Functional Definition and 380 Threads

Having some trouble determining the most mathematically correct way to express something I understand only numerically and physically. Basically I am modeling radiation within a volume. 1. For each point dV within a volume V there is a scalar value e(dV), which is an amount of radiation emitted...

Hey guys, should the -OH group in glycolic acid be considered a functional group or a substituent due to the presence of the carboxylic acid group?

Homework Statement [/B] Hi I am looking at this action: Under the transformation ## \phi \to \phi e^{i \epsilon} ## Homework Equations [/B] So a conserved current is found by, promoting the parameter describing the transformation- ##\epsilon## say- to depend on ##x## since we know that...

Hi, I have a blog oriented on computational physics: https://compphys.go.ro For many posts I have a GitHub project. Lately I started some DFT oriented ones, the latest being a DFT (with plane waves basis) project for a 'quantum dot'. Currently I started working on a project that will use the...

Hi, I am working on a home-task to analyse the properties of a ODE and its solution in a Hilbert space, and in this context I have: 1. Generated a matrix form of the ODE, and analysed its phase-portrait, eigenvalues and eigenvectors, the limits of the solution and the condition number of the...

Homework Statement "The functional ## J[f] = \int [f(y)]^pφ(y)\, dy ## has a functional derivative with respect to ## f(x) ## given by: $$ \frac {δJ[f]} {δf(x)} = \lim_{ε \rightarrow 0} \frac 1 ε \left[ \int[f(y) + εδ(y-x)]^pφ(y)\, dy - \int [f(y)]^pφ(y)\, dy\right] $$ $$ =...

Hello, I've a following question: Is necessary know well func. analysis, and all its theorems to handle well quantum physics...?

Homework Statement Let f((x+y)/2)= {[f(x)+f(y)]/2} for all real x and y {f'(x)=first order derivative of f(x)} f'(0) exists and is equal to -1 and f(0)=1. Find f(2) Homework Equations Basic formula for differentiablilty: f'(x)=limit (h tends to 0+) {[f(x+h)-f(x)]/h} The Attempt at a...

I am working on a lab where we were measuring how magnetic force between two parallel conductors varies with current and the separation between the two conductors. I need to find a formula for the dependence of Force on current to create a fit line on the data in my graph that shows the relation...

I have read the wiki page (https://en.wikipedia.org/wiki/Functional_(mathematics) but it is not helping. I understand what a regular function is input > do something to that input > output. but not what functional is, Wikipedia says "from a vector space into its underlying field of scalars"...

I'm studying QFT in the path integral formalism, and got stuck in deriving the Schwinger Dyson equation for a real free scalar field, L=½(∂φ)^2 - m^2 φ^2 in the equation, S[φ]=∫ d4x L[φ] ∫ Dφ e^{i S[φ]} φ(x1) φ(x2) = ∫ Dφ e^{i S[φ']} φ'(x1) φ'(x2) Particularly, it is in the Taylor series...

Hi PF I try to understand how we get get a Taylor expansion of a non linear functional. I found this good paper here F maps functions to scalars. F[f] is defined. It has not scalars as arguments. I agree with A13 and A18. In another paper (in french) skip to page 9 the fisrt term is ##\int dx...

I have ## \int_{t = 0}^{t = 1} \frac{1}{x} \frac{dx}{dt} dt = \int_{t = 0}^{t = 1} (1-y) dt ## [1] The LHS evaluates to ## ln \frac{(x(t_0+T))}{x(t_0)} ##, where ##t_{1}=t_{0}+T## My issue is that, asked to write out the intermediatary step, I could not. I am unsure how you do this when the...

Homework Statement theres a table x - 1;2;3;u;u;u;u;8 y - 370;160;90;u;24;u;u;u u - unknown value i have to complete this table with this equation y=(k/x)+b you have to find k and b values Homework Equations y=(k/x)+b x is relevant to y...

Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...

I'm starting out on DFT right now. I'm an experimental Physics student, so I'm not very familiar with theories. Can you recommend any good textbooks or resources that I can utilize for my study?? Thanks in advance.

In the calculus of variations, the integral itself is a "functional." It depends on the form of the function of the Lagrangian: q and q-dot But I have seen this word "functional" used elsewhere in different contexts. I have seen: "A functional is a real valued function on a vector space." I...

