# Tagged Questions

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### Fréchet normal cone

Given $x\in \Omega(\subset X)$ (X: Banach space) and $\varepsilon\geq 0$, the set of $\varepsilon-$normals to $\Omega$ at $x$ by \begin{align} \widehat N_\varepsilon(x;\Omega):=\left\{x^*\in X^*\mid ...
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### Prove that the maximizing point configuration on the unit circle for a Vandermonde like functional is a picket fence

For $\lambda_i \in S^1 \subset \mathbb{C}$, consider the functional $H(\{\lambda_1, \ldots, \lambda_n\}):= \sum_{j < k} | \lambda_j - \lambda_k |$. I want to show that $H$ is globally maximized by ...
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### Mathematical question concerning Lagrange multipliers of a Lagrangian

In Lagrangian Mechanics I have in general holonomic constraints of the form $f(q_1,...,q_n,t)=0$ and then I am able to use the method of Lagrange multipliers, where I go from a Lagrangian $L$ to a ...
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### I need clarification on $\delta$ - derivative

Please can someone tell me more about $\delta$ -derivative ($\delta=x\dfrac{d}{dx}$) as it appears in the Hadamard definition of frational derivative or elsewhere. Why, when or where we use it. Does ...
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### Does setting derivative to zero suffice always for minimization of convex functions?

I have this convex function in $X$, given by $Trace(AX^TBX)$ where $A$, $B$ are p.s.d and all entries are real. Now if I had a linear function $l(X)$ that prevents a trivial zero-matrix solution for ...
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### Vector Field Generating Variation Along Curve

I'm learning a proof of the fact that length extremising curves are geodesics of the Levi-Civita connection, and have found something I don't understand. The argument states the following. Suppose ...
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### Curvature and Torsion problem

Calculate the curvature and torsion of $$x= e^t\sin(t),\quad y= e^t\cos(t),\quad z= e^t$$ I'm not sure if I am doing this correctly since I am getting quite complicated results. But I understand ...
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### Multiple Integral Equation

$$f(x) = 2a \int_{0}^{x}{f(t)\;dt} - \left(\frac{b^2}{2}\right)\int_{0}^{1}{|x-t|f(t)\;dt}$$ where $0<a<b$ My task is to solve for $f(x)$. I'm having difficulty solving this integral equation. ...
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### Derive the solution to the Lagrangian $\mathcal L= y(x)\sqrt{1+y'(x)^2}$

I am supposed to derive the solution to the Lagrangian $$\mathcal L= y(x)\sqrt{1+y'(x)^2}$$ Unfortunately I am unable to solve both, the Euler Lagrange equation or the Beltrami equation. It may be ...
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### Optimizing a functional with a differential equation as a constraint

I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense. We have a parametric ...
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### Extremal condition calculus of variations

if I have a functional with a Lagrangian $L(t,x(t),y(t),x'(t),y'(t))$, meaning two functions x and y of one parameter t. And want to solve the minimization problem $$\int_0^t L \, dt.$$ Then I get ...
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### Calculus of Variations-question on rotating curve of max volume

My calc of variations is still rusty. I'm assuming implementation of arclength revolution formula is necessary, but how to find y(1/2a)?
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### Calculus of Variations - Circles with soapy membrane problem [duplicate]

Calculus of variations is coming to me at a crawl pace. Here is a problem on my agenda that I wanted to get solved, but am not quite sure how to approach. I've been thinking about it for a while ...
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### Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$

Suppose that we have a function $f(x,y)$ of two variables: $$f(x,y) = g(x) + h(y) + 5(x-y) = x^2 + y^2 + 5(x-y)$$ where $g(x) = x^2$ and $h(y) = y^2$ are also functions of $x$ and $y$, respectively. ...
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### derivatives manipulations

How to show this: $f(x,y+en,y'+en')-f(x,y,y')= en(df/dy)+e(dn/dx)(df/dy')+O(e^2)$ y and n are functions of x, e small constant And y is smooth. What identities or properties are used here?
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### Proving that this function must be even (II)

Suppose $g:\mathbb{R}^d\rightarrow\mathbb{R}$ is continuous. Also let $\mathbf{x}=(x_1,\ldots,x_d)\in\mathbb{R}^d$. I'd like to prove the following: If ...
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### Proving that this function must be even

Let $u:\mathbb{R}^d\rightarrow\mathbb{R}$ be a continuous function that is not identically equal to zero. Suppose further that $u$ is an odd function (ie. $u(\mathbf{x})=-u(-\mathbf{x})$). Let ...
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### Differentiating with respect to a function using variable transformation

Earlier I asked a question about differentiating $f(x,y)$ with respect to $x-y$. I am working on the solutions trying to use the hints from earlier questions. Is it correct to do the following: ...
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### Pf. of weak lower semicontinuity for convex Lagrangians

This question is about the proof of Theorem 1 in Chapter 8 of Evans' PDE book (p. 468 in the 2nd edition). Let $u,u_k\in\mathrm{W}^{1,q}(U)$ for all $k\in\mathbb{N}$, $U\subset\mathbb{R}^n$ be open, ...
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### Pre-requisites for the Calculus of Variations

I'm interested in working through the book : "Calculus of Variations" by Gelfand and Fomin. However, I lack the pre-requisites to do so (I'm familiar with linear algebra and one-variable calculus ...
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### Lagrange multipliers — What have I done wrong?

I am trying to find the stationary points of the potential $U(x,y)=x^2+y^2$ with constraint $x^2-2y^2=1$ So I set the Augmented potential $U^*=x^2+y^2+m(x^2-2y^2)$ where $m$ is the Lagrange ...
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### What's the largest possible volume of a taco, and how do I make one that big?

Let $f$ be a continuous, even, positive function over some interval $I=[-a,a]$ such that the total arc length of $f$ over $I$ is at least $2$, $f(0)=0$, and $f$ is increasing on $(0,a)$. View the ...
I am attempting to calculate the functional derivative of a functional $$E[\rho] = \int G(\rho(\mathbf{r}),\nabla\rho(\mathbf{r}),\mathbf{r})d\mathbf{r},$$ where ...
I am trying to understand a proof for critical points of certain energy functions being harmonic functions. It goes as follows: For a function $u(x_1,..,x_n)$, a functional E(u) is defined as \$E(u) ...