1
vote
0answers
28 views

Finding maximum rate of change of total derivatives

consider $PV =nRT , P,V,T =$ pressure , volume , temperature respectively. $nR =$ constant let $n=R=1$ differentiate with respect to $t$ (time) $dP/dt = ∂P/∂T * dT/dt + ∂P/∂V * ...
1
vote
0answers
17 views

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 ...
0
votes
0answers
57 views

Derivation of Euler Lagrange Equation

I was reading on the derivation of the Euler Lagrange Equations (in the link: http://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation focusing on: "Derivation of one-dimensional Euler–Lagrange ...
0
votes
1answer
27 views

how to introduce time into calculus of variations for image processing?

I'm studying some topics about calculus of variation applied to image processing. I'd like to understand how to introduce time parameter to evolve an image in an iterative way. For example, let's ...
2
votes
0answers
33 views

Sufficient conditions for the use of the Beltrami identity

For reference, I shall use the notation used in the wikipedia article for the Beltrami identity in the application section (http://en.wikipedia.org/wiki/Beltrami_identity) In the article, the ...
0
votes
1answer
39 views

Counterexample for the Chain rule for the Gateaux-derivative

I'm reading the book of Drabek, Milota - Methods of Nonlinear Analysis, and at page 121, they state: but I can't manage to find such counterexample. For clarity the Gateaux derivative is defined ...
1
vote
0answers
42 views

Can the Euler-Lagrange equations be derived from a variation over a time of order $dt$ rather than $t$?

In the calculus of variations, the solution of the Euler-Lagrange equations gives those functions for which a given functional is stationary. Now all derivations I've come across up to now, carry out ...
0
votes
1answer
37 views

Restricted function

Let $A=\{(x,y): x,y\in(-1,1)\}$. Is there a function $f:A\mapsto A$ such that $f(x,0)=(x,x^2)$ $f$ differentiable and bijective on $A$. I have tried a lot of constructions but the problem is in ...
0
votes
1answer
44 views

How to compute $\frac{\partial F_x}{\partial F}$?

If $F$ is a function of $x$ and $y$, and $F_x$ denotes $\dfrac{\partial F}{\partial x}$, how would you compute $\dfrac{\partial F_x}{\partial F}$? Can I use the chain rule to write this as: ...
2
votes
0answers
39 views

Cancelling some dx's

How can you prove $$\frac{\partial f}{\partial y} = \frac{d}{d x} \left( \left( \frac{\partial}{\partial \frac{dy}{dx} } \right) \right)f ?$$ It's tempting to cancel the two $dx$'s, but can ...
1
vote
0answers
29 views

Multivariable transversality conditions for infinite time horizon in variational calculus?

I would like to find the optimal paths of $d[t]$ and $k[t]$ that maximize the following function: $$J(x)=\int_0^\infty e^{-pt}(d[t]-d[t]^2-d'[t]^2+k[t]-k[t]^2-k'[t]^2+d[t]k[t])$$ Using Mathematica ...
5
votes
1answer
56 views

Show that $ M$ is constant on $[a,b]$ (variational calculus)

Let $F:\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}$ be $C^2$ on $[a,b]$ and $u$ be a solution for the Euler-lagrange equations for the functional given by $$J(u) = \int F(u(t),\dot{u}(t)).dt, $$ ...
1
vote
1answer
61 views

To show that $ J(u) = \frac{1}{2}\int_\Omega |\nabla u|^2 -\frac{1}{p+1}\int_\Omega |u|^{p+1} $ is not bounded above for $1 < p < 2^*-1$

For a bounded $ \Omega\subset\mathbb{R}^n $ with smooth boundary, and for $ 1 < p < \frac{n+2}{n-2} = 2^* -1 $ where $ \frac{1}{2^*} = \frac{1}{2}-\frac{1}{n}$, I have the functional $ J : ...
1
vote
1answer
70 views

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 ...
0
votes
1answer
61 views

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 ...
2
votes
1answer
103 views

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 ...
2
votes
2answers
94 views

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 ...
1
vote
1answer
312 views

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 ...
3
votes
1answer
157 views

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. ...
3
votes
2answers
104 views

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 ...
4
votes
1answer
294 views

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 ...
1
vote
0answers
139 views

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 ...
1
vote
1answer
98 views

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)?
3
votes
0answers
54 views

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 ...
1
vote
1answer
901 views

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. ...
1
vote
1answer
45 views

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?
1
vote
0answers
66 views

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 ...
5
votes
1answer
90 views

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 ...
1
vote
0answers
188 views

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: ...
2
votes
1answer
66 views

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, ...
0
votes
0answers
200 views

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 ...
2
votes
1answer
84 views

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 ...
54
votes
1answer
2k views

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 ...
2
votes
1answer
507 views

Troublesome functional derivative: second term of Euler-Lagrange equation

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 ...
3
votes
2answers
426 views

Divergence Theorem, Laplacian, Energy Minimization

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) ...