Homework Statement Homework EquationsThe Attempt at a Solution I counted 4 functional groups. I got: -Carboxylic acid -Ketone -Alcohol -Ether However, this combination is not available. I was wondering if phenol is a functional group as C seems the most likely option. I thought phenyl is a...

Homework Statement To show that ##\rho(p',s)>\rho(p',s') => (\frac{\partial\rho}{\partial s})_p\frac{ds}{dz}<0## where ##p=p(z)##, ##p'=p(z+dz)##, ##s'=s(z+dz)##, ##s=s(z)## Homework Equations I have no idea how to approach this. I'm thinking functional derivatives, taylor expansions...

Hello everyone, i just finished a course of analysis(2)\vector calculus.Now iI'm interested in doing Gd of curves and surfaces(Do Carmo), and functional analysis(Rudin'sbook), but do not know what may have precedence between the two, on which i should start before you think?

Consider a two-point function $$f(\mathbf{r}_{1},\mathbf{r}_{2})$$ If one requires homogeneity, then this implies that for a constant vector ##\mathbf{a}## we must have $$f(\mathbf{r}_{1},\mathbf{r}_{2})=f(\mathbf{r}_{1}+\mathbf{a},\mathbf{r}_{2}+\mathbf{a})$$ How does one show that if this is...

Homework Statement Let ##u## and ##v## be differentiable functions of ##x,~y## and ##z##. Show that a necessary and sufficient condition that ##u## and ##v## are functionally related by the equation ##F(u,v)=0## is that ##\vec \nabla u \times \vec \nabla v= \vec 0## Homework Equations (Not...

Hi, In a problem I have been working on for a while now I have found that I want to find the function satisfying the functional relation f(x)n f(a - x) = 1 for n = 1 I believe I have proven that f(x) = x/(a - x). On this page is an answer I do not quite understand. One of the prerequisits...

I'm looking for examples of some interesting compounds that contain alcohol functional group (please, no joke suggestions about various liquor). Bonus points for creativity/thinking-outside-the-box. Thanks.

Homework Statement Consider the functional ##S(a,b) = \int_0^∞ r(1-b)a' \, dr ## of two functions ##a(r)## and ##b(r)## (with ##a' = \frac{da}{dr}##). Find the ##a(r)## and ##b(r)## that extremize ##S##, with boundary conditions ##a(∞) = b(∞) = 1##. Homework EquationsThe Attempt at a Solution...

Hi, Could someone post a solution to the following questions : 1. Let R be the real numbers and A a collection of all groups that are either bound or their complement is bound. a. Show that A is an Algebra. Is it a sigma algebra? b. Define measure m by m(B) = {0 , max(on B) x <...

Hi - my professor in functional analysis posted 4 prior years tests just 4 days before the test without solutions. I'd appreciate if anyone can help send solutions for the following with the following questions : 1. $\mu$ is a sigma additive measure over sigma algebra $\Sigma$. A $\in...

Hi, I'm taking a course in functional analysis and having some trouble with the following questions : 1. L1(R) is the space of absolutely integrable functions on R with the norm integrate(abs(f(x)) over -inf to +inf. Define a linear operator from L1(R) to L1(R) as A(f)(x)=integrate...

Homework Statement This is a problem from Haim Brezis's functional analysis book. Homework EquationsThe Attempt at a Solution I'm assuming (e)n is the vectors like (e)1 = (1,0,0), (e)2=(0,1,0) and so on. We know every hilbert space has an orthonormal basis. I also need to know the...

Hallo Everybody, I am searching for a book (or lecture notes) that details the calculations that lead me from a given Lagrangian to the Feynman rules of the theory. It should not be rigouros, just the main steps to get the Feynman rules. Thanks for your help!

There are two forms of Riemann functional equation. One is more symmetric and follows from the other and the duplication theorem of the Gamma function. At least, that's been claimed here...

Urs Schreiber submitted a new PF Insights post Higher Prequantum Geometry III: The Global Action Functional - Cohomologically Continue reading the Original PF Insights Post.

Homework Statement Let ##f,g## be two real valued functions, defined on the segment ##[a,b]## and continuous on ##[a,b]##, such that ## 0 < g < f ##. Show there exist ##\lambda > 0 ## such that ## (1+\lambda) g \le f ## Homework Equations The Attempt at a Solution Set ##h = f/g##. Since...

