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. I

    Fundamental Theorem of Calculus: Part One

    I am a little confused over part 1 of the fundamental theorem of calculus. Part 2 makes perfect sense to me. I guess my confusion is if we have an integral g(x) defined from [a, b], and we are looking at point x, how do we know that g'(x) = f(x)? It makes sense in the idea that they are...
  2. F

    String theory and fundamental physical constants

    I have heard that ST can calculate the masses of the particles species of some vacuum and also its decay rate, charge and so on, but I have not seen any clear paper on the subject. Can anyone point me to one? Are "hbar" and "c" the same in all the vacuums?why or why not? Can ST predict the...
  3. D

    Proving the Fundamental Group of SO(2) is Z: How Can it Be Done Explicitly?

    Good morning. I was wondering how do you prove explicitly that the fundamental group of SO(2) is Z?
  4. S

    Fundamental matrix vs Wronskian

    I have just learned the first order system of ODE, i found that the Wronskian in second order ODE is |y1 y2 ; y1' y2'| but in first order system of ODE is the Wronskian is W(two solution), i wonder which ones is the general form? thank you very much
  5. moenste

    Find fundamental freq. and 1st overtone closed each end pipe

    Homework Statement Calculate the frequency of: (a) the fundamental, (b) the first overtone, produced by a pipe of length 40 cm which is closed at each end. (Velocity of sound in air = 340 m s-1.) Answers: (a) 425 Hz, (b) 850 Hz 2. The attempt at a solution L = 0.4 m, v = 340 m s-1 (a) f1 = ...
  6. moenste

    Fundamental frequency if string halved and tension * 4

    Homework Statement The fundamental frequency of vibration of a particular string is f. What would the fundamental frequency be if the length of the string were to be halved and the tension in it were to be increased by a factor of 4? Answer: 4 f 2. The attempt at a solution We have f = f1...
  7. H

    Origins of the Fundamental Interactions/Forces

    Are the four fundamental interactions (or "forces") e.g. gravity, weak, color, electromagnetic believed to be the result of "phase changes" as the universe expanded from the quark-gluon plasma?
  8. B

    Anthropomorfic Fundamental Forces

    Hello everyone, I'm a Belgium based writer, currently working on my debut novel that mixes scientific fact with Biblical mythology. My main characters are the physical manifestations of concepts. Now, Amber, the lead, is Koved, the Malach ("angel"/manifestation) of gravity. Her predecessor...
  9. T

    Book recommendation on fundamental physics in Graphene

    I am a condensed matter grad. student, but do not work on Graphene. I am interested in learning about the fundamental physics in graphene (e.g. the need to use Dirac equation to represent the motion of electrons). I know that many books are there, but not all are geared towards fundamental...
  10. paulo84

    Electromagnetic spectrum/(dark)matter/4 fundamental forces

    OK, I need some help understanding some stuff. The way I see it: you've got 4 fundamental forces in physics right? I believe these are gravity, electromagnetic, strong interaction and weak interaction. The electromagnetic spectrum is basically waves with photons (photons in all of the...
  11. goonking

    Fundamental frequency of oscillation in 2 ropes

    Homework Statement Homework EquationsThe Attempt at a Solution if the material are the same in both strings, then the density should be the same. v = sqrt (tension/μ) tension in the first string should be 30 kg x 9.8 m/s^2 = 294 N next, v = λƒ and string#2 needs to have twice the...
  12. A

    What is a fundamental particle according to QFT?

    In quantum field theory, a fundamental particle is an excitation in the underlying field, but what does that mean? Do fundamental particles have any physical existence according to QFT?
  13. N

    M Theory -- can a fundamental open string end on an M5 brane?

    In M Theory, can an elementary open 1d string attach its ends on an M5 brane? According to M theory, the elementary string are created when an M2 brane is wrapped tightly around a circular dimension. When the open string is created can its end point attach on an M5 brane??
  14. Vishwasks001

    Gravitational Constant: Why Is It Fundamental?

    Hi, I was wondering if there is any need in the theory of relativity for Newton's Gravitational Constant 'G' to remain a fundamental constant. Constant in Coulomb's Law can be expressed in 'c' and 'pi', then why Constant in Newton's Law needs to be a fundamental constant??
  15. A

    What field deals with the following fundamental questions?

    Physics deals with finding out the basic rules by which the universe works. For me, physics is the most fundamental understanding of the universe. So theoretically, everything could be explained by the laws of physics. Is there any field that deals with the idea? Similarly, chemistry is...
  16. S

    I want to find out the fundamental truths in this world

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  17. C

    The Fundamental Theorem of Calculus

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  18. pellman

    Fundamental theorem of algebra and factoring?

