# Tagged Questions

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### Integral invariant under parametrization

Consider a continuous function $F(z,p)\colon \Omega\subset\mathbb{R}^N \times \mathbb{R}^N \to \mathbb{R}$ and the functional $$\mathcal{F}(u)=\int_{a}^{b}{F(u(t),u'(t))\,dt}.$$ Prove that ...
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### Partial Derivative of Integration

Suppose that I have some set of weight functions, $W = \{w_1(i,j), w_2(i,j),..., w_k(i,j)\}$, where each weight function is a Taylor polynomial in $\mathbb{R}^2$ with constants $c_{kn}$ where $n$ is ...
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This question was inspired a previous question of mine. If we are given that $\Omega \subset \mathbb{R}^{n}$ is open and bounded and $$\int_{\Omega}fv dx = 0$$ where $f \in C(\Omega)$ and $v \in ... 0answers 43 views ### First variation of convolution of two nonlinear functions, how to reexpress$\left[x \delta x * x^2 \right]$? A new variational principle is presented in this paper: Mixed Convolved Action When trying to derive something like the equation of motion of a Duffing oscillator, I take the following approach: Set ... 2answers 81 views ### Calculus of Variations In the Calculus of Variations there is a passage from Euler's characteristic equation: $$\frac {\partial F}{\partial y} - \frac {d}{dx} \left(\frac {\partial F}{\partial y'} \right)=0$$ in ... 1answer 73 views ### How to prove$\int_a^b f(x)\varphi(x)dx=0\Rightarrow f(x)=0$I am doing some reading on the calculus of variations and one of the first examples uses the following theorem: Let$f\in C[a,b]$. If$\int_a^b f(x)\varphi(x)dx=0$for all$\varphi\in C[a,b]$, then ... 1answer 95 views ### Fundamental lemma of calculus of variations, gradients Let$D \subset \mathbb{R}^d$be a smooth bounded domain. Let$C_c^\infty(D)$denote smooth and compactly supported functions on$D$. Let$f \in [C_c^\infty(D)]^d$be a smooth, compactly supported ... 0answers 61 views ### Extremal of functional$ I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy} I have the following functional: $$I\left[ y(x) \right] = \int_{0}^{\frac{\pi}{2}} {\left((y')^2 - y^2 + 2xy\right)dy}$$ subject to boundary conditions: \begin{align} y(0) &= 0 \\ ... 2answers 78 views ### Euler-Lagrange problem solution Hi, Can anyone solve this question? I have no clue. 1answer 94 views ### Maximizing score in number-guessing game. This is inspired by a puzzle (related to the two-envelopes problem) that I've seen in several places, including unbounded generalizations. The basic premise is that Alice chooses two real numbers ... 0answers 157 views ### How to take the limit of some integral? f\left( x^{\prime },t+\varepsilon \right) = \int_{-\infty }^\infty dx\int_{-i\infty }^{i\infty } \frac{d\tilde{x}}{2\pi i} \left(1+\varepsilon \left[ \tilde{x}D_{1}\left( x,t\right) ... 1answer 95 views ### Representation for function (“null-Lagrangian”) LetL(t,x,p) \in C^m([0,1] \times \mathbb{R}^n \times \mathbb{R}^n;\mathbb{R})$,$m\geqslant1$and define for any$u \in C^1([0,1];\mathbb{R}^n)$$\mathcal{L}u = ... 0answers 62 views ### How can I integrate this? Let \Omega\subset\mathbb{R}^N be a bounded domain and \phi_1,v,\phi\in W_0^{1,p}(\Omega) with p\in (1,\infty). How can I evaluate the integral:$$\int_0^1F(s)dswhere ... 0answers 195 views ### How to find \kappa to minimize integral I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) \mathrm{d}x I am trying to find such value \kappa \in (0,1) that would minimize the integral \begin{aligned} I = \frac{1}{\kappa}\int\limits_{0}^{T} \mathrm{exp}\left(-f(\kappa,x)\right) ... 0answers 29 views ### variation of a final state due to changes in period (where the period is a parameter) I have a simple ordinary differential equation \frac{dx}{dt}=f(x,t,p,T) x(0) = x_0, x(T) = x_T where p and T are constant parameters. How do I compute \frac{dx_T}{dT} ? Thanks! NOTE: I ... 1answer 273 views ### Separation of variables and substituion; first integral from the Euler-Differential Equation for the minimal surface problem Let P_1=(a,y_a),P_2=(b,y_b), y\in C^1 (a,b), y_a>0,y_b>0 And the area integral: \int^b_a y(x) \sqrt{1+y'(x)}dx From the Euler differential-equation we obtain:y'=1/\alpha ... 0answers 399 views ### Finding a proper solution of a given functional It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ... 1answer 202 views ### Find function to make maximum value Let{f : [0, 1] \rightarrow [-1, 1] }$is a continuous function such that${ \int_{0}^{1} x f \left(x\right) dx =0}$Find$f(x)$such that${ \int_{0}^{1} \left(x ^{2 } + \frac{1}{4} \right) f ...
Suppose I wish to find the Euler-Lagrange equation for an integral $\int_V f(u,\mathop{\mathrm{grad}} u)\,dV$ where $V$ is a volume given by some equation, for example say $x^2+y^2+z^2\le 1$, and ...
I came across the following "functional" at work: $$\Pi [b]=\int_0^\infty\int_0^{\lambda b(v,\lambda)} vf(v,\lambda) \; dv \; d\lambda$$ it's part of an optimization problem that tries to find ...