Today, in my advanced particle physics class, the professor reminded the time-dependent perturbation theory in NRQM and derived the formula: ##\displaystyle \frac{da_m(t)}{dt}=-i \sum_n e^{-i(E_n-E_m)} \int_{\mathbb R^3}d^3 x \phi^*_m (\vec x) V(\vec x,t) \phi_n(\vec x)##. Then he said that...

Hi, hopefully this is in the right place. I have big plans for making a Velociraptor costume and I've got a handle on how I'm putting most of it together, but there is one issue and it's a big one; comfort. I know that I'm going to be uncomfortable in the costume regardless of what I do and I'm...

Hi What is more beneficial as a graduate course? Functional Ceramics vs Principles of Metal Forming? Please tell me what you think Thanks a lot

Let's call our functional $$F[f]=\int dx\:A\left(x,f,f',f''...\right)$$ We know that the variation of F can be written as $$\delta F=\int dx\:\left[\frac{\partial A}{\partial f}\delta f+\frac{\partial A}{\partial f'}\delta f'+...\right]$$ If i wanted to get everything in terms of delta f in...

Homework Statement [/B] Consider the following action: $$\begin{align}S = \int \mathrm{d}^4 z \; \bar\psi_i(z) \, (\mathrm{i} {\not{\!\partial}} - m)_{ij} \, \psi_j(z)\end{align}$$ where ##\psi_i## is a Dirac spinor with Dirac index ##i## (summation convention for repeated indices). Now I would...

I'm in my last semester of my undergraduate majoring in mathematics (focusing on mathematical physics I guess - I'm one subject short of having a physics major) and am wondering, largely from a physics perspective if it would be better to do a functional analysis course or a differential...

I've been trying to fill in my mathematical blanks of things I just took as dogma before. Especially, not having a background in functional analysis, the functional derivatives often seem to me mumbo jumbo whenever things go beyond the "definition for physicists". In particular I tried looking...

I'm reading Rudin's principles and so far I really like it. I find charm I'm his terseness, and I think having that motivation to do a lot of the stuff myself makes it pretty fun (like only using the outline of the Dedekind cuts section and prove all the steps myself). However, I have heard not...

I'm trying to solve the exercise below in a book I'm reading. I inverted equation 1.3 to get ## \phi_{\mathbf k}(t)=\int \frac{e^{-i \mathbf k \cdot \mathbf x}}{(2\pi)^{\frac 3 2}} \phi(\mathbf x,t) d^3 \mathbf x ##. Then I put it in I to get: ## I=\int \int d^3 \mathbf x d^3 \mathbf y...

I am looking for reliable information about the functional dependence of the diameter ##d(t)## of the visible universe on the time ##t## since the big bang singularity, based on the different hypotheses currently deemed competitive.

Hello! (Wave) Let $V=C^1([a,b])$. Show that if $J$ is a continuous functional in respect to the norm $||y||_1:=||y||_{\infty}+||y'||_{\infty}, y \in V$ then it is also continuous in respect to the norm $||y||:=||y||_{\infty}$. Also, show that the inverse of the above claim does not hold. Let...

Hello! (Wave) Is the following functional over $C^1([a,b])$ linear? $$J(y)= \int_a^b (y')^2 dx+ G(y(b))$$ That's what I have thought:if $J$ would be linear it would have to hold: $\forall y \in C^1([a,b]), \forall \lambda \in \mathbb{R}$ $J(\lambda y)=\lambda J(y)$ or equivalently...

In my textbook (see attached picture) there appears a functional derivative, but I honestly don't know how to evaluate a quantity like this. What should I do? I have tried to google but all I could find was how to take functional derivatives, where polynomials appeared under the integral, while...

I can't convince myself whether the following functional derivative is trivial or not: ##\frac \delta {\delta \psi(x)} \big[ \partial_x \psi(x)\big],## where ##\partial_x## is a standard derivative with respect to ##x##. One could argue that ## \partial_x \psi(x) = \int dx' [\partial_{x'}...

Hi all, hope this is the right forum. Please feel free to move it otherwise: I am confused on whether functional dependence can be determined uniquely by the particulars of a given table, or if it is determined in a more general sense So I have my relational table. Let A,B be attributes ...

I have the opportunity to pursue an independent study in functional analysis (using Kreyszig's book) or calculus on manifolds (using Tu's book) next semester. I think that both of the subjects are interesting and I would like to study them both at some point in my life, but I can only choose one...

By properties I mean the +I or -I effect as compared to other functional groups along with whether it induces a +R or -R effect?