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. L

    Calculate Fundamental Groups of X_1-X_5

    I need to calculate the fundamental group of the following spaces: X_1 = \{ (x,y,z) \in \mathbb{R}^3 | x>0 \} X_2 = \{ (x,y,z) \in \mathbb{R}^3 | x \neq 0 \} X_3 = \{ (x,y,z) \in \mathbb{R}^3 | (x,y,z) \neq (0,0,0) \} X_4 = \mathbb{R}^3 \backslash \{ (x,y,z) \in \mathbb{R}^3 | x=0,y=0...
  2. FeDeX_LaTeX

    Fundamental definition of entropy

    Hello; "S\propto\ln\Omega, where \Omega is the number of microstates" is what a user told me was the fundamental definition of entropy. What is S? Is it the number of macrostates? And where does the ln come from? Can I see an example of where this formula would be used in practice? Thanks.
  3. N

    Accuracy of Fundamental thereom compared to MVT

    Hello! The Mean Value thereom gives F(b)-F(a) = f(c).(b-a) Where f is F' and c is the value of x at which it's derivate is equal to the average rate of change over the interval a to b for F. The Fundamental thereom of calculus also gives F(b)-F(a) = (\frac{1}{b-a} \left \left \int...
  4. T

    Fundamental frequency and temperature

    1. Balboa Park in San Diego has an outdoor organ. When the air temperature increases, the fundamental frequency of one of the organ pipes _____. a) goes down b) stays the same c) goes up d) is impossible to determine 2. v=331sqrt(1+t/273)/ 3. goes up?)
  5. R

    How Does Motion Affect Sound Frequency and Wavelength Calculation?

    Fundamental frequency! 1. Ultrasound with a frequency of 4.079 MHz can be used to produce images of the human body.If the speed of sound in the body is the same (1.97 km/s) as in salt water, what is the wavelength in the body? Answer in units of m. and 2. On a day when the wind is...
  6. K

    Fundamental frequency and tension

    Homework Statement A guitar string has a fundamental frequency of 429 Hz when its tension is 259 N. The string is being tuned to a fundamental frequency of 388 Hz. What is the required tension? Homework Equations v = sqrt(T/u): where v is the speed, T is the tension and u is the...
  7. A

    The fundamental thermodynamic relation

    Why is it that the The fundamental thermodynamic relation dU = Tds - PdV works in general even though Tds > dQ for irreversible processes. Likiewise PdV >/= dW for irreversible processes. Do the two irreversible effects cancel out. How does this happen physically.
  8. M

    ICL MSc in Quantum Fields and Fundamental Forces - how hard?

    Hello, I am thinking of applying for the above course and just trying to gauge how difficult it is really. I have a 2:1 in physics from Cambridge, and although at times I felt like I was knocking my head against a brick wall overall I think I might have pulled to a first if the exams had...
  9. C

    How Do You Determine the Fundamental Groups of Specific Matrix Spaces?

    I am reading Munkres and know exactly how to find the fundamental groups of surfaces, using pi_1 and reducing it down to simpler problems. However, I'm completely lost when looking at my final exam it says to find the fundamental groups of matrices! How do you go about doing that! There are...
  10. T

    Fundamental Theorem of Arithmetic

    Homework Statement Using the Fundamental Theorem of Arithmetic, prove that every positive integer can be written uniquely as a power of 2 and an odd number. Homework Equations The Attempt at a Solution Since the FTOA states that any integer can be written as a product of primes...
  11. N

    Fundamental theorem of calculus (something isn't right)

    The 2nd part of fundamental theorem of calculus says: Over what open interval does the formula F(x)=\int_{1}^{x}\frac{dt}{t} represent antiderivative of f(x)=1/x ? By looking at the theorem I would say that f(x) is continuous only for x \neq 0 So I would say that F(x) is defined on...
  12. E

    Verlinde, LQG, entropy and gravity as a fundamental force vs emergent

    Verlinde and Jacobson's early work, strongly implies that gravity is emergent. Anyhow, one of Jacobson's and Verlinde's claim in his paper is that since gravity is not a fundamental force, it does not make physical sense to quantize it canonically. So the LQG program is misguided...
  13. H

    Fundamental Resonant Frequency of a Waveguide

    Homework Statement For a given rectangular waveguide the cut off frequency of the fundamental mode is 6.5GHz. What is the fundamental resonant frequency of a 30mm long cavity made from the same waveguide? Homework Equations Unsure The Attempt at a Solution I would have...
  14. C

    Finding Fundamental Group: Step-by-Step Guide

    I'm studying for an exam which is a couple months away and I found an old exam which asks the following: Find the fundamental group of: a) The closed subset in R3 given by the equation x - y^2 -z^2 = in the standard coordinates. b) The closed subset in R3 given by the equation x - y^2 -z^2...
  15. L

    The Fundamental Theorem of Calculus, Part I

    According to Wikipedia, "the first fundamental theorem of calculus shows that an indefinite integration can be reversed by a differentiation." Am I wrong or is this theorem very simple? Indefinite integrals are the same as antiderivatives. So isn't this theorem simply stating that the...
  16. L

    Fundamental Theorem of Calculus and Line Integrals: Does it Apply?

    If I draw a random curve over a scalar field, then it is not generally true that the line integral of the scalar field over the curve equals the difference between the value of the antiderivatives of the scalar field at the beginning and finishing points of the curve, as one can clearly see by...
  17. M

    Fundamental theorem of calculus

    Homework Statement Does the function F(x)=int(sin(1/t)dt,0,x) (integral of sin(1/t) with lower limit 0 to upper limit x) have a derivative at x=0? Homework Equations The Attempt at a Solution I was thinking that F(x) shouldn't have a derivative at x=0 because the integrand isn't even...
  18. Phrak

    What is the Fundamental Theorem of Computer Science?

    What is the Fundamental Theorem of Computer Science? No such formally named theorem exists if I do a google search. But I'm very curious as to what students and professors of Computer Science might think it should be. Anyone?
  19. James889

    Question about the fundamental theorem of Calculus

    Hi, Suppose we're asked to find the derivative of the integral f(x)~=~\int_{-13}^{sin~x} \sqrt{1+t^2}~dt Now, the solution apparently looks like this: f(x)^{\prime} = \sqrt{1+sin^2(x)}\cdot~cos(x) Why? Why does the solution contain the upper limit `plugged in` ? A more sensible...
  20. M

    Pilot wave theory, fundamental forces

    I read a short high-level article about the pilot wave interpretation of quantum mechanics and I have some questions. Is there a good way to formulate that theory so that the only force on a particle is from the pilot wave (inertia, gravity, EM, ... move/effect the wave which in turn...
  21. P

    Disparity of fundamental forces

    Homework Statement interested in getting reliable information on the actual strength of the fundamental forces,including the strong nuclear, weak nuclear, electromagnetic and gravitational forces.What is their relative strengths one to another? Homework Equations The Attempt at a...
  22. marcus

    What is emergent ; what is fundamental ?

    What is "emergent"; what is "fundamental"? Bee Hossenfelder had an interesting discussion of the term "emergent" back in mid 2008: http://backreaction.blogspot.com/2008/04/emergence-and-reductionism.html There are apparently several ways the term is used in theoretical physics. Several...
  23. Q

    Forces by Proxy: Exploring a New Theory of the 4 Fundamental Forces

    So there is a theory out there that seems to model the "four" fundamental forces based upon a higgs-like particle, which the author calls k-particles. He seems to believe that these particles act upon electrons, protons, fermions, etc. to produce forces by the means of k-particle density. It's...
  24. Pythagorean

    What are the fundamental forces involved in the chemical reaction?

    I'm assuming the standard reductionist viewpoint: that chemistry is simplified quantum physics. Of the four fundamental forces in modern physics, what forces play a role in chemical reactions? Is it safe to assume it's solely electromagnetics?
  25. Ranku

    How is the strength of fundamental forces measured?

