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. O

    Gap in fundamental knowledge of differential equations

    This is a question where the answer is given but I don't understand how to derive it. The answer to the homogeneous eq. is: A cosh(x) + B sinh(x) with no explanation of how it is arrived at and the given answer then goes on to describe how the non-homogeneous part is solved. Diff. eq. ...
  2. Mk

    Zero point energy as 5th fundamental force

    I heard it suggested that the casmir effect could be related to a 5th fundamental force. I thought it was interesting, but can't explain my reasoning to thinking it was. Why couldn't it be? I almost stutter. Input?
  3. A

    Fundamental process of a photon elastically scattering from an electon

    I learned a lot from the discussion of the converse problem, photon absorption. What about this case? Perfectly elastic.
  4. A

    Fundamental Process of an electron absorbing a photon

    Hello I was thinkng of this question and I found it best expressed in a clear manner in another site. I quote the question and waiting for an asnwer: "Is it a fundamental property of electrons that they can absorb photons? Geometry requires that two points define a straight line. Geometry...
  5. B

    Part 2 of the Fundamental Theorem of Calculus

    Hello, I have a problem that I am getting stuck simplifying further. The problem asks me to find the integral if it exists using Part 2 of the FTC. I know that the second part of the FTC says:\int_{a}^{b} f(x)dx = F(b) - F(a) Where F is the anti-derivative of f. Here is the problem...
  6. A

    Current: Why It's the Fundamental Unit & What Defines It

    Why the current is fundamental quantity? Why the unit of fundamental quantity current is defined in terms of force?
  7. F

    How does the concept of fundamental velocity relate to parallel rates?

    I hope you can follow my train of thought (I'm not being elitist here, I am open to the possibility that my post may be unintelligible) 1. Velocity is a parallel rