# Questions tagged [gateaux-derivative]

This tag is for questions regarding to the Gateaux differential or, Gateaux derivative, a generalization of the concept of directional derivative in differential calculus. It is often used to formalize the functional derivative commonly used in Physics, particularly Quantum field theory.

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Definition. Let $X,Y$ be Banach spaces and $f:U\rightarrow Y$ a map on an open subset $U\subset X$. The map $f$ is called Gâteaux differential at $x\in U$ if there exists a continous linear map $A:X\... 1 vote 1 answer 37 views ### Gateaux Diff and Frechet diff yesterday I took a test and I can't answer this question. “Let$E,F$be two normative spaces and f be a function on E to F with Gateaux differential and all limits $$\lim_{t \to 0}\dfrac{f(x+tv)-f(x)}{... 2 votes 1 answer 64 views ### Is this equivalent to the definition of directional derivative? Let f=f(x_1,x_2) be a C^1 scalar valued function of two variables at the point \vec{x}_0. We know the directional derivative of f at \vec{x}_0 in the direction of \vec{v} (unit vector) is ... 2 votes 0 answers 21 views ### Characterization of Optimal Payoff (under Expected Utility) via Gateaux-Derivative/Fréchet Derivative Background: Let (\Omega, \mathcal{F}, \mathbb{P}) model a financial market and T>0. Denote by (S_t)_{t\in[0,T]} the price process of the risky asset in the financial market. Assume that the ... • 53 1 vote 0 answers 31 views ### Iterated Gateaux derivative Let F[h] be a generic functional of the function h(x):{\mathbb{R}^d}\mapsto \mathbb{R}. Consider the following definition of Gateaux derivative in direction g(x):{\mathbb{R}^d}\mapsto \mathbb{R}... • 627 2 votes 0 answers 56 views ### Understanding Gateaux derivatives Background I'm studying Gateaux derivatives and I find some difficulties to underestand the general case where the order of the derivative is n>1. First, I'm considering the following definition ... • 627 2 votes 1 answer 37 views ### Question on the Gateaux derivative (being the principal part of a system of PDEs) Let me start with a disclaimer. This is the first time ever dealing with Gateaux derivative, hence my knowledge around this topic is poor. As I was going through this paper, I got stuck in the ... • 2,924 0 votes 0 answers 29 views ### Index notation for Gateaux derivative of matrix function confined to subspace of traceless, symmetric matrices Suppose I have a function \Lambda: S^\text{tr} \to S^\text{tr} where S^\text{tr} = \{Q \in \mathbb{R}^{n \times n} \: |\: Q = Q^T \: \text{and} \: \text{tr}(Q) = 0\} is the space of traceless, ... • 272 0 votes 0 answers 20 views ### Gateaux derivative and stricly differentiable function F:X->Y , suppose that F is Gateaux differentiable at point x,and also strictly differentiable at point x,then F is continiously differentiable at point x 1 vote 0 answers 81 views ### Difference between Gateaux and Fréchet derivative Suppose I have the operator$$T:L_2[0,1]\to L_2[0,1],T[x(t)] = \sin(x(t)).$$My first question: Is it true that the Gateaux derivative of this operator equals to the ordinary derivative of \sin(... 0 votes 1 answer 125 views ### How should I derive the Euler-Lagrange equation by using the Gateaux differential? Find the Gateaux differential for the functional S[y]=y(0)+\frac{1}{2}\int_{0}^{1}(4y'^2+x^2y^2)dx, y(1)=2 and use it to derive the Euler-Lagrange equation. Be sure to specify all boundary ... • 89 2 votes 0 answers 61 views ### Taking limit inside integral sign I am reading through the wiki page of the Gateaux derivative. In an example, they let \Omega be some Lebesgue measurable subset of \mathbb{R}^n and consider the integral operator$$ E: L_2(\Omega) ... • 291 0 votes 0 answers 73 views ### On Marsden's 'Introduction to Mechanics and Symmetry' Exercise 1.6-1. As I don't have access to the solutions manual of this book I would by grateful for any comments regarding my solution to this exercise. Is there a mistake somewhere? Or can I improve / shorten it in ... 