Fundamental Definition and 947 Threads

Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits.
There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells.

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  1. Math Amateur

    MHB What is Stoll's definition of the natural logarithm function?

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's definition of the natural logarithm function (page 234 -235) The relevant section of Stoll reads as follows: In this section we read: " ... ... To prove (a), consider the function L(ax), x \gt 0. By...
  2. P

    Proving fundamental set of solutions DE

    Homework Statement Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.The...
  3. G

    Fundamental and contravariant representations

    The invariant of SL(2,C) is proven to be invariant under the action of the group by the following \epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta} The existence of an invariant of this form (with two indices...
  4. Math Amateur

    MHB Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2 Stoll's statement of Theorem 6.3.2 and its proof reads as follows: In the above proof we read: Since \mathscr{L}( \mathscr{P} ...
  5. G

    Fundamental frequencies of square wave and sine wave

    Homework Statement What are the fundamental frequencies for a 50 kHz square waveform of 50% duty cycle and a 25 kHz sinusoidal waveform, respectively? (The duty cycle of a square waveform is the ratio between the pulse duration and the pulse period.) Homework Equations My teacher then gave...
  6. T

    The Fundamental Theorem of Calculus

    Homework Statement Evaluate the definite integral of the algebraic function. Use a graphing utility to verify your result. Integral from [1 to 2] of (3/x^2 - 1) Homework Equations The answer is 1/2 f(x)dx= F(b) - F(a) The Attempt at a Solution I tried taking it to make it -x^-3 - 1x as the...
  7. K

    Solve for y(x) using the Fundamental Theorem of Calculus

    Homework Statement Solve the integral equation for y(x): y(x) = 1 + ∫ { [y(t)]^2 / (1 + t^2) } dt (integral from 0 to x) See attached image for the equation in a nicer format. Homework Equations Fundamental Theorem of Calculus The Attempt at a Solution dy/dx = y(x)^2 / (1 + x^2) ∫ dy/y^2 = ∫...
  8. DrPapper

    Classical Fundamental Principles of Classical Mechanics - Kai S. Lam

    Hello all, I'm currently taking an upper undergraduate two part Mechanics course using the above mentioned book by its author. He's a great professor and I was wondering if anyone else has checked out this book? It's very math heavy and I'm struggling with some of the language since I haven't...
  9. P

    Wave Problem -- Total amplitude of fundamental and first three harmonics

    Homework Statement An acoustic signal is composed of the first three harmonics of a wave of fundamental frequency 463 Hz. If these harmonics are described, in order, by cosine waves with amplitudes of 0.100, 0.300, and 0.760, what is the total amplitude of the signal at time 0.401 seconds...
  10. PcumP_Ravenclaw

    Understanding the fundamental theorem of algebra

    Dear all, I am trying to understand the fundamental theorem of algebra from the text (Alan F. Beardon, Algebra and Geometry) attached in this post. I have understood till the first two attachments and my question is from the 3rd attachment onwards. I will briefly describe what...
  11. J

    Why is the fundamental theorem of arithmetic special?

    Why is it significant enough to be fundamental? Some people say that it is fundamental because it establishes the importance of primes as the building blocks of positive integers, but I could just as easily 'build up' the positive integers just by simply iterating +1's starting from 0.
  12. I

    Why the muon is considered a fundamental particle?

    This question arises from the fact that the muon has a mass close to the first excited state of the radial vibration of the electron (Prog. Theor. Vol. 47 (1972), No. 3 Cohesive force of electron and Nambu's mass-formula).
  13. abram

    Fundamental kinematics question

    Homework Statement I seem to have a fundamental misunderstanding of the kinematic principles in this question. A constant force of 8.0N is exerted for 4.0s on a 16-kg object initially at rest. What will the change in speed of this object be? Homework Equations F = ma Δx = v0t + 1/2 at2 Δv =...
  14. samgrace

    Which areas of maths are from which fundamental areas

    Hello, I am a physics student and have catagorised most of physics, e.g classical mechanics, relativistic mechanics, quantum mechanics and quantum field theory, and have also identified all the mathematics involved in each of these catagories. For example classical mechanics involves Calculus...
  15. P

    MHB Fundamental theorem of galois theory

    In fundamental theorem of galois theory,(statement): given that K be galois extension of F, G(K/F) be its galois group, S(K) be the set of all subfields of K containing F & S(G) be the set of all subgroups of G(K/F), mapping g: from S(K) to S(G) defined by g(H)=G(K/H), mapping h: from S(G) to...
  16. M

    Can all fundamental forces be fictitious force ?

