In mathematics, the Gateaux differential or Gateaux derivative is a generalization of the concept of directional derivative in differential calculus. Named after René Gateaux, it is defined for functions between locally convex topological vector spaces such as Banach spaces. Like the Fréchet derivative on a Banach space, the Gateaux differential is often used to formalize the functional derivative commonly used in the calculus of variations and physics.
Unlike other forms of derivatives, the Gateaux differential of a function may be a nonlinear operator. However, often the definition of the Gateaux differential also requires that it be a continuous linear transformation. Some authors, such as Tikhomirov (2001), draw a further distinction between the Gateaux differential (which may be nonlinear) and the Gateaux derivative (which they take to be linear). In most applications, continuous linearity follows from some more primitive condition which is natural to the particular setting, such as imposing complex differentiability in the context of infinite dimensional holomorphy or continuous differentiability in nonlinear analysis.
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In mathematics, the Gateaux differential or Gateauxderivative is a generalization of the concept of directional derivative in differential calculus....
those generalizations are the Gateauxderivative and the Fréchet derivative. One deficiency of the classical derivative is that very many functions are...
coordinates being constant. The directional derivative is a special case of the Gateauxderivative. The directional derivative of a scalar function f ( x ) = f (...
notion, like the Gateauxderivative is preferred. In many practical cases, the functional differential is defined as the directional derivative δ F [ ρ , ϕ...
infinite-dimensional spaces of functional analysis. For instance, the Fréchet or Gateauxderivative can be used to define a notion of a holomorphic function on a Banach...
Hadamard directional derivative exists, then the Gateauxderivative also exists and the two derivatives coincide. The Hadamard derivative is readily generalized...
of derivative is not quite strong enough, and one requires strict differentiability instead. The Gateauxderivative extends the Fréchet derivative to...
and h are functions, and ε is a scalar. This is recognizable as the Gateauxderivative of the functional. Compute the first variation of J ( y ) = ∫ a b...
where d x F ( x ; δ x ) {\displaystyle d_{x}F(x;\delta _{x})} is the Gateauxderivative of F {\displaystyle F} with respect to x {\displaystyle x} in the...
Several concepts of a derivative may be defined on a Banach space. See the articles on the Fréchet derivative and the Gateauxderivative for details. The Fréchet...
spaces, ultimately giving rise to such notions as the Fréchet or Gateauxderivative. Likewise, in differential geometry, the differential of a function...
_{t\rightarrow 0^{+}}{\frac {T(tG+(1-t)F)-T(F)}{t}}} , which is the one-sided Gateauxderivative of T {\displaystyle T} at F {\displaystyle F} , in the direction of...
The directional derivative δ S {\displaystyle \delta {\cal {S}}} on the left is known as variation in physics and Gateauxderivative in mathematics. Lagrangian...
C ( X , R ) {\displaystyle \Phi \in C(X,\mathbf {R} )} and have a Gateauxderivative Φ ′ : X → X ∗ {\displaystyle \Phi '\colon X\to X^{*}} which is continuous...
0^{+}}{\frac {P_{K}(x+\delta v)-x}{\delta }}.} Which is just the GateauxDerivative computed in the direction of the Vector field Given a closed, convex...
order variation of the vector field. The directional derivative calculates the Gateauxderivative as calculated in Beg's original paper[49] and. First...
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