Functionals Definition and 40 Threads

In section 1.12 Variational Approach to the Solution of the Laplace and Poisson Equations, Jackson mentions that in electrostatics, we can consider "energy type functionals". He gives, for Dirichlet Boundary Conditions, $$I[\psi]=\frac{1}{2}\int_{V}\nabla\psi\cdot\nabla\psi d^3x-\int_{V}g\psi...

I not very good at using the LaTex editor, so I took a photo of my HW questions. For the first question, I'm not really sure how to get started, should I write out a specific case? Like what would $$\varphi (P)$$ be when m=1? For the second question, I know that a linear functional have two...

All these concepts belong to the toolbox of physicists. I read them quite often on our forum and their usage is sometimes a bit confusing. Physicists learn how to apply them, but occasionally I get the impression, that the concepts behind them are forgotten. So what are they? Especially when it...

Taken from Emmy Noether's wonderful theorem by Dwight. E Neuenschwander. Page 28 1. Homework Statement Under what circumstances are these definite integrals functionals; a) Mechanical work as a particle moves from position a to position b, while acted upon by a force F...

Homework Statement Let V be a vector space over R. let Φ1, Φ2 ∈ V* (the duel space) and suppose σ:V→R, defined by σ(v)=Φ1(v)Φ2(v), also belongs to V*. Show that either Φ1 = 0 or Φ2 = 0. Homework Equations N/A The Attempt at a Solution Since σ is also an element of the duel space, it is...

Homework Statement Suppose u,v ∈ V and that Φ(u)=0 implies Φ(v)=0 for all Φ ∈ V* (the duel space). Show that v=ku for some scalar k. Homework Equations N/A The Attempt at a Solution I've managed to solve the problem when V is of finite dimension by assuming u,v are linearly independent...

Hi, I have a probably very stupid question: Suppose that there is an expression of the form $$\frac{d}{da}ln(f(ax))$$ with domain in the positive reals and real parameter a. Now subtract a fraction ##\alpha## of f(ax) in an interval within the interval ##[ x_1, x_2 ]##, i.e. $$f(ax)...

I am very much struggling with this problem: The set $\{\sin x, \cos x, x \sin x, x \cos x, x+2, x^2-1 \}$ on interval of $[0, \pi]$ is linearly independent and generates vector space $V$. Find the dimension of the kernel of the Dirac functionals in $V$. Here are what I know of the definitions...

I have reading through various sources on linear functionals, but all seem somewhat inconsistent with regard to denoting the set of all linear functionals and the set Also, what is the standard definition of a continuous linear functional? I really couldn't find much besides this Let ##f : V...

Consider the following problem: ##A## is a functional (some integral operator to be more specific) of a (complex) function ##F##. We want to minimize ##A[F]## wrt. to a constraint ##B[F]=\int (|F|²)=N## If I read around online I find that in general such extremization problems are done by...

So yesterday I learned about functionals, which my book claims are "machines that take a function and return a number", in contrast to functions, which take a number and return another number. I immediately thought of definite integration: it's an operation that takes a function, and returns a...

Homework Statement I have been given a functional $$S[x(t)]= \int_0^T \Big[ \Big(\frac {dx(t)}{dt}\Big)^{2} + x^{2}(t)\Big] dt$$ I need a curve satisfying x(o)=0 and x(T)=1, which makes S[x(t)] an extremum Homework Equations Now I know about action being $$S[x(t)]= \int_t^{t'} L(\dot x, x)...

First of all, apologies if this isn't quite in the right section. I've been studying functionals, in particular pertaining to variational calculus. My query relates to defining a functional as an integral over some interval x\in [a,b] in the following manner I[y]= \int_{a}^{b} F\left(x, y(x)...

I have been studying calculus of variations and have been somewhat struggling to conceptualise why it is that we have functionals of the form I[y]= \int_{a}^{b} F\left(x,y,y' \right) dx in particular, why the integrand F\left(x,y,y' \right) is a function of both y and it's derivative y'? My...

My question is on using a form of the single variable Noether's theorem to remember the multiple variable version. Noether's theorem, for functionals of a single independent variable, can be translated into saying that, because \mathcal{L} is invariant, we have \mathcal{L}(x,y_i,y_i')dx =...

Hello guys, I posted this question in the classical mechanics forum (thinking stochastic mechanics falls into the classical mechanics category) but I haven't gotten an answer there. I was told I'd be better off posting the question here. I don't know how to move a thread, so I'll just copy and...

Are functionals a special case of operators (as written on Wiki)? Operators are mappings between two vector spaces, whilst a functional is a map from a vector space (the space of functions, say) to a field [or from a module to a ring, I guess]. Now, the field is NOT NECESSARILY a vector...

Hello guys, Recently I came across a definition to which I'd never given much thought. I was reading through Gelfand and Fomin's "Calculus of variations" and I read the part about weak and strong extrema, and I really can't manage to wrap my head around these definitions. They can be found in...

I read that B3LYP is a hybrid functional which uses some HF method and some DFT method for its calculations. According to this page: my professor told me that you can set the proportions of each method that the B3LYP uses yourself, so for example you can make it so it uses 70% HF and 30% DFT or...

Is this "theorem" true? Relationship between linear functionals and inner products Suppose we have a finite dimensional inner product space V over the field F. We can define a map from V to F associated with every vector v as follows: \underline{v}:V\rightarrow \mathbb{F}, \ w \mapsto \langle...

