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. T

    Fundamental reason behind Aufbau principle

    This principle states that the electron starts filling from the lowest energy up. Why?
  2. T

    Problem With fundamental theorum of Calculus

    Homework Statement The question: F'(\pi/2) if F(X)= \int^{cosx}_{0} e^{t^{2}}The Attempt at a Solution I thought I thought F'(X) = f(t) = e^{t^{2}} replacing t with cos^{2}x But my book writes: F'(x) = (-sinx)e^{cos^{2}x}
  3. R

    Using the Fundamental Theorm of Calculus II to evaluate indefinite integrals

    Homework Statement Evaluate the following definite integrals using the Fundamental Theorem of Calculus: 1. 1 \int 4 ( (5x2+7x+5)dx ) 2. -5pi/6 \int 4pi/6 (−6sinx+7cosx) dx ) 3. 2 \int 4 ( (e^-4x)/((e^(-4x))+7) dx ) If it's unclear, the number on the...
  4. J

    Maxwell Eqns - Prob converting from fundamental to time-harmonic form

    Hi all Im trying to derive the time-harmonic form of the Maxwell equations from the original, time-dependent form, however I am not sure if my working and logic is correct e.g. for \nabla\times E = -dB/dt I want to be able to get \nabla\times \vec{E} = -j\omega\vec{B} where...
  5. P

    Fundamental Theorem of Calculus problem

    Homework Statement The FTC states that {d\over{dx}}\int_a^x f(t)dt = f(x) Now, how do I do something like {d\over{dx}}\int_a^{g(x)} f(t)dt = ?Homework Equations The Attempt at a Solution I know that it has to do with the chain rule, but I forgot my textbook at school and I can't seem to find it...
  6. K

    Fundamental therom of calculus

    Hi.guys i have some confusion on the fundamental therom of calculus. here is my question. 1. somethimes when i use the fundamental therom of calculus which is F(a) - F(b). the out come will be negative .in the same question when i type in my calculator its only give me positive i guess is...
  7. H

    What Is the Fundamental Frequency of a Wire Under Modified Conditions?

    Homework Statement A stretched wire vibrates in its fundamental mode at a frequency of 400Hz. What would be the fundamental frequency if the wire were half as long, with twice the diameter and with four times the tension? Homework Equations fn = nV / 2L where n is the harmonics...
  8. R

    How Do the Two Theorems of the Fundamental Theorem of Calculus Imply Each Other?

    There are two theorems: The fundamental theorem of calculus: \int_{a}^{b}F'(x) = F(b) - F(a) And the theorem that states if f is continious on [a,b]and g:[a,b]->R is defined by g(x) = \int_{a}^{x}f(t) dt, then g is differentiable on (a,b) and; g'(x) = \frac{d}{dx}(\int^{x}_{a}f(t)) =...
  9. F

    Fundamental Frequency of an organ

    Homework Statement An open organ pipe (i.e., a pipe open at both ends) of length L0 has a fundamental frequency f0. Part A If the organ pipe is cut in half, what is the new fundamental frequency? 4f0 2f0 f0 f0 f0 Part B Part C This part will be visible after you complete...
  10. B

    Fundamental question about 'smell'

    As I understand it, the sensation of smell is caused by small amounts of matter leaving the body before being incident on olfactory receivers in our nose. Now, since the 'smell' has mass and it is leaving the body, could there eventually arise a situation where some body loses all its mass as...
  11. N

    Unable to do fundamental factoring

    I have been trying this problem for a long time now. Factor the polynomial t^7+2t^6-24t^5. So far I came up with t^n(t+x)(t+y). I have no clue on how to work this out because i am only able to do them to the 2nd degree. I can really use the help. Thanks, Neil
  12. P

    Why ampare is fundamental unt in S.I rather than charge

    hii i have a problem that why ampare{unit of current} is fundamental unit, ampare is I=Q/t and it is drived.while electric charge is not a drived unit and it also measured by millican mattod.and if we place coulomb{unit of electric...
  13. J

    A question on fundamental dimensions

    Well i do understand about the 8 fundamental dimensions in physics, but i cannot understand what makes only the length, mass and time to commute with one another to produce derived units like force or acceleration. Why cannot we have a unit something like K^2 or sqrt(candella)?
  14. R

    Bell Labs Kills Fundamental Physics Research

    http://blog.wired.com/gadgets/2008/08/bell-labs-kills.html" Six Nobel-prize worthy research came from that lab, including the invention of the silicon transistor, and the laser! It was a business decision by the parent company but, I think it's a very myopic business decision to abandon...
  15. J

    Elegant Universe: No fundamental physical laws beyond strings?