    Is the fundamental theorem of algebra (for polynomials on the complex plane) equivalent to the statement that any polynomial p of degree n>0 can be written p(z) = c(z - a_1 ) (z- a_2) \cdot \cdot \cdot (z - a_n ) or am I missing some subtle distinction? And if not equivalent, does the theorem...
  19. R

    Fundamental Poisson Bracket - Canonical Transformation

    When proofing the poisson brackets algebraically, what is the tool of choice. Can one use the muti dimensionale chain rule or how is it usally done?
  20. N

    Fundamental point like particles?

    Is it possible that fundamental point like particles (ie electrons) are actually particles with shape size and structure that are too small for any of our current technology to detect?
  21. P

    Antiderivatives and the fundamental theorem

    I know that according to the first fundamental theorem of calculus: $$\frac{d}{dx} \int_a^x f(t) dt = f(x)$$ I also know that if ##F## is an antiderivative of ##f##, then the most general antiderivative is obtained by adding a constant. My question is, can every single antiderivative of ##f## be...
  22. P

    Why is ##g## differentiable only on ##(a,b)## and not on ##[a,b]##?

    "If ##f## is continuous on ##[a,b]## and: $$g(x) = \int_a^x f(t) dt$$ Then ##g## is continuous on ##[a,b]##, differentiable on ##(a,b)##, and ##g'(x) = f(x)##." This is the first fundamental theorem of calculus. I'm curious as to why ##g## is only differentiable on ##(a,b)##, but not ##[a,b]##.
  23. B

    Fundamental theorem of calculus for double integral

    I was reading about double integral when a doubt came to my mind: how to find the antiderivative of the function f(x,y), like bellow, and compute the fundamental theorem of calculus for double integral? \int_{2}^{8} \int_{2}^{6} f(x,y) dx \wedge dy = ? OBS: It's not an exercise. I know how...
  24. K

    Electrical Designing Your Own CPU Cooler Fan: Fundamental Tips

    Hi folks.I'm 3rd grade mechanical engineering student.I want to make cpu cooler.I will make pid controlled fan with an arduino. PID path is easy but I could not find anything about fan dynamics. I want to design my own fan.I need something like "fundamentals of fans" that should answer my...
  25. Hunny

    Fundamental Theorem of Calculus Problem

    1. If g(x) = ∫ f(t) dt = xln x, find f(1) The ∫ has x^2 on top and 0 on bottom. 2. g'(x) = f(x) <--FTC1 The Attempt at a Solution g'(x) = f(x) u=x^2 g'(x) = u*lnu * 2x(derivative of inner function) g'(x) = 2x(x^2)ln(x^2) f(1) = 2(1)(1^2)ln(1^2) f(1) = 0, since ln(1) = 0 I...
  26. kreil

    Comments - How Fundamental is the Arrow of Time?

    kreil submitted a new PF Insights post Retrocausality: How Fundamental is the Arrow of Time? Continue reading the Original PF Insights Post.
  27. andrewkirk

    Thermodynamic systems: spot the invalid fundamental equation

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  28. M

    EM: Vector potential vs. Field tensor: Which is fundamental?

    In my lecture we were discussing the Lagrangian construction of Electromagnetism. We built it from the vector potential ##A^\mu##. We introduced the field tensor ##F^{\mu \nu}##. We could write the Langrangian in a very short fashion as ##-\frac{1}{4}F_{\mu \nu}F^{\mu \nu}## In the end we...
  29. A

    What is the most *Fundamental* theory in physics?

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  30. Omega0

    Interaction - Fundamental question

    Hi, What is the basic idea about particle interaction in QM? Say you have first one single charged particle in an "empty universe". Here I think the first issue is given: That's impossible, you'll need masses (or better: energy) from a GRT view. Okay, say we watch this single charged...
  31. S

    Projection stereographic and second fundamental form

    Let r:R2 →R3 be given by the formula Compute the second fundamental form with respect to this basis (Hint: There’s a shortcut to computing the unit normal n). I can't find thi shortcut, does anyone help me? I'm solving it with normal vector and first and second derivate, but I obtained...
  32. B

    Fundamental Theorem of Field extensions

    Suppose F is a field and that ## f(x) ## is a non-constant polynomial in ##F[x]##. Since ##F[x] ## is a unique factorization domain, ## f(x) ## has an irreducible factor, ## p(x) ##. Then the fundamental theorem of field theory says that the field ## E = F[x]/<p(x)> ## contains a zero of ## f(x)...
  33. H

    Why are quarks fundamental particles?

    Are quarks really considered fundamental particles that cannot be divided further? If an up quark can transmute to a down quark and release a W+ boson which decays to a positron and a neutrino (for example) - doesn't this mean that there is substructure to a quark? What exactly is it that makes...
  34. Calpalned

    Green's Theorem vs Fundamental

    Homework Statement 1) How do I know when to use Green's Theorem, the Fundamental Theorem for Line Integrals or the regular method of using parametrization? 2) Assuming that the three methods above are all used to solve line integrals, why do the Fundamental Theorem and Green give different...
  35. O

    Homework Question - Fundamental Theorem of Calc Example

    Homework Statement Hi, I've been working through a practice problem for which I used the fundamental theorem of calculus, or one of its corollaries. The setup is a population changing over time. The population, P(t) at t = 0 is 6 billion. The limiting population as t goes to infinity is given...
  36. C

    Which interactions proceed by which fundamental force?

