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Questions tagged [frechet-derivative]

The Fréchet derivative of a function from an open subset of a Banach space into another Banach space at a point is a linear map from the first Banach space into the second one which approximates particularly well the function near the given point. It generalizes the concept of derivative of a real function of one real variable.

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Frechét Derivative of F: $g \mapsto \int_0^n g(x)p(x)dx$

I've been wondering what the Frechét Derivative of F: $g \mapsto \int_0^n g(x)f(x)dx$ is. By definition, the Frechét Derivative $D_{F,g}(g)$ is such that (omitting limits on integrals): $lim_{||g||\...
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Using chain rule to calculate Fréchet derivative of $F(X) = \det(A^T (I - X) A)$

Let $\mathbb{M}^n$ be the set of real $n \times n$ matrices, and let $A$ be a fixed real $n \times n$ matrix. Define the function $F: \mathbb{M}^n \rightarrow \mathbb{R}$ by $$ F(X) = \det( A^T (I-X) ...
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Check whether $F(x,y)$ is differentiable or not at $(0,0)$

So...I am having trouble checking whether $F(x,y)=e^{x+y}$ is differentiable or not on $(0,0)$ The partial derivatives for $x$ and $y$ on $(0,0)$ (if no mistakes were made) are $1$ for both of them. ...
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Directional derivative and Jacobian matrix

I have a problem with an exercise that goes as follows: Let $\mathbf{f}$ be a $\mathbb{R}^n\rightarrow \mathbb{R}^m$ function and $\mathbf{a}$ an interior point of the domain of $\mathbf{f}$. ...
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Differentiability, linear operators

Let $Y$ be a complete normed linear space, and let $M$ denote the space of bounded linear operators from $Y$ to itself. Let $L : M → M$ be the map defined by $L(A) := A^2$. I am supposed to show that ...
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Fréchet derivative of the energy functional

Let $\Omega \subset\mathbb{R}^n$ be an open set and $$E(u)=\frac{1}{2}\int_{\Omega} | \nabla u|^2 \quad (u \in H_0^1 (\Omega)). $$ Then, what is the Fréchet derivative of the functional $E$? And why? ...
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Frechet Derivative of a Function Containing an Integral Operator Applied to the Differentiated Term

I'm new to Frechet and Gateaux differentiation, this likely has some mistakes. I was hoping someone could check this and offer some guidance. Suppose we have the following functional: \begin{equation}...
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Mean Value Inequality in Banach Space without Hahn-Banach or Integrals

If $f : E \to F$ is a continuous map of Banach spaces, with bounded Fréchet derivative. Then $x_0,x_1 \in E\Rightarrow \|f(x_1) − f(x_0)\| ≤ M\|x_1 − x_0\|$ where $M = \sup \|f'(x)\|.$ The most ...
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Spivak Change of Variable

I was a bit confused about the following statement. On page 69, Spivak says "in fact, if T is the linear transformation $Dg(a)$, then $(T^{-1}\circ g)'(a)=I $. I may be applying the chain rule ...
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Going from the differential to the derivative (Frechet and matrix calculus)

For a function $f: A \rightarrow B$, Frechet differentiability tells us that we want to find a linear operator that satisfies $$\lim_{H\rightarrow 0} \frac{||f[X+H] - f[X] - G[H]||}{||H||} = 0$$ ...
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Confused about chain rule in Frechet derivative

I've recently started to learn about Frechet derivatives and now have a simple example which I'm not sure if I've solved correctly. To be honest, I've only got a poor understanding of how it works so ...
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Basic questions about Frechet derivatives

I had a couple of basic questions about the Frechet derivative. 1) The chain rule: If $F: X\rightarrow Y$ and $G: Y\rightarrow Z$, then $D(G\circ F)(x) = DG(F(x))DF(x)$. The left hand side has a ...
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Gateaux Derivative of Fourier Transform/Characteristic Function

Suppose that $X$ is a square integrable random-variable defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$. It's characteristic function/Fourier transform is defined to be $$ \mathfrak{...
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About the Frechet derivative of a functional

How can I compute the Frechet derivative of the functional $$ I(u)=\frac{1}{2}\int_{\Omega} \vert \nabla u\vert ^2\ dx \ + \ \int_{\Omega}\left[1-|u|^2\right]^2\ dx$$ for $u$ in a functional ...
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Differentiability of partial function function

Let $E,F$ be Banach spaces. If $f\colon [0,1]\times E\to F$ is continuously differentiable, can we say something about the differentiability of $$g\colon E\to C^1([0,1],F),\quad y\mapsto f(\cdot,y)?$$ ...
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Computing Frechet derivative

Let $F:C^{([-1,1])}\rightarrow C^{([-1,1])}$ defined as $F(q)(x)=q^{2}(-x)$. I want to find the derivative of this function my current stratergy is to use the directional derivative so $dF(q)(x)h= \...
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Differentiability at the origin of Norm

For $\alpha > 1$, show that $f:\mathbb{R}^n \to \mathbb{R}^n$, given by $$f(x)=\rVert x\rVert^\alpha$$ is differentiable at the origin. What I did: What is necessary is to find a linear ...
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Understanding Differentiability of an Inner Product map

I have been trying to solve the following question without success. Question: Show that $f:\mathbb{R^n} \times \mathbb{R^n}\to \mathbb{R}$, defined as $$f(x,y) = x\cdot y$$ is differentiable and ...
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Is the functional $w \mapsto \int_0^1 | \ \| w(t) \| - 1 | \ dt$ $C^1$ or even smooth?

Let $H:= H(I;\mathbb{R}^3)$ be the space of $L^2$ + absolutely continuous functions with $L^2$ derivative. For $w \in H$ consider the functional $$\psi(w) = \int_0^1 | \ \| w(t) \|^2 - 1 | \ dt$$ ...
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L'Hospital Rule for Vector Valued Functions

Question: Using Rudin's formulation of the L'Hospital Rule, show that it remains true if a) $f:(a,b)\to \mathbb{R^k}$ is vector-valued and $g:(a,b)\to \mathbb{R}$ b) $f:(a,b)\to \mathbb{C}$ and $g:(...
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How to prove that Frechet derivative exists and coincides with the Gateau derivative?

Let $X$ and $Y$ be Banach spaces and $U \subseteq X$ be open.Let $F:U \to Y$ be Gateaux differentiable and let the mapping $x \to F′(x)$ be continuous from $U \in L(X,Y)$. How can I prove that the ...
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Fréchet Derivative of functionals which depend on the derivative

Let $\Omega:=[x_a,x_b]\times[t_a,t_b]\subset \mathbb{R}^2$ and consider the functional $D:C^2\left(\Omega,\mathbb{R}\right)\rightarrow \mathbb{R}$ such that $$D[y]= \iint_{\Omega}d(x,t,y) \, dx \, dt$$...
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Existence of continuous derivatives of partial functions and total differentiability

We know that for a function $\mathbb{R}^m\to \mathbb{R}^n$ the existence and continuity of partial derivatives implies the differentiability of that function. Will this hold true for functions on ...
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Partial Derivative of Frechet-differentiable Function

Let $X,Y,Z$ be Banach-Spaces and $F:X \times Y \rightarrow Z$ Frechet-differentiable. Then it holds $$ F'(x,y)(u,v) = F_x (x,y)u+ F_y(x,y)v .$$ How do I prove this?
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What does the reparameterization mean in Fréchet distances?

I am trying to understand the definition of frechet distance but I am struggling to understand the reparameterization part in the definition. I got the following definition from wikipedia Let A and ...
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Help showing derivative is compact

I'm struggling with proving the following theorem: $\textit{Let }A:U\subset X\rightarrow Y\textit{ be a completely continuous operator from an open subset U of a normed}$ $\textit{space X into a ...
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Calculation of Gateaux Derivative

I would like to ask some details for the calculation (and the justification) of the following problems. Let us denote $u^{+} =\max\{u,0\}$. Take $u,v \in H_{0}^{1}(\Omega)$ for a bounded domain $\...
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Why is there an “implication” rather than and “and” in this definition of the derivative?

I am readig Pugh's Analysis book: Definition Let $f:U \to \mathbb{R}^m$ be given where $U$ is an open subset of $\mathbb{R}^n$. The function $f$ is differentiable a $p \in U$ with derivative $(...
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Frechet derivative of a function on matrix-valued $L^2$ functions

$\newcommand{\tr}{\text{Tr}}$ Let us denote $L^2(X, \mathbb C^{n \times n})$ be the space of matrix-valued $L^2$ functions. That is, if $f \in L^2(X, \mathbb C^{n \times n})$, each entry of $f$ is an $...
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How to find Frechet derivative of $f(x)=\|Ax-b\|^2$ at any $x^*$?

Given a real $m \times n$ matrix $A$ and $b \in \mathbb{R}^m$, let $f(x)=\|Ax-b\|^2$ for any $x \in \mathbb{R}^n$. Find Frechet derivative of $f(x)=\|Ax-b\|^2$ at any $x^*$? Actually I am wondering ...
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Showing that a given function is Frechet differentiable

I'd like to do the following exercise from my book. The statement is as follows. Let $\gamma \epsilon { C }^{ 1 }(R\times { R }^{ n })$,$c>0$ and ${ \gamma }_{ 0 }\epsilon { L }^{ 1 }({ R }^{ n })...
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Showing a function is Frechet Differentiable?

