Differential operators

thumb|300px|The gradient, represented by the blue arrows, denotes the direction of greatest change of a scalar function. The values of the function are represented in greyscale and increase in value from white (low) to dark (high).

500px|thumb|upright=1.75|alt= A vector field with diverging vectors, a vector field with converging vectors, and a vector field with parallel vectors that neither diverge nor converge|The divergence of different vector fields. The divergence of vectors from point (x,y) equals the sum of the partial derivative-with-respect-to-x of the x-component and the partial derivative-with-respect-to-y of the y-component at that point: \nabla\!\cdot(\mathbf{V}(x,y)) = \frac{\partial\, {V_x(x,y){\partial{x+\frac{\partial\, {V_y(x,y){\partial{y

derivative of a function of several variables with respect to one variable, with the others held constant

differential operator describing the rotation at a point in a 3D vector field

divergence of the gradient

right|100px|thumb|Del operator,represented bythe nabla symbol Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by ∇ (the nabla symbol). When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a field (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vec

the matrix of all first-order partial derivatives of a vector-valued function

matrix of second derivatives

function with multiplicative scaling behaviour

second-order differential operator that is the Laplace operator of Minkowski space

typically linear operator defined in terms of differentiation of functions

theorem that the difference between the dimensions of the kernel and cokernel of a differential operator on a manifold is the integral of a characteristic class

symbol used in mathematics, physics and engineering to indicate a differential operator in a Cartesian vector space

derivative of a function of several variables with respect to one variable, without the others held constant

derivative of a tensor field along the flow defined by a vector field

in differential geometry, a differential operation defined in differential forms that increases the form degree by 1

Type of differential operator

differential operator

concept in calculus of variation

linear map from p-forms on an n-dimensional manifold to (n−p)-forms

operator on functions, defined by the composition of Fourier transformation, multiplication with a certain smooth function of both position and momentum, and inverse Fourier transformation

differential algebra

first-order differential linear operator on spinor bundle, whose square is the Laplacian

Partial differential operator

linear partial differential operators (first order constant coefficients) on f(ℂⁿ or ℝ²ⁿ)

infinitesimal version of a Lie groupoid: manifold M with vector bundle E, vector bundle map ρ: E→TM, and a Lie bracket on sections of E, so that [s,ft] = (ρ(s)f)t+f[s,t] for any function f: M→ℝ and sections s, t of E

elliptic differential operators in geometry mathematics