    I have been getting ready for my Intro to Modern Physics final. The professor likes to ask a lot of questions like Which of the following neutron reactions proceeds via the electromagnetic force? Or something like this... I have been looking all over the internet and my textbook. All I can...
  37. D

    What should I study or read to learn physics in a very fundamental way?

    Hello I'm a 15year old boy living in Korea, and lived in the U.S. for a couple of years. I want to learn physics in a very fundamental(?) way, for example, knowing the reason why from Newton's law of gravitation F = GmM/r^2, F is proportion to the SQUARE of r (not cube or power of 4??)...
  38. Math Amateur

    Multiplication of Path Classes and the Fundamental Group

    In Chapter 7 of John M. Lee's book on topological manifolds, we find the following text on composable paths and the multiplication of path classes, [f] ... ... Lee, writes the following:In the above text, Lee defines composable paths and then defines path multiplication of path classes (not...
  39. Puma

    Any signs of fundamental blocks of energy?

    I would just like to ask a basic question as to whether there is any sign of there being a minimum energy block size which may hint at a fundamental particle building unit, or whether there is just a continuum of energies as far as we know so far.
  40. Mr Davis 97

    The first fundamental theorem of calculus

    Say I have the statement ##\int \frac{\mathrm{d} y}{\mathrm{d} x}\mathrm{d}x = y##. How does the fundamental theorem of calculus make this necessarily true? When I see the formal statement of the theorem, it is usually in terms of a definite integral such as ##F(x) = \int_{a}^{x}f(t)dt##. How...
  41. J

    Can a harmonic be louder than the fundamental frequency?

    This is not a homework question per se, but rather something I have come across during a homework project. Using Audacity, I recorded a few different instruments playing the same notes (investigating timbre). I noticed that (using a steel string acoustic guitar) the first harmonic at 131 Hz...
  42. I

    If electrons are fundamental particles then why do they -

    If electrons are fundamental particles and leptons then what happens when they emit or absorb photons or Z particles? What is known about the mechanism of these aborptions or emissions for any particles? Do the absorbed electrons gain mass as particles are described to do as they move nearer to...
  43. N

    Fundamental Differences: MINOS and DUNE

    Curious if anyone has insight into why a new long baseline neutrino experiment, DUNE, is being funded when the MINOS infrastructure has been in place for a few years. The experiments seem approximately the same, traveling similar distances to detectors in the Soudan Mine and to the underground...
  44. zonde

    QM laws fundamental vs effective

    I have impression that physicists consider QM laws as fundamental, meaning that there should be a way how QM mathematical form is realized in reality. On the other hand QM laws could be effective laws resulting from some process of evolution. In that case it should be possible to characterize...
  45. D

    MHB Fundamental theorem and limit proofs

    Prove that the limit as n approaches infinity of ((2^n * n!)/n^n) equals to zero. The hint is to use Stirling's approximation. What is this?
  46. ElectricKitchen

    Fundamental Relationship Between Time and Space Derivatives

    Many physical laws involve relationships between time derivatives to space derivatives of one or more quantities. For example, thermal conduction relates the thermal energy time rate of change [dQ/dt] to temperature space rate of change [dT/dx]. In fluid flow, the Navier-Stokes Theorem relates...
  47. C

    The Fundamental Assumption of Statistical Mechanics

    Fundamental assumtption: "a closed system is equally likely to be in any of its g accessible micro- states, and all accessible micro- states are assumed to be equally probable." There's just a few things I don't understand about this, 1. Isn't saying that a closed system is equally likely to...
  48. D

    Fundamental theorem of calculus - question & proof verifying

    I understand that the fundamental theorem of calculus is essentially the statement that the derivative of the anti-derivative F evaluated at x\in (a,b) is equal to the value of the primitive function (integrand) f evaluated at x\in (a,b), i.e. F'(x)=f(x). However, can one imply directly from...
  49. andyrk

    Why Must f(x) Be Continuous in the Fundamental Theorem of Calculus?

    In the fundamental theorem of calculus, why does f(x) have to be continuous in [a,b] for F(x) = \int_a^x f(x) dx ?
  50. D

    Proving the fundamental theorem of calculus using limits

    Would it be a legitimate (valid) proof to use an \epsilon-\delta limit approach to prove the fundamental theorem of calculus? i.e. as the FTC states that if f is a continuous function on [a,b], then we can define a function F: [a,b]\rightarrow\mathbb{R} such that F(x)=\int_{a}^{x}f(t)dt Then F...
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