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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
47 views

On definition of Gâteaux derivative

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\...
Hải Nguyễn Hoàng's user avatar
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)}{...
Esteban LS's user avatar
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 ...
Soumya Ganguly's user avatar
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 ...
MWilk's user avatar
  • 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 $g(x):{\mathbb{R}^d}\mapsto \mathbb{R}$...
matteogost's user avatar
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 ...
matteogost's user avatar
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 ...
kaithkolesidou's user avatar
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, ...
Lucas Myers's user avatar
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
VadimStacheff's user avatar
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(...
VadimStacheff's user avatar
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 ...
Purity's user avatar
  • 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) ...
Abm's user avatar
  • 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 ...
Alfons Winkel's user avatar
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 $u$ is defined as follows: \begin{align} \Delta_u f(...
Lisa's user avatar
  • 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 $\theta_{1,2}$...
Vogtster's user avatar
  • 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\...
hoh hoh's user avatar
  • 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 $F \in \mathcal{F}$...
improbable_probabilist's user avatar
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 $...
Yongyi Yang's user avatar
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 ...
lesssugarlightice's user avatar
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^{\...
Akira's user avatar
  • 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 ...
amilton moreira's user avatar
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 &...
MarcoDJ01's user avatar
  • 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 $X$...
WillG's user avatar
  • 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 $\det'(I)=\operatorname{tr}$,...
WillG's user avatar
  • 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 ...
FuncAna09's user avatar
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 ...
Kolodez's user avatar
  • 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 ...
stowo's user avatar
  • 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. ...
Markus Klyver's user avatar
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 \} \...
blamethelag's user avatar
  • 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 &...
Akira's user avatar
  • 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: \begin{equation} df(u,\psi)=\dfrac{d}{d\epsilon}f(u+\epsilon\psi)\Bigg{|}_{\epsilon=0} \end{equation} Why is the ...
Johann Wagner's user avatar
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 ...
ViktorStein's user avatar
  • 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 ...
Smilia's user avatar
  • 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{\...
Silentmovie's user avatar
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 $...
Arbiter's user avatar
  • 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) =...
Arbiter's user avatar
  • 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 ...
Curious Prob's user avatar
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 (...
NDewolf's user avatar
  • 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 $$F(x)(\psi)=f(x(\psi))$$ ...
carlos85's user avatar
  • 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 ...
Falcon's user avatar
  • 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' ...
Idontgetit's user avatar
  • 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 $ \...
Ken Hung's user avatar
  • 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}{...
manuel459's user avatar
  • 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 $$\...
wlad's user avatar
  • 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 \...
Anacardium's user avatar
  • 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 ...
Leon Avery's user avatar
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 $a \in I$, ...
Ulivai's user avatar
  • 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(...
Pazu's user avatar
  • 1,077
0 votes
1 answer
102 views

Derivative of magnitude of complex trace (function of matrices with Gateaux derivative)

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 ...
Vogtster's user avatar
  • 663
1 vote
0 answers
71 views

How to find the Gateaux derivative when F(a)=b

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.
Real struggle's user avatar