    After reading many questions, , I wonder: is it possible to consider also the other fundamental forces, the electroweak interaction and the strong interaction or ultimately the unification of these, to be fictitious forces like gravity in the framework of general relativity? If we want a final...
  17. C

    The Fundamental Theorem for Line Integrals

    Homework Statement Determine whether or not f(x,y) is a conservative vector field. f(x,y) = <-3e^(-3x)sin(-3y),-3e^(-3x)cos(-3y) > If F is a conservative fector field find F = gradient of f Homework Equations N/A The Attempt at a Solution Fx = -3e^(-3x)(-3)cos(-3y) Fy =...
  18. nicolauslamsiu

    Fundamental theorem of calculus

    Homework Statement Using Fundamental Theorem of Calculus to find the derivative 2. Homework Equations upper limit=x^2, lower limit=4x ∫ { 1 / [1+ (sin t)^2] }dt The Attempt at a Solution two independent variables are involved, how should i find the derivative? [/B]
  19. E

    Fundamental capacitor discharge question

    Hi All, I understand that when a capacitor is subjected to a DC voltage, it stores charge. However, when the DC voltage is removed (open circuit) why does the capacitor not internally discharge? Shouldn't the charge just go back to where it was, leaving the capacitor with 0 stored charge? I am...
  20. T

    Finding the Fundamental Solutions of a Third Order ODE

    Homework Statement Find the fundamental solution to ty''' - y'' = 0 Homework EquationsThe Attempt at a Solution I think I'm missing something really obvious, but I have the characteristic polynomial: t\lambda^3 - \lambda^2 = 0 Solving the equation: \lambda^2 (t\lambda - 1) = 0 I get zero...
  21. avito009

    Why is the strong nuclear force the strongest of the four fundamental forces?

    Is more force required at smaller distances like the nucleous of an atom? Since strong force applies itself at the level of the atom does that prove that more force is required at smaller distances to attract according to the inverse square law? Does strong nuclear force obey the inverse square...
  22. PcumP_Ravenclaw

    Proof of fundamental theorem of arithmetic

    Dear all, Please help me understand the proof by induction for only one way of expressing the product of primes up to the order of the factors. Please see the two attachments from the textbook "alan F beardon, algebra and geometry" A is a set of all natural numbers excluding 1 and 0?? r and s...
  23. S

    Fundamental question about thermodynamics

    Hopefully this is the right place to post this question. This is a very fundamental question on the applicability and limitations of classical equilibrium thermodynamics (CET). I've been learning non-equilibrium thermodynamics (NET), and a few sources mentioned that NET is needed because CET is...
  24. kmm

    C and h are not fundamental constants?

    I was looking through Zee's 'Quantum Field theory in a Nutshell" and he says that c and \hbar are "not so much fundamental constants as conversion factors." I've heard other physicists say this as well. I understand that these constants are used in some equations to give units of energy so...
  25. P

    Why Is Gravity Considered a Fundamental Force in Physics?

    This is what I got after watching some documentaries: Acording to the general theory of relativity, gravity is curvature of space-time caused by the massive objects. It was explained that it is not a force in a sense that it acts between two bodies, but it curves the space and the bodies are...
  26. C

    Really fundamental cross/dot product questions

    So I've been working on physics homework and we have some vector/dot product questions. This is really long, but the questions I have really are rudimentary at best. I have seven total questions. You're given two vectors that only have an x and y component, A, and B, and the positive Z axis is...
  27. K

    Fundamental of Physics 10th Edition Problem 52

    Homework Statement A bolt is dropped from a bridge under construction, falling 83 m to the valley below the bridge. (a) How much time does it take to pass through the last 20 % of its fall? What is its speed (b) when it begins that last 20 % of its fall and (c) just before it reaches the...
  28. S

    Engineering Physics degree, taking Fundamental Engineering test

    Just wanted to let the forum know that through some diligence I have been approved to take the Fundamental Engineering test in South Carolina with a B.S. in Physics. I took about a extra year worth of advanced engineering classes along with skipped all freshman and sophomore engineering...
  29. atyy

    Are PVMs more fundamental than POVMs from the view of decoherence?

    In quantum mechanics, POVMs can be considered more fundamental than PVMs because PVMs can be considered a special sort of POVM. However, because of the Naimark extension, one can formally consider POVMs to always be derived from PVMs. Accordingly, one could argue that choosing PVMs or POVMs to...
  30. T

    So-called Fundamental Theorem of Algebra

    This came up in one of my readings: "Neither the so-called fundamental theorem [of algebra] itself nor its classical proof by the theory of functions of a complex variable is as highly esteemed as it was a generation ago, and the theorem seems to be on its way out of algebra to make room for...
  31. S

    Fundamental note vs fundamental frequency of string

    Homework Statement A guitr player changes the frequncy of the note produced by a guitar string by pressing his fingers along the string. The fundamental frequency of the string is 264hz. What are the frquncies of the fundamental note if the player plucked the string at 1/4 of the way from one...
  32. jegues

    Calculating RMS of Fundamental Component for Unknown Periodic Input Signal

    Every 50μs I have access to one sample of an unknown peroidic input signal and I am looking to calculate the RMS of the fundamental component. Is there any convenient ways one can think of doing this?
  33. B

    Do Waves Truly Exist in the Physical World? A Fundamental Question

    Caution:This is probably an extremely stupid question but I couldn't find any answers on the web. So, do waves really exist in the physical world the way we plot them on a graph? or is it just our way of representing an entity? and if they truly physically exist, how do we know for sure that...
  34. ugenetic

    B, H, MMF, which one is the fundamental "driving" force?