    What is the unit of 'strength' of the fundamental forces? When it is said that gravitational force is weaker than the strong nuclear force, how are they compared? What is meant when it is said that the fundamental forces approach each other in strength at higher energies in the early universe?
  26. J

    The Fundamental frequency of an open organ pipe

    Homework Statement The fundamental frequency of an open organ pipe corresponds to the note middle C (f = 261.6 Hz on the chromatic musical scale). The third harmonic (f3) of another organ pipe that is closed at one end has the same frequency. Compare the lengths of these two pipes...
  27. G

    Fundamental Quantum Probability

    Let |\psi_n\rangle\in\mathcal{H}, where \mathcal{H} is Hilbert space, be orthonormal states forming a complete set, and n\in\mathbb{N}. Let |\Psi\rangle=\sum_{n=1}^N c^{(1)}_n|\psi_n\rangle, where c_ns are normalized coefficients and N is either finite or infinite. Let m be an eigenvalue of...
  28. D

    Finding General Solution / Fundamental matrix

    Morning everyone, Studying for a test and having a problem on a practice question he gave us to study with. Here's the question along with the answer: Y' = AY + [e^t e^-t 0] with A = [-1 0 4 -0 -1 2 0 0 1] the...
  29. S

    What is the fundamental identity used to prove csc2α - 1 = cos2α / csc2α?

    Homework Statement Okay, so this is some trig I learned last year but have since forgotten. If you can give me the first step, I can solve the rest on my own. The given statement is true and you have to prove why using Pythagorean Identities. csc2\alpha-1 = cos2\alpha ________ csc2\alpha
  30. W

    Fundamental Theorem of Calculus concept

    I just learned about the fundamental theorem of calculus. I can see that this ties together differentiation and intergration, but I was wondering what kind of problems can be solved by using this theorem? In other words, what can the theorem be applied to?
  31. S

    Fundamental Matrix: Proving Vectors Solutions

    Homework Statement Show that the vectors \overline{x}(t) = \begin{bmatrix} e^t \\ -t \end{bmatrix} and \overline{y}(t) = \begin{bmatrix} t \\ e^(-t) \end{bmatrix} are solutions for \overline{x}' = \frac {1} {1+t^2} \begin{bmatrix} 1+t & e^t(1-t) \\ -e^-(t) (1+t) & t-1...
  32. R

    Tension and fundamental frequency

    Homework Statement A particular violin string plays at a fundamental frequency of 294Hz. If the tension is increased 15%, what will be the new fundamental frequency? Homework Equations f=v/2L v=sqrt(T/(m/L)) The Attempt at a Solution 294 = sqrt(T/(m/L))/2L so...
  33. D

    Proving Periodicity for a Function with Two Elements and a Constant Period

    Homework Statement The elements of f(t) are 1+kt and t here are 0<t<1 the other element is 1 and t here are 1<t<2. f(t+2)=f(t) prove if k is not equal to zero, the period f is 2. i know the graph of element 1 but how should I graph the other element with k. Based from the graph...
  34. W

    Current Loops and fundamental group

    The Law of Biot and Savart Law tells us how to find a differential form that generates the first de Rham cohomology of S3- embedded loop. Run a steady current through the loop. This form is just the dual of the induced magnetic field (using the Euclidean metric). Ampere's Law tells us...
  35. F

    Is the Fundamental group of the circle abelian?

    Homework Statement Is the Fundamental group of the circle (S^1) abelian? Not a homework question, just something I want to use. Homework Equations The Attempt at a Solution Intuitively it appears to be and it is isomorphic to the additive group of integers which is abelian. I...
  36. D

    What is the Fundamental Period of sin(5t + π) on the Interval (-∞, +∞)?

    Homework Statement The function is defined on (-infinity, +infinity) Find the fundamental period of sin(5t+\pi) Homework Equations The Attempt at a Solution f(x)=sin(5t+\pi) f(x)=f(x+2p) f(x+2p)=sin(5t+\pi+2p) f(x+2\pi/5)=sin(5t+\pi+2\pi/5) The period i got was...
  37. T

    Can the Fundamental Group of a Genus 2 Surface Embed That of a Genus 1 Surface?

    Can someone please explain this to me. Let X = fundamental group of a genus 2 surface Let Y = fundamental group of a genus 1 surface Can X embed Y? Ty
  38. G

    Fundamental Group of the projective plane after we remove n points?

    So I have been wondering, what is the fundamental group of a projective plane after we remove n points? I tried doing this using Van Kampens Theorem, maybe I am applying in incorrectly, I am getting that it is the Free group on n generators. However, when I think of RP^2 as a quotient of...
  39. C

    Is conservation laws are more fundamental than Newton's second law?

