mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named...

In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal...

rise to such notions as the Fréchet or Gateaux derivative. Likewise, in differential geometry, the differential of a function at a point is a linear function...

spaces. The Fréchet derivative should be contrasted to the more general Gateaux derivative which is a generalization of the classical directional derivative...

in 1922, Wiener immediately saw that he could use Gateaux' definition to define his "differential space" and construct a measure of Brownian motion (later...

a {\displaystyle df_{a}} is called the (total) derivative or (total) differential of f {\displaystyle f} at a {\displaystyle a} . Other notations for the...

after Gottfried Wilhelm Leibniz, is represented as the ratio of two differentials, whereas prime notation is written by adding a prime mark. Higher order...

R exists as a real number. Semi-differentiability is thus weaker than Gateaux differentiability, for which one takes in the limit above h → 0 without...

{\displaystyle \Omega _{0}\mapsto T_{s}(\Omega _{0})=\Omega _{s}.} Then the Gâteaux or shape derivative of F ( Ω ) {\displaystyle {\mathcal {F}}(\Omega )}...

In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical...

certain coordinate systems Differential form – Expression that may be integrated over a region Ehresmann connection – Differential geometry construct on fiber...

derivatives which solve the Cauchy–Riemann equations, a set of two partial differential equations. Every holomorphic function can be separated into its real...

programming. The Hamilton–Jacobi equation is a first-order, non-linear partial differential equation − ∂ S ∂ t = H ( q , ∂ S ∂ q , t ) . {\displaystyle -{\frac {\partial...

applications in nonlinear analysis and differential geometry. Formally, the definition of differentiation is identical to the Gateaux derivative. Specifically, let...

dynamical world of ordinary differential equations. A projected dynamical system is given by the flow to the projected differential equation d x ( t ) d t...

this notion of functional differential is so strong it may not exist, and in those cases a weaker notion, like the Gateaux derivative is preferred. In...

(2): 275–292. doi:10.1137/140963510. Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2...

) = 0 {\displaystyle D_{v}(u)=0} (usually the weak form of a partial differential equation), thus the considered objective is j ( v ) = J ( u v , v ) {\displaystyle...

{x}})\leq \limsup \Phi (x_{n}).} Evans, Lawrence C. (1998). Partial Differential Equations. Providence, Rhode Island: American Mathematical Society. ISBN 0-8218-0772-2...

sur une nouvelle formule de calcul differentiel" [On a new formula of differential calculus], The Quarterly Journal of Pure and Applied Mathematics (in...

the Gateaux derivative for details. The Fréchet derivative allows for an extension of the concept of a total derivative to Banach spaces. The Gateaux derivative...

linearity does not. If the Hadamard directional derivative exists, then the Gateaux derivative also exists and the two derivatives coincide. The Hadamard derivative...

method of a priori estimates. Suppose that we wish to solve the linear differential equation P u = f {\displaystyle Pu=f} for u , {\displaystyle u,} with...

curves defined by differential equations within which he built a new branch of mathematics called "qualitative theory of differential equations". Poincaré...

{\displaystyle P:U\to Y} is a continuously differentiable function, then the differential equation x ′ ( t ) = P ( x ( t ) ) , x ( 0 ) = x 0 ∈ U {\displaystyle...

TVS-valued) functions which, in particular, are used in the definition of the Gateaux derivative. They are fundamental to the analysis of maps between two arbitrary...

partial differential equation – In mathematics, a hyperbolic partial differential equation of order n {\displaystyle n} is a partial differential equation...

Eulerian specification of the flow field associated to the ordinary differential equation, d d t φ t = v t ∘ φ t ,   φ 0 = i d , {\displaystyle {\frac...