    I've just read in the first pages of The Elegant Universe that Greene believes that all fundamental laws should be contained in the microscopic domain, all other laws are derived. Is it really so evident that the laws of thermodynamics are derived and not fundamental? The second law, is it...
  16. M

    Is the Schroedinger equation not fundamental in BCS theory?

    Can the Schroedinger equation be derived from first principles in BCS theory? Zapperz suggests the Madelung and Schroedinger equations are not fundamental in BCS theory. If such is the case, how exactly does such a derivation work?
  17. Y

    Organ Pipe and Fundamental Frequency

    Homework Statement The organ pipe is 2.0 m long, was open at both ends, and was originally tuned to a fundamental frequency of 128 Hz (C below middle C). a) what is the wavelength of the fundamental? b)if the note you now hear is closer to 262 Hz (middle C), where is the blockage with...
  18. J

    Integral and Fundamental Theorem of Calculus

    Homework Statement Find 2 \int 0 \ (2-x^2)dx using def of the definite integral and using FTC Homework Equations The Attempt at a Solution any help on how to start would be great?
  19. I

    Find Local Max of f(x) w/ FTC2

    Homework Statement Question One: Find a continuous function f and a number a such that 2 + \int_{a}^{x} \frac {f(t)} {t^{6}} \,dt = 6 x^{-1} Question Two: At what value of x does the local max of f(x) occur? f(x) = \int_0^x \frac{ t^2 - 25 }{ 1+\cos^2(t)} dt The attempt at a solution I...
  20. S

    Fundamental theorem of calculus

    Homework Statement Evaluate each definite integral. ok I'm not sure how to do the integration sign, but... b=4, a=1 for (5yy^.5)+3y^.5)dy Homework Equations The Attempt at a Solution I'm not really sure what I'm doing wrong. i integrated...[tex]5y^2/2*y^(3/2)/(3/2)+3y^(3/2)/(3/2) then i...
  21. P

    Wavelength and Fundamental Frequency of Standing Wave on 2m String

    A standing wave resonates at 400 Hz with three antinodes on a string tied tightly between two posts 2.0 meters apart. What is the wavelength of this standing wave? What is the fundamental frequency of this string? I tried to draw the picture out, and it looks like the wavelength is just...
  22. S

    Some Fundamental Interactions Of Matter Found To Be Fundamentally Different

    Some Fundamental Interactions Of Matter Found To Be Fundamentally Different Than Thought ScienceDaily (July 2, 2008) — Collisions have consequences. Everyone knows that. Whether it's between trains, planes, automobiles or atoms, there are always repercussions. But while macroscale collisions...
  23. O

    What is the intuitive idea behind the fundamental theorem of calculus?

    Can somebody explain to me, geometrically and intuitively, the fundamental theorem of calculus? I understand that i can find the area between the graph of f'(x) and the x-axis where b>x>a by finding f(b)-f(a), but i don't understand why. Suppose something is too hard to integrate. Can i...
  24. D

    Quantum Mechanics: Fundamental Question

    Hi, I'm extremely new to quantum mechanics (my only knowledge of quantum mechanics is that taught in physics C, ergo none, and a bit of wave mechanics... however I have a pretty strong mathematics background... diff-eqs + linear alg. + vector calc.), and was wondering as to how the derivation of...
  25. humanino

    Utility of fundamental research ?