2 votes 0 answers 104 views ### How to use second-order Gateaux derivative for convex optimization? Consider a functional$f:V \to \mathbb{R}$where$V$is some vector space. The first-order Gateaux derivative of$f$at point$v$in the direction$uis defined as follows: \begin{align} \Delta_u f(... • 2,941 0 votes 0 answers 32 views ### Trying to derive the adjoint equation of a ODE system and having problem understanding chain rule on a vector using gateaux differentiation. So suppose I have a part of the lagrangian $$\mathcal{L}_{\text{part}} = \int_{0}^{T_{\text{max}}} -\dot{\lambda}^T u - \lambda^T F(u)$$ where $$F(u) = \max(0,\theta_1u+\theta_2)$$ where\theta_{1,2}$... • 663 1 vote 1 answer 104 views ### Chain rule for a functional derivative. Given$L:\mathbb{R}\times \mathcal{P}(\mathbb{R}^n) \to \mathbb{R}$, where$\mathcal{P}(\mathbb{R}^n)$is the space of probability densities on$\mathbb{R}^n$. I want to calculate $$\frac{d}{d\... • 11 0 votes 0 answers 25 views ### Gateaux derivative chain rule varying the argument and the argument of the argument Suppose I have a functional G: \mathcal{F} \to \mathbb{R} and a functional F: \mathcal{H} \to \mathbb{R}. Let F_\tau = F + \tau \tilde{F} and H_\tau = H + \tau \tilde{H} for F \in \mathcal{F}... 0 votes 1 answer 55 views ### Why Gauteaux derivative is homogeneous? According to wikipedia, given a locally convex topological vector space X and a functional f: X \to \mathbb R (let's assume f is a real value function here to make things simpler), and a vector ... • 318 2 votes 0 answers 87 views ### Taylor's theorem with Gateaux derivatives I'm trying to figure out the rate of the remainder term in the Taylor expansion of some functional F using its Gateaux derivative. I found the following from Wikipedia, where U is an open subset ... 2 votes 1 answer 131 views ### An example of f and a such that f is Gâteaux but not Fréchet differentiable at a I'm reading this lecture note about differentiability. Let (X, |\cdot|_X) and (Y, |\cdot|_Y) be normed spaces. Let A be an open subset of X and f: A \to Y. The directional derivative f^{\... • 17.2k 0 votes 0 answers 38 views ### Why Gateaux derivative is a distribuition? Let E be a vector bundle , E^* the dual bundle and D a density bundle. Denote by \Gamma(E) the space of section of the bundle E. By definition the a distribution \omega in a vector ... 0 votes 1 answer 62 views ### Calculate first/second variation of the functional f : V \rightarrow \mathbb{R}, y \rightarrow \sin(y(1)) Hey I have a problem with this exercise. I have to calculate the first and second variation of the functional f : V \rightarrow \mathbb{R} y \rightarrow \sin(y(1)) for V = C^0([a, b]) with a &... • 680 1 vote 1 answer 119 views ### If F has bounded linear Gateaux derivative at x_0, what condition guarantees a Fréchet derivative? Let X and Y be real normed spaces and suppose F:X\to Y has a bounded linear Gateaux derivative F' at x_0\in X. (In the following, F' also denotes a Gateaux derivative at other points in X... • 6,467 3 votes 0 answers 127 views ### Flaw in proof of Jacobi's formula on Wikipedia? The Wikipedia article on Jacobi's formula (which gives the differential of the determinant function) contains two proofs, the second of which begins with a lemma claiming \det'(I)=\operatorname{tr},... • 6,467 1 vote 3 answers 195 views ### Composition between Frechet and Gateaux derivative I am currently dealing with the two terms "Gateaux-derivation" and "Frechet-derivation" and would like to know if there is a Frechet differentiable function f and Gateaux ... 