I just started learning the Frechet Derivatives. So I have a function $H:\mathbb{R}^{N\times n}\to\mathbb{R}^{N\times n}$, i.e. $U^T\in\mathbb{R}^{N\times n}$ and $$H(U^T)=GW\times (F(U))^T+S\times U^...
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Derivative of a functional with respect to another functional

I am trying to make sense of functional derivatives and have a couple of questions bothering me: Let $F[X]$ be a functional of $X(t)$ and $G[X]$ another functional of $X(t)$. By chain rule in the ...
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Fréchet derivative: dependency of the choice of norm

Let $ f_1: L^2(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $, $ f_2: H^1(\Omega)\rightarrow\mathbb{R}, u\mapsto \Vert u\Vert_{L^2 (\Omega)}^2 $ and $f_2: H^1(\Omega)\...
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Does $[F'(u^*)]^{-1}$ exist if $u^*$ is a solution of $\partial_t u + [u+1-t] \partial _x u -1 = 0$?

Given $$F(u):=\partial_t u(x,t) + [u(x,t)+1-t] \partial_x u(x,t) -1 $$ we can calculate the derivative of $F$ and get that the derivative of $F'(u)$ applied to $v$ gives us $$F'(u)(v)= \partial_t ...
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minimize norm $||C-A^TB ||^2$ by taking derivative with respect to vector

Let's try to minimimize a matrix function $f(B)=||C-A^TB ||$ with respect to $B$. Here $C_{k \times 1}$ and $A_{3 \times k}$ and $B_{3 \times 1}$ matrices. One solution presented to me was to take ...
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Computing Fréchet derivative of $F(f)(x) = \int^{x}_{0} \cos(f(t)^{2})dt, x \in [0,1]$

Let $X = \mathcal{C} \left( [0,1] \right)$ be the Banach space of continuous functions on $[0,1]$ (with the supremum norm) and define a map $F : X \rightarrow X$ by $$F(f)(x) = \int^{x}_{0} \cos(f(...
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Proving the product rule for Fréchet Derivative.

Let $X$ be a normed vector space, $U\subset X$, and $F,G:U\rightarrow \mathbb{R}$ differentiable at $x\in U$. Show that the map $F\cdot G:U\rightarrow \mathbb{R}$, $F\cdot G(x)=F(x)G(x)$ is also ...
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Frechet derivative of basic derivative

I was reading about functional analysis and it interested me greatly applying the techniques of analysis to function spaces but it seems to have abruptly stopped when getting to the derivative. I know ...
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Higher total derivatives (Frechét derivatives)

This issue I just cannnot resolve, so I'd highly appreciate your help. Let $a_1, ... , a_n \in \mathbb{R}^k$ with $k$ a natural positive number. If we consider the function $$ W: \mathbb{R}^k \to \...
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Fréchet derivative; open set

I have some problems in understanding the definition of the Fréchet derivative of an operator $F: X \rightarrow X$. In fact, most authors report that $F$ must be defined in some open neighborhood of ...
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Directional derivatives of matrix trace functionals

Let $P$ be an $n\times n $ positive semidefinite matrix over $\mathbb{C}$, let $p\in\mathbb{R}$ be in the range $0<p<1$. Consider the function $g:[0,\infty)\rightarrow\mathbb{R}$ defined by $g(x)...
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Explaination of the Frechet Derivative

Let $E$ and $F$ be Banach spaces and let $U$ be an open subset of $E$. Suppose $g:U \to F $. $g$ is continuous at $x_0$ if there exists a linear transformation, $T_{x_0}$, such that $$ \lim \limits_{...
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The matrix $I_n - v^t x$ is invertible when $\langle v, x \rangle \neq 1$

Let $v \neq 0$ be any vector in $\Bbb R^n$ and let, $U_v = \{x \in \Bbb R^n : \langle v,x \rangle \neq 1\}$. Then: (i) Show that the matrix $I_n - v^t x$ is invertible $\forall x \in U_v$. (...
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Lagrange multiplier method on Banach spaces (in Dirac notation)

I want to prove Cauchy–Schwarz' inequality, in Dirac notation, $\left<\psi\middle|\psi\right> \left<\phi\middle|\phi\right> \geq \left|\left<\psi\middle|\phi\right>\right|^2$, using ...
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113 views

Application of Higher Order Derivatives

If $V = \mathbb{R}^{n}$ and $W = \mathbb{R}^{m}$, and $f:V\rightarrow W$ is smooth, then higher order derivatives at $x\in V$, $D^{n}_{x}f$ can be thought of as symmetric, multilinear functions $V\...
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The relation between Frechet derivative and differentiation in $\mathbb{R}^2$

In the book of Complex Made Simple by Ullrich, at page 4, it is given that However, if we are considering $\mathbb{C} = \mathbb{R}^2$, then $Df$ is a $2\times 2$ matrix, and not a complex number, but ...
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If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [closed]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
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Linearity of the derivative operator of polynomials

Suppose $f(x)=x^i$ as a simple polynomial function of power $i$. As we know from the definition of the derivative operators, such an operator must be linear. How can we prove that the derivative ...
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“FTC”-type relationship between Frechet Derivative and Bochner integral?

Let $X,Y$ be Banach spaces and $(X,\Sigma,\mu)$ be a measure space. Then, for a function $f: X\to Y$ under suitable conditions, we can define the Frechet derivative $\text df: X\to B(X,Y)$ or $\text ...