    Hi everyone, Can scientific research be founded on utility ? I expect this to be a contreversial topic. I certainly do not want to bias the answers by my own opinion. I was surprised to discover a few other researchers' opinions. Sorry if this does not belong to GD, please move to the...
  26. M

    About the origen of the fundamental theories

    Hi , i have this question - it's not an idea , and it's not a theory indeed , it's merely a question - we all know that the theory of relativity is inspired by maxwell theory of electromagnetism , from which the constancy of speed of light in any frame of reference is derived . and we know that...
  27. M

    Fundamental trignometry problem

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  28. M

    Exploring the Discontinuity of Fundamental Quantities and Calculus

    i wondered at the idea that calculus works for continuous functions and in reality fundamental quantities are discontinuous. For example energy or an electron can't asume all values. Therefore isn't there a conflict when we work on a equation like dE/dt or something similar which involve...
  29. L

    What is charge? Why do fundamental particles possess it?

    I attended a colloquium a few days ago concerning the LHC and the search for the Higgs Boson. After some thought, I have the following question. Among other things, there are two measurable properties of all of the fundamental particles in the Standard Model: mass and charge. It is...
  30. W

    Why Do Fundamental Trig Identities Confuse Me?

    I just don't get this stuff. I've been trying on my own with the book. Also, is there a better way to post this? Homework Statement tanx 1 + secx _________ + _________ = 2csc x 1 + secx tanx I need to prove that this side equals the other. Homework...
  31. M

    How Do You Find a Fundamental Set of Solutions Using Jordan Canonical Form?

    Homework Statement Find a fundamental set of solutions to the system of differential equations: y1' = 3*y1 - y2 y2' = y1 + y2 by reducing the problem to Jordan Canonical Form. Homework Equations y' = Ay J = P^{-1}AP The Attempt at a Solution I have found the following...
  32. C

    How Does Opening a Parachute Affect a Falling Parachutist's Motion?

    Homework Statement A parachutist of mass 70kg, in free fall at 136 km/h, opens a parachute. if the chute creates a 895 N air resistance force and the parachutist is being pulled down by a force of gravity at 686 N a) find hte acceleration of the parachutist b) how far does she fall in 5.0 s...
  33. A

    Discovering the Fundamental Ideas of Physics: From Atoms to Conservation Laws

    Here is an interesting question that my urge for simplification and basic/fundamental understanding has lead me to ask: -------------------------------------------------------------- What would you say are the most fundamental ideas of Physics, the most fundamental science? By fundamental what...
  34. C

    Fundamental Solutions for ODEs with Continuous Coefficients

    Suppose that p and q are continuous on some open interval I and suppose that y1 and y2 are solutions o the ode y''+(t)t'+q(t)y=0 a. Suppose that y1 , y2 is a fundamental set of solutions. Prove that z1, z2 given by z1=y1+y2, z2=y1-y2 is also a fundamental set of solutions. b. prove that...
  35. C

    Can z1 = y1 + y2 and z2 = y1 - y2 Form a Fundamental Set of Solutions?

    Suppose that p and q are continuous on some open interval I and suppose that y1 and y2 are solutions o the ode y''+(t)t'+q(t)y=0 a. Suppose that y1 , y2 is a fundamental set of solutions. Prove that z1, z2 given by z1=y1+y2, z2=y1-y2 is also a fundamental set of solutions. b. prove that...
  36. J

    Finding the Fundamental Frequency and Length of an Open Pipe at 20°C

    My question goes as follows: An open pipe in air is designed to produced 2 successive harmonics at 235 Hz and 275 Hz at 20 degrees Celsius. What is the fundamental frequency? What is the length of the pipe (m)? I'm not sure how to solve this problem, as I can't find the equation to...
  37. J

    Fundamental Frequency and Pipe Length Calculation for Open Pipes

    My question goes as follows: An open pipe in air is designed to produced 2 successive harmonics at 235 Hz and 275 Hz at 20 degrees Celsius. What is the fundamental frequency? What is the length of the pipe (m)? I'm not sure how to solve this problem, as I can't find the equation to...
  38. R

    Fundamental Matrices in Differential Equations

    The question: Find the fundamental matrix for the given system of equations, and then find the specific solution to satisfy x(0) = I. x' = \left(\stackrel{1}{5}\stackrel{ -1}{ -3}\right)x My solution: I found the eigenvalues to be r_{1}= -1 + i and r_{2}= -1 - i I then found the...
  39. H