2 votes 0 answers 175 views ### Conditions for chain rule for Gateaux derivatives Let X,Y,Z be locally convex topological vector spaces over \mathbb R (not necessarily Banach), D_X \subseteq X, D_Y \subseteq Y and let f \colon D_X \to D_Y, g \colon D_Y \to Z. Let us ... • 589 3 votes 2 answers 485 views ### Gâteaux differentiable function (with linear continuous Gateaux derivative) that is not Fréchet differentiable Currently, I am following a portion of the text Methods of Nonlinear Analysis. For normed real vector spaces (X,\|\cdot\|_X),(Y,\|\cdot\|_Y) and a \in X, consider the following definitions from ... • 555 0 votes 2 answers 59 views ### Property of the Gâteaux derivative Let D \colon X \to Y be a map of Banach spaces and define$$(\mathrm d_f D) g := \displaystyle\lim_{\varepsilon \to 0} \frac{D(f+\varepsilon g) - Df}\varepsilon$$given that the limit exists. ... • 1,285 0 votes 2 answers 57 views ### f' has a limit \Longrightarrow f has a limit without codomain Banach? Using Hahn Banach and Cauchy criterion I am able to prove the following: Let X normed, Y Banach and U \subset X open. Suppose that for x_0 \in U we have f : U \setminus \{x_0 \} \... • 1,999 0 votes 1 answer 101 views ### f is Gâteaux differentiable at a and the limit \lim _{t \rightarrow 0} \frac{f(a+t v)-f(a)}{t}=f^{\prime}(a)(v) is uniform for \|v\|=1 I'm reading this lecture note about differentiability of convex function. Let X be a normed space, A \subset X an open set, f: A \rightarrow \mathbb{R} a function, and a \in A a point. For a &... • 17.2k 0 votes 0 answers 38 views ### Definition of the Gateaux derivative Pretty simple question. This definition of the Gateaux Derivative presents it as: $$df(u,\psi)=\dfrac{d}{d\epsilon}f(u+\epsilon\psi)\Bigg{|}_{\epsilon=0}$$ Why is the ... 2 votes 0 answers 89 views ### Definition of certain Gateaux differentials of the norm in E. R. Lorch: "A Curvature Study of Convex Bodies in Banach Spaces" In the paper E. R. Lorch: A Curvature Study of Convex Bodies in Banach Spaces from 1953, the following assumptions and definitions are stated in section II (p. 107-108): Let (B, \| \cdot \|) be a ... • 4,788 3 votes 1 answer 564 views ### Gateaux derivatives is a linear continuous operator In Clarke's book p.61, it is said that the Gâteaux derivatives F'(x;v) at x in the direction v of F:X\to Y (X,Y being normed spaces) imply that v\mapsto F'(x;v) is linear continuous. But ... • 2,014 0 votes 1 answer 205 views ### Fast way to compute Fréchet/Gateaux Derivatives I was wondering if there is any fast way to compute Fréchet/Gateaux Derivatives, or at least a reasonable guess in most cases, say from the usual derivatives table. The Fréchet derivatives \dfrac{\... • 307 1 vote 1 answer 133 views ### Derivation of Noether's theorem by Gateaux derivative Noether's theorem states that if: \ \int_{a}^{b} F(x, y, y') \,dx = \ \int_{a_{new}}^{b_{new}} F(x_{new}, y_{new}, y_{new}') \,dx_{new} for any a, b and y(x), and when x and x_{new} and ... • 23 1 vote 1 answer 255 views ### What does 𝛿 mean in calculus of variation? In a calculus of variation problem, let functional J = \ \int_{x_1}^{x_2} F(x, y, y') \,dx . When x_1 and x_2 are all fixed, the variation of J are defined by Gâteaux variation: \delta J(y, h) =... • 23 1 vote 0 answers 225 views ### Reference for the Gateaux gradient of a functional on a Hilbert manifold The entry for Gâteaux gradient in the Encyclopedia of Mathematics wiki reads: Gâteaux gradient of a functional f at a point x_0 of a Hilbert space H. The vector in H equal to the Gâteaux ... 