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Question 1

1vote

1answer

91views

The Gateaux derivative and the Jacobian

QuestionIs it accurate to claim that the Gateaux derivative is the continuous linear operator represented by the Jacobian matrix? That is, the Jacobian is the finite dimensional equivalent of the ...

Jared

267

askedNov 21 at 13:58

Question 2

1vote

0answers

35views

Under what condition can we commute Gâteau-derivatives?

Let $\lambda \in X \subset C^2(\mathbb R^+)$, where $X$ is a closed subspace of $C^2$. Let us consider a function $\psi[\lambda]$ defined on $\mathbb R^2 \times \mathbb R^+$ such that$$(x, t) \mapsto ...

Falcon

4,536

askedOct 1 at 10:04

Question 3

2votes

2answers

233views

Bounded linear functional and Gateaux derivative

The following concerns Theorem 3.6.1 in Hsing and Eubank's Theoretical Foundations of Functiona Data Analysis, with an Introduction to Linear operators.Suppose $f: \mathbb{X}_1 \to \mathbb{X_2}$, ...

cmperez024

173

askedAug 20 at 18:05

Question 4

5votes

0answers

60views

Is the Gâteaux derivative of the Levi-Civita connection a smooth $(1,2)$-tensor field?

DefinitionSuppose $X,Y$ are locally convex topological vector spaces and $U\subseteq X$ is open. We say a functional\begin{align} J:\ U &\longrightarrow Y\\ u &\longmapsto J[u]\end{align}...

PermQi

905

askedAug 18 at 16:51

Question 5

2votes

0answers

80views

Symmetrization of functional derivatives

I'm dealing with a mathematical problem stemming from quantum field theory (QFT). However, at the moment, I'm not concerned with the physics aspect of it and, hence, I wish to view it in purely formal ...

user724377

askedAug 6 at 23:18

Question 6

2votes

1answer

98views

Gateaux vs Fréchet differentiability in FEM

I am trying to figure out the practical relevance of having only Gateaux differentiability vs Fréchet differentiability. As an example consider the Dirichlet energy$$E(u) = \frac{1}{2}\int_{\Omega} \|...

lightxbulb

2,438

askedMay 5 at 9:33

Question 7

1vote

0answers

41views

Stationary points - Gateaux vs Frechet derivative

In basic vector calculus one terms a point $f$ stationary for $E$ if $\nabla E(f) = 0$. On the other hand, in variational calculus we term $f$ stationary for $E$ if the first variations are zero at $f$...

lightxbulb

2,438

askedApr 25 at 17:19

Question 8

4votes

2answers

117views

What is the inverse operation of the Gateaux derivative?

ContextThis is purely a question of curiosity, I can't provide much context.I am taking a course in computational fluid dynamics to say that a class of problems can be classified as "saddle ...

Math Attack

5,677

askedApr 17 at 15:42

Question 9

2votes

0answers

92views

Looking for references on Gateaux derivatives, especially for physics applications

I am interested in finding literature on Gateaux derivatives, particularly in the context of variational methods in physics. I believe that by reformulating physical variational principles using ...

neconoco

71

askedMar 22 at 2:47

Question 10

5votes

2answers

401views

Proving the Chain Rule for Gateaux Differentiation

For functions $f:\mathbb{R}^n\to\mathbb{R}^m$, the Chain Rule can be stated both in terms of the total derivative or in terms of partial derivatives. $\def\bU {\textbf{U}} \def\bV {\textbf{V}} \def\...

Sam

5,300

askedMar 12 at 18:06

Question 11

0votes

0answers

31views

Gâteaux derivative of an integral for the Levelset method

I'm trying to understand a step in the paper Distance Regularized Level Set Evolution and its Application to Image Segmentation. Given is a domain $\Omega \subset R^2$ and a continuously ...

Pablo Jeken Rico

253

askedFeb 27 at 11:58

Question 12

0votes

0answers

19views

Choice of terminology (Critical Points v.s Stationary Points or etc.) in the calculus of variations

Given a Banach space $X$ and an open subset $U\subset X$, a functional $\mathscr{F}:U\mapsto \mathbb{R}$. What is the standard terminology of the point $u\in U$ such that the first variation $\delta\...

Silentmovie

348

askedJan 16 at 5:20

Question 13

0votes

1answer

81views

The meaning of partial derivatives with respect to Gateaux derivative

I've seen the Gateaux Derivative defined as:Let $X$ be a Banach space, $u,v \in X$, and $E: X \to \mathbb{R}$. The Gâteaux Derivative of $E$ at $ u \in X$ is defined as:$$ \delta E(u ; v) = E'(u ...

blz

657

askedNov 18, 2024 at 12:23

Question 14

1vote

0answers

61views

Fréchet differentiability question

I have quite a difficult question regarding Fréchet and Gateaux differentiability. Suppose we have $T: L_2[0,1] \to L_2[0,1]$. $Tx(t) = \sin{x(t)}$. How to show that T is Gateaux differentiable ...

ed199957

67

askedOct 13, 2024 at 13:07

Question 15

3votes

1answer

86views

Gateaux differentiability of squared $\ell_p$ norm

Consider the Euclidean vector space $\mathbb{R}^n$ and the squared $p$-norm $||\cdot||_p^2$ for ($1<p<\infty$). I'm trying to understand if $||\cdot||_p^2$ is Gateaux differentiable at any point ...

user224369

askedAug 1, 2024 at 1:21

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Questions tagged [gateaux-derivative]

The Gateaux derivative and the Jacobian

Under what condition can we commute Gâteau-derivatives?

Bounded linear functional and Gateaux derivative

Is the Gâteaux derivative of the Levi-Civita connection a smooth $(1,2)$-tensor field?

Symmetrization of functional derivatives

Gateaux vs Fréchet differentiability in FEM

Stationary points - Gateaux vs Frechet derivative

What is the inverse operation of the Gateaux derivative?

Looking for references on Gateaux derivatives, especially for physics applications

Proving the Chain Rule for Gateaux Differentiation

Gâteaux derivative of an integral for the Levelset method

Choice of terminology (Critical Points v.s Stationary Points or etc.) in the calculus of variations

The meaning of partial derivatives with respect to Gateaux derivative

Fréchet differentiability question

Gateaux differentiability of squared $\ell_p$ norm

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