    Ignoring Fundamental Theorem of Polynomials (a^n + b^n)

    Homework Statement Okay, basically why does (a_{}n + b_{}n) ignore the Fundamental Theory of Polynomials? Homework Equations ... I could post them here, but basically when n is odd (a_{n} + b_{n}) = a series that looks like this: (a+b) (a_{n-1} b_{0} - a_{n-2}b +a_{n-3}b_{2} + ... +...
  40. C

    Linear Algebra: 4 Fundamental Subspaces

    Homework Statement Without computing A, find the bases for the 4 fundamental subspaces. [1 0 0][1 2 3 4] [6 1 0][0 1 2 3]=A=LU [9 8 1][0 0 1 2] Homework Equations N/A The Attempt at a Solution There was an "example" in the book. It just showed the answers. It was: [1 0...
  41. S

    What Happens When Atoms Age and Decay: Fundamental Forces at Play?

    IDK if my understanding is right...but as far as I understand physics...all the fundament forces shares 2 characteristics - attraction and repulsion...now, here are my questions.. 1. Assume there are only 2 uncharged atoms in the universe...would they come together or go apart or stay where...
  42. C

    Thermodynamics Finding a fundamental equation.

    Thermodynamics! Finding a fundamental equation. This is the question: "While he Gibbs free energy G is the fundamental function of the atral variables (T,p,N), (T=temperature, p=pressure, N=number of molecules), growing biological cells often regulte not the numbers of molecules N, but the...
  43. E

    Fundamental theorem of algebra

    [SOLVED] fundamental theorem of algebra Homework Statement Theorem: If F is a field, then every ideal of F[x] is principal. Use the above theorem to prove the equivalence between these two theorems: Fundamental Theorem of Algebra: Every nonconstant polynomial in C[x] has a zero in C...
  44. A

    Fundamental Lemma of Variational Calculus

    I was just going through the derivation of the euler - lagrange equation that rests on the proof of the fundamental lemma of the calculus of variations which states the following: If a function g(x) vanishes at the endpoints i.e g(a) = 0, g(b) = 0 and is continuously differentiable within...
  45. DaTario

    Four or Five fundamental forces

    Hi all In Paul Tipler's book on modern physics (with Ralph LLewelyn) I read an explanation for the formation and stability of a molecule, which is based on Pauli's exclusion principle. This principle was responsible for a term in the energy equation, which yields also (naturally) a term in...
  46. I

    What are the fundamental symbols and concepts in logic?

    what are the fundamental symbols in logic? maybe this is a vague question but that's cause i don't know anything about logic. are they \neg \wedge \vee? do we define the material implication in terms of these symbols? basically I'm having trouble understanding the paradoxical nature of...
  47. D

    Why Isn't the Inverse of My Fundamental Matrix Correct?

    Hello, I have a little problem. I´ve calculated the fundamental matrix of a EDO system, such that: M(t) = P * exp( J*t) where J is a diagonal matrix: J = [-3 , 0 ; 0 , 1] and P = [1 , 1 ; 3 , -3] The problem arise when I try to find the inverse matrix of M. What I do is this As we know the...
  48. marcus

    Only two fundamental constants? (Matsas in Nature News)

    http://www.nature.com/news/2007/071220/full/news.2007.389.html this just came out in Nature News, comment? To be sure, they are talking about how many DIMENSIONAL fundamental constants you need. The main bunch of constants that people always want to measure and to explain are the...
  49. H

    Help w/First Fundamental Theorem of Calculus

    For the first fundamental theorem of calculus, I don't quite understand it... I think I got the integral part, but not the interval [a,x]... can you guys help me? thx
  50. L

    Definite Integral w/ Second Fundamental Theorem of Calculus

    Homework Statement Find F'(x) if F(x)=\int_{0}^{x^3}(\sin (t^2))dt The Attempt at a Solution Here's what I did: F(x)= -\cos (t^2)\biggr]^{x^3}_{0} and I get: F(x)= -\cos (x^6) +1 F'(x)= sin (x^6)(6x^5) However, the book's answer is F'(x)= 3x^2 \sin(x^6) How did...
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