and science - More

Read more here: » List of letters used in mathematics and science: Encyclopedia II - List of letters used in mathematics and science - Latin

Nonlinearity - Linear systems. In mathematics, a linear function f(x) is one which satisfies the following properties: Additivity: f(x + y) = f(x) + f(y) Homogeneity: Systems that satisfy both additivity and homogeneity are considered to be linear systems. These two rules, taken together, are often referred to as the principle of superposition. Important exa ...

See also:

Nonlinearity, Nonlinearity - Background, Nonlinearity - Linear systems, Nonlinearity - Nonlinear systems, Nonlinearity - Specific nonlinear equations, Nonlinearity - Tools for solving certain non-linear systems, Nonlinearity - Examples of nonlinear equations

Read more here: » Nonlinearity: Encyclopedia II - Nonlinearity - Background

We have a differential equation of the form and we define the linear operator where D represents the differential operator. We therefore have to solve the equation Lu(x) = f(x) for u(x), where L ...

See also:

Method of variation of parameters, Method of variation of parameters - Technique, Method of variation of parameters - Example usage

Read more here: » Method of variation of parameters: Encyclopedia II - Method of variation of parameters - Technique

Partially defined operators A, B on Hilbert spaces H, K are unitarily equivalent iff there is a unitary operator U:H → K such that U maps dom A bijectively onto dom B, A multiplication operator is defined as follows: Let be a σ-finite countably additive measure space and f a real-valued measurable function on X. An operator T of the form whose domain is the space of ψ for which the right-hand side above is in ...

See also:

Self-adjoint operator, Self-adjoint operator - Symmetric operators, Self-adjoint operator - Self-adjoint operators, Self-adjoint operator - Spectral theorem, Self-adjoint operator - Borel functional calculus, Self-adjoint operator - Resolution of the identity, Self-adjoint operator - Formulation in the physics literature, Self-adjoint operator - Extensions of symmetric operators, Self-adjoint operator - Von Neumann's formulas, Self-adjoint operator - Examples, Self-adjoint operator - Spectral multiplicity theory, Self-adjoint operator - Example: structure of the Laplacian, Self-adjoint operator - Pure point spectrum

Read more here: » Self-adjoint operator: Encyclopedia II - Self-adjoint operator - Spectral theorem

To prove the power rule for differentiation, we use the definition of the derivative as a limit: Substituting f(x) = xn gives One can then express (x + h)n by applying the binomial theorem to obtain The i = n term of the sum can then be written independently of the sum to yield Canceling the xSee also:

Calculus with polynomials, Calculus with polynomials - The power rule, Calculus with polynomials - Proof of the power rule, Calculus with polynomials - Differentiation of arbitrary polynomials, Calculus with polynomials - Generalization

Read more here: » Calculus with polynomials: Encyclopedia II - Calculus with polynomials - Proof of the power rule

Given the equation for the damped harmonic oscillator: the expression in parentheses can be factored out: first obtain the characteristic equation by replacing D with λ. This equation must be satisfied for all y, thus: Solve using the quadratic formula: Use these data to factor out the original differential equation: This implies a pair of solutions, one corresponding to and another to The solutions are, respec ...

See also:

Linear differential equation, Linear differential equation - Homogeneous linear differential equation with constant coefficients, Linear differential equation - Inhomogeneous linear differential equation with constant coefficients, Linear differential equation - Other meanings, Linear differential equation - Example I, Linear differential equation - Example II, Linear differential equation - Example III, Linear differential equation - Generalization

Read more here: » Linear differential equation: Encyclopedia II - Linear differential equation - Example III

If M and N are two smooth manifolds, how do we define the jet of a function ? We could perhaps attempt to define such a jet by using local coordinates on M and N. The disadvantage of this is that jets cannot thus be defined in an equivariant fashion. Jets do not transform as tensors. In fact, jets of functions between two manifolds belong to a Jet bundle. This section begins by introducing the notion of jets of functions from the real line to a manifold. It proves that such jets form a vector bundle, analog ...

See also:

Jet mathematics, Jet mathematics - Jets of functions between Euclidean spaces, Jet mathematics - Example: One-dimensional case, Jet mathematics - Example: Mappings from one Euclidean space to another, Jet mathematics - Example: Algebraic properties of jets, Jet mathematics - Jets at a point in Euclidean space: Rigorous definitions, Jet mathematics - An analytic definition, Jet mathematics - An algebro-geometric definition, Jet mathematics - Taylor's theorem, Jet mathematics - Jet spaces from a point to a point, Jet mathematics - Jets of functions between two manifolds, Jet mathematics - Jets of functions from the real line to a manifold, Jet mathematics - Jets of functions from a manifold to a manifold, Jet mathematics - Jets of sections, Jet mathematics - Differential operators between vector bundles

Read more here: » Jet mathematics: Encyclopedia II - Jet mathematics - Jets of functions between two manifolds

Purely functional programs have no side-effects. Since functions do not modify state, no data may be changed by parallel function calls. For this reason, pure functions are always thread-safe, a fact which is exploited by languages that use call-by-future evaluation. Because ordering of side-effects does not have to be preserved in their absence, some languages (such as Haskell) use call-by-need evaluation for pure functions. "Pure" functional programming languages typically enforce referential transparency, which is the notion ...

See also:

Functional programming, Functional programming - History, Functional programming - Higher-order functions, Functional programming - Comparison with imperative programming, Functional programming - Pure functions, Functional programming - Monads, Functional programming - Expansion of functional programming, Functional programming - Speed and space considerations, Functional programming - Functional languages

Read more here: » Functional programming: Encyclopedia II - Functional programming - Pure functions