    I was looking at a problem regarding solenoid with an iron core inside. the iron core has an air gap . once the MMF was turned on, a constant flux will flow thru the core and air gap (ignore fringing), so B's are the same for air gap and core but H's are different. I am guessing some similar...
  35. C

    Best books on fundamental Statistics?

    Hi awesome Physicsforum members, I’m looking for a good introdoctury book for learning Statistics (self-study) which succeeds in explaining the concepts very clearly . What I’m looking for in a statistics book My goal with the book is not to achieve some abstract mathematical skills, but...
  36. marcus

    Is Unitarity Limited by Real Clocks in Quantum Gravity?

    Bei Lok Hu at the University of Maryland has a review article on the "fundamental decoherence" research topic, which devotes detailed attention to the treatment by Gambini and Pullin. I think it's an interesting topic for several reasons, so I'll give some links. Here's the paper by B.L. Hu et...
  37. H

    Fundamental forces and Pauli's exclusion principle

    To which type of Fundamental forces below the Pauli's exclusion belong? Strong nuclear forces Coulomb forces Weak nuclear forces Gravitational forces
  38. Q

    Deriving Coulomb's law and other fundamental equations from F=ma?

    I saw a joke crib sheet for a physics class that said "F=ma, derive the rest," but is it actually possible to derive at least electricity and magnetism questions from Newton's second law?
  39. F

    A question about fundamental constants in Multiverse idea

    Has there been a debate as to whether fundamental constants can be chosen freely for a theory or some constraints make it unlikely that to be the case. for example the dimentionful parameters don't have a specific value, only combinations give FSC, that means(to my mind) that there is unlikely...
  40. O

    Do Fundamental Particles Experience Heat?

    Heat is a product of excited energy states of the fundamental particles that make up atoms, correct? So do the particles, themselves, get "hot" - or is heat just experienced as radiation on the macroscopic scale? Do neutrons, for example, have a thermal property at all?
  41. M

    On Fundamental Closed String Solutions

    I am doing a project on string theory and my first task is to work out fundamental string solution from the I(10) string action in NS sector.I am following A.Dahbolkar, G.Gibbons, J.A.Harvey & F.Ruiz Ruiz, Nuclear Physics B340 (1990) 33—55. Why is it called fundamental string? I have not taken...
  42. J

    Fundamental theorem of calculus for double integral

    The popular fundamental theorem of calculus states that \int_{x_0}^{x_1} \frac{df}{dx}(x)dx = f(x_1)-f(x_0) But I never see this theorem for a dobule integral... The FTC for a univariate function, y'=f'(x), computes the area between f'(x) and the x-axis, delimited by (x0, x1), but given a...
  43. Wes Tausend

    Fundamental Valid Frame of Rest

    ... Three scientists are familiar with Special Relativity and it's rules. The three scientists decide to conduct a Special Relativity experiment. The plan is for two of them to get into two separate rocket ships and pass one another nearly head-on overhead while the third scientist...
  44. D

    Higgs field as the 5th fundamental force?

    Greetings, I am brand new to quantum physics. By that I mean I have no idea of the mathematics and know terribly little about the physics. In fact my best background comes from the book QED by Richard Feynman. Anyway, I do know that there are four fundamental forces which have force-carrying...
  45. C

    Fundamental groups and arcwise connected spaces.

    If a space X is arcwise connected, then for any two points p and q in X the fundamental groups ##\pi_1(X,p)## and ##\pi_1(X,q)## are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class ##\pi_1(X)##. I started to think about the generality...
  46. Spinnor

    B-mode plots, spherical harmonics?, fundamental modes?

    If the B-mode sky plots could be Fourier transformed what would be a plot of the lowest order B-mode harmonic plotted on a sphere look like? I guess we need two functions of spherical coordinates, one function for amplitude at points on a sphere and one function for the orientation at the...
  47. P

    Fundamental theorem of Calculus

    Suppose a is a constant. If G(x) = \int_a^x \Big [ f(t) \int_t^x g(u) du \Big ] dt, what is G\,'(x)? My attempt, G\,'(x) = f(x) \int_x^x g(u) du = 0, and I am sure this is wrong.
  48. evinda

    MHB First fundamental theorem of Calculus

    Hello! :) I am looking at the theorem: "$f:[a,b] \to \mathbb{R}$ integrable We suppose the function $F:[a,b] \to \mathbb{R}$ with $F(x)=\int_a^x f$.If $x_0$ a point where $f$ is continuous $\Rightarrow F$ is integrable at $ x_0$ and $F'(x_0)=f(x_0)$". There is a remark that the theorem stands...
  49. L

    Fundamental theorem of calculus

    Homework Statement Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx##...
  50. D

    MHB Fundamental Matrix of \(\mathbf{A}\) - Find the Solution

    Given \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -5 & -4 \end{bmatrix} \] find the fundamental matrix. If I had real eigenvectors, I could simply do \[ e^{\mathbf{A}t} = \mathbf{S}e^{\mathbf{D}t}\mathbf{S}^{-1}, \] but I have complex eigenvalues and eigenvectors. Also, if the...
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