Hello, Problem, let B={a_1,a_2,a_3} be a basis for C^3 defined by a_1=(1,0,-1) a_2=(1,1,1) a_3=(2,2,0) Find the dual basis of B. My Solution. Let W_1 be the subspace generated by a_2=(1,1,1) a_3=(2,2,0), let's find W*, where W* is the set of linear anihilator of W_1. Consider the system...

Homework Statement Show that ## \displaystyle B_1(u,v)=\int_a^b (p(x) u \cdot v + q(x) \frac{du}{dx} \cdot v)dx## is a bilinear functional and is NOT symmetric Homework Statement Bilinear relation ##B(\alpha u_1+\beta u_2,v)=\alpha B(u_1,v) +\beta B(u_2,v)## (1) ##B(u, \alpha v_1+...

Homework Statement Why does this not qualify as a linear functional based on the relation ##l(\alpha u+\beta v)=\alpha l(u)+\beta l(v)##? ##\displaystyle I(u)=\int_a^b u \frac{du}{dx} dx## Homework Equations where ##\alpha## and ##\beta## are real numbers and ##u## , ##v## are...

Homework Statement Let C be the class of C1 functions on interval [0,1] satisfying u(0)=0=u(1). Consider the functional F(u)= 1 ∫[(u')2 + 3u4 + cosh(u) + (x3-x)u] dx 0 (note: u is a function of x.) Analyse the functional F term by term. Decide for each term whether it is convex or...

Homework Statement Prove that a continuous linear functional, f is bounded and vice versa. Homework Equations I know that the definition of a linear functional is: f( \alpha|x> + \beta|y>) = \alpha f(|x> ) + \beta f( |y> ) and that a bounded linear functional satisfies: ||f(|x>)) ||...

Hi By some googling it seems like there exist some kind of expansion of the Taylor series for statistical functionals. I can however, not sort out how it is working and what the derivative-equivalent of the functional actually is. My situation is that I have a functional, say \theta which...

I have a question: If x\in X is a normed vector space, X^* is the space of bounded linear functionals on X, and f(x) = 0 for every f\in X^*, is it true that x = 0? I'm reasonably confident this has to be the case, but why? The converse is obviously true, but I don't see why there couldn't be an...

I have a question about mappings that go from a vector space to the dual space, the notation is quite strange. A linear functional is just a linear map f : V → F. The dual space of V is the vector space L(V,F) = (V)*, i.e. the space of linear functionals, i.e. maps from V to F. L(V,F)=...

Let M be an n-dimensional manifold, with tangent spaces TpM for each point p in M. Let F(M) be the vector space of smooth functions M --> R, over R, with the usual definitions of addition and scaling. Tangent vectors in TM can be defined as linear functionals on F(M) (Fecko: Differential...

Suppose that \mathcal H is a Hilbert space, and that A:\mathcal H\rightarrow\mathcal H is linear and unbounded. Is it safe to conclude that y\mapsto\langle x,Ay\rangle is unbounded for at least one x\in\mathcal H? How do you prove this? (My inner product is linear in the second variable).For...

Are functionals united with the vector space which they operate on? For example, Physics is a functional of Behavioral Psychology. However, Behavioral Psychology does not include Physics. Am I correct? Thank you, Jake

Homework Statement Prove that if m<n, and if y_1,\cdots,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j]=0 for j=1,\cdots,m. What does this result say about the solutions of linear equations? Homework Equations...

Hello all, does anybody know what means duality pairing in connection with functional. For example limE\rightarrow0\frac{\partial}{\partialE}F(u+Ev)=<DF(u),v>. Where F is functional F:K\rightarrowR. Thank You for answers.

How do I solve this problem- I know it has something to do Riesz represenation but am having difficulty connecting dots Conside R4 with usual inner product. Find the linear funcitonal associated to the vector (1,1,2,2). What am I missing- is this problem complete or is there something...

Suppose we have a bounded linear functional f defined on L1 (the sequence space of all absolutely summable sequences) and we take the natural (Schauder) basis for L1, that is, the set of sequences (E1,E2,...,En,...) that have 1 in the n th position and everywere else zero. Pick x in L1. Then...

Assume that m<n and l_1,l_2,...,l_m are linear functionals on an n-dimensional vector space X . Prove there exists a nonzero vector x \epsilon X such that < x,l_j >=0 for 1 \leq j \leq m. What does this say about the solution of systems of linear equations?This implies l_j(x)...

I am studying for a final I have tomorrow in linear algebra, and I am still having trouble understanding linear functionals. Can someone help me out with this example problem, walk me through it so I can understand exactly what a linear functional is? Is the following a linear functional? \ y...

Here is the problem I have been asked to solve: Assume that m < n and l1, l2, . . . , lm are linear functionals on an n-dimensional vector space X. (a) Prove there exists a non-zero vector x in X such that the scalar product < x, lj >= 0 for 1 <= j <= m. What does this say about the solution of...

1) let S:U->V T:V->W be linear operators, show that: (ToS)^t=S^toT^t. 2) let T:V->U be linear and u belongs to U, show that u belongs to Im(T) or that there exist \phi\inV* such that T^{t}(\phi)=0 and \phi(u)=1 about the first question here what i tried to do: (ToS)^{t}(\phi(v))=\phi...

I'm not quite sure if this is a linear functional but the question asks: if L=D^2+4xD-2x and y(x)=2x-4e^{5x} I am to find Ly=? My first impressions to solve this is the take Ly=y''(x)+4xy'(x)-2x i'm not quite sure how to solve this but I got: y''(x)=-100e^{5x} y'(x)=-20e^{5x}+2...