Functional Definition and 417 Threads

Shown below is the graph of The graph interests the axis at 3 points, B, C and D. Given the points of intersection, and the brackets below, form and expand an equation for the graph of ATTEMPT: I've assumed to involve the points of intersection for the x-axis...

Homework Statement Prove that snell's law ## {n_1}*{sin(\theta_1)} ={n_2}*{sin(\theta_2)} ## is derived from using euler-lagrange equations for the time functionals that describe the light's propagation, As described in the picture below. Given data: the light travels in two mediums , one is...

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: $$ \frac {δJ[f]} {δf(x)} = \lim_{ε \rightarrow 0} \frac 1 ε \left[ \int[f(y) + εδ(y-x)]^pφ(y)\, dy - \int [f(y)]^pφ(y)\, dy\right] $$ $$ =...

Homework Statement I am currently working on an exercise list where I need to calculate the second functional derivative with respect to Grassmann valued fields. $$ \dfrac{\overrightarrow{\delta}}{\delta \psi_{\alpha} (-p)} \left( \int_{x} \widetilde{\bar{\psi}}_{\mu} (x) i \partial_{s}^{\mu...

Homework Statement Question Use the functional equation to show that for : a) ##k \in Z^+ ## that ## \zeta (-2k)=0## b) Use the functional equation and the euler product to show that these are the only zeros of ##\zeta(s) ## for ##Re(s)<0## . And conclude that the other zeros are all located...

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...

Homework Statement Hi, Question attached: inserting ##\phi (x)= \int \frac{d^4k}{(2\pi)^2}\phi(x)e^{-i k_u x^u}## and similar for ##J(x) ## / ##J(k)## into the action and then integrating over ##k## gives: Solution attached: I AM STUCK on this part, completing the square ; so I see...

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...

Homework Statement I am attempting to derive Caldeira-Leggett's influence functional found in their paper "Path Integral Approach To Quantum Brownian Motion". If you find my following statements confusing, then pages 16-18 of http://web.science.uu.nl/itf/Teaching/2006/MxWakker.pdf show the...

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...

Consider the evaluation of the following functional determinant: $$\text{log}\ \text{det}\ (\partial^{2}+m^{2})$$ $$=\text{Tr}\ \text{log}\ (\partial^{2}+m^{2})$$ $$= \sum\limits_{k} \text{log}\ (-k^{2}+m^{2})$$ $$= VT \int\frac{d^{4}k}{(2\pi)^{4}}\ \text{log}\ (-k^{2}+m^{2})$$ $$= iVT...

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 <...

Hello! (Wave) What is Applied Functional Analysis about? What knowledge is required? Which is the difference between this and Functional Analysis ?

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...

In QFT, we can work with functionals of fields ##\Psi[\phi(x)]=\langle \phi | \Psi \rangle## that give us the probability amplitude for the field to be ## \phi(x) ##. It seems to me that the Born rule we get here i.e. ##P(\phi(x))=|\Psi[\phi(x)]|^2=|\langle \phi | \Psi \rangle|^2## is not of...

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...

1./2. Homework Statement In my QFT lecture we were introduced to the 1PI effective action ##\Gamma[\varphi]## for a scalar theory (in Euclidean space-time). In one-loop approximation we've found \Gamma^{(\text{1-loop})}[\varphi] = S[\varphi] + \frac{1}{2} \operatorname{Tr} \log D^{-1} where...

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...