4 votes 0 answers 199 views ### Generalizing complex derivative as Fréchet/Gateaux derivative So it is well-known that complex differentiability of a function f:\mathbb{C}\rightarrow\mathbb{C} is equivalent to the function being Fréchet/Gateaux differentiable and the component functions (... • 1,653 2 votes 1 answer 325 views ### Gateaux derivative of the following operator Let f \colon \mathbb{R} \to \mathbb{R} be \mathcal{C}^1[a,b] with bounded derivative. Let H=L^2[a,b] and consider the Nemytskii operator F\colon H \to H defined by$$F(x)(\psi)=f(x(\psi))$$... • 123 3 votes 1 answer 630 views ### Example of function that is Gâteaux-differentiable but not Fréchet-differentiable I am looking for an example of a function that is Gateaux-differentiable but not Fréchet-differentiable. I know that there is a lot of example of function f: \mathbb R^2 \to \mathbb R that satisfies ... • 3,958 3 votes 0 answers 459 views ### First variation and Gâteaux derivatives I have two questions about the relation between Gâteaux and functional derivatives. In calculus of variation textbooks, the first variation is usually defined as follows. Let V be a 'function space' ... • 1,871 1 vote 0 answers 44 views ### How to compute the functional derivative of the following functional encountered in electrical impedance tomography? In the context of electrical impedance tomography, one encounters the following partial differential equation. Let \Omega be a bounded Lipschitz domain in \mathbb{R}^2 with C^1 boundary \... • 1,184 0 votes 1 answer 43 views ### How to calculate a differential of an Integral squared? could anyone help me calculating the derivative of this function: G:C[0,1]\rightarrow\mathbb{R},x\mapsto \left(\int_0^1 x(s)ds\right)^2 I tried using the Gateaux Differential - working out \frac{1}{... • 257 0 votes 0 answers 145 views ### Does the Euler-Lagrange functional actually have a functional derivative? It's well known that given a functional$$S(\boldsymbol q) = \int_a^b L(t,\boldsymbol q(t),\dot{\boldsymbol q}(t))\, \mathrm{d}t,$$the directional (or otherwise known as the Gateaux) derivative$$\... • 8,175 1 vote 1 answer 1k views ### Mean value inequality in the framework of Banach spaces. Let$E,F$be Banach spaces and$U \subseteq F$be an open set. Let$f : U \longrightarrow F$be a Gateaux differentiable function then$$\|f(x) - f(y)\|_F \leq \|x - y\|_E \sup\limits_{0 \leq \theta \... • 2,363 1 vote 1 answer 242 views ### Citation for the Gateaux derivative FTC The Wikipedia page for the Gateaux Derivative contains the following statement: A version of the fundamental theorem of calculus holds for the Gateaux derivative of$F$, provided$F$is assumed to be ... • 354 0 votes 0 answers 282 views ### Prove Gateaux derivative implies Fréchet derivative in some condition Q: Let$f: I \subseteq \mathbb{R} \rightarrow \mathbb{R^n}$, where$I$is an interval with some element (not empty), be a continuous function. Prove that if$f$is Gâteaux differentiable in$a \in I$, ... • 599 1 vote 1 answer 101 views ### First Variation of$L_2$with Linear operator Let$\Omega \subset \mathbb{R}^2$be open and bounded,$P:C^1(\Omega) \to L_2(\mathbb{R})$be a linear and bounded operator. I want to calculate the first variation of$\|Pu - f\|^2_{L_2}$,$u \in C^1(...
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So if I define $y(U)=|Tr(U^*V)|^2$ If I do the Gateaux derivative: $y_U[\tilde{U}] =\frac{d}{d\epsilon}|Tr((U^*+\epsilon\tilde{U})V)|^2$, Here is where I get confused because of how things are nested ...
How can i approach finding the Gateaux-derivative $F'(a)(r)$ where $F(a)=b$? I tried $F'(a)(r)=\frac{F(a+rt)-F(a)}{t}=\frac{F(a+rt)-b}{t}$ but I'm not sure how to continue or how to simplify this.