    Is conservation laws are more fundamental than Newton's second law in Newtonian mechanics? I know from the point of view of Noether's theorem conservation laws are more fundamental. But all the conservation laws can be derived from the F= ma equation. And from these conservation laws I can't...
  40. S

    Tripling fundamental frequency

    So the question goes like this- The fundamental frequency of a bass violin string is 1045 Hz and occurs when the string is 0.900 m long. How far from the lower fixed end of the bass violin should you place your fingers to allow the string to vibrate at a fundamental frequency 3 times as great...
  41. A

    Vector calculus fundamental theorem corollaries

    Homework Statement Prove \int_{V}\nabla\ T d\tau\ = \oint_{S}Td\vec{a} Homework Equations Divergence theorem: \int_{V}(\nabla\bullet\vec{A})d\tau\ = \oint_{S}\vec{A}\bullet\ d\vec{a} The Attempt at a Solution By using the divergence theorem with the product rule for...
  42. M

    What is the fundamental period of 3cos(1.3PiN) - 4sin(0.5piN + 0.5Pi)?

    Hi all, First post here for me. 1. Problem Statement Find the fundamental period of: 3cos(1.3PiN) - 4sin(0.5piN +0.5Pi) 2. Relevant equations: T = 2PI / w 3. My attempt: Not sure how to proceed. By themselves the fundamentals period would be: 3cos(1.3PiN) w = 1.3Pi...
  43. A

    The Fundamental Relation of Thermodynamics

    What does it mean to say that the fundamental relation of thermodynamics is homogeneous first-order? I struggle with the abstraction of mathematical definitions and, I'm really seeking more to understand the relation of the physical variables of thermodynamics. Why are all of the variables...
  44. J

    How do you determine a f(x) is periodic and fundamental period

    OKay so here are two examples: let say 1) f(x) = cos 2x + cos x how do you determine whether this function is periodic? I know I have to find period for each cosine. The first one is pi and second is 2pi. Then what do you do? And eventually, how do you determine the fundamental period...
  45. G

    Question on finding the Fundamental Period of a periodic FM signal

    Homework Statement s(t)=13*cos(2000*pi*t + 100*sin(8*pi*t)) Determine the fundamental period of this signal, i.e., the shortest period. Homework Equations x(t)=A*cos(2*pi*f(c)*t + a*sin(2*pi*f(m) + phi)) A=13; f(c)=1000; a=100; f(m)=4; phi=0 Inverse Euler's - A*cos(omega*t + phi) =...
  46. S

    Beta decay fundamental question

    Can someone please clarify beta decay for me. As I understand it, in B- decay, a neutron turns into a proton, electron, and anti-neutrino, then the electron and anti-neutrino are ejected. But from this example, it looks like there are two additional electrons as a result of decay. 19/8 O ->...
  47. D

    Fundamental Theorems of Calculus and Chain Rule: Computing h'(t)

    Homework Statement Heres the question: f(t) = Integral from c to t of a(s)ds, and h(t) = ef(t) Compute h' (t) using: 2 fundamental theorems of Calculus and The Chain Ruleany help please ? Homework Equations Fundamental Theorems of calc g(x) = Integral from a to x of f(t) dt and...
  48. C

    What are the fundamental information-processes in physics?

    I’ve been thinking about comments made by Fra in a number of threads, where he raises questions like – what does an observer “see” at the sub-atomic scale? We could make a long list of more and less reputable ideas about the fundamental information-processes in physics, going back to Wheeler’s...
  49. F

    Fundamental Theorem of Arithmetic Problem

    Problem number 4 on the image has me stumped. I understand the problem (obviously not enough) and what its saying, I'm just having trouble putting it into a proof. Can i get a hint to get me started? Thanks http://img40.imageshack.us/i/asdasdjql.jpg/"
  50. facenian

    Little fundamental question

    Hi when a system C is composed of two subsystems A and B each described by states spaces Ha and Hb then we are told that the state space corresponding to C is the direc produc space H_a \otimes H_b. Shouldn't this be considered a postulate? or it is a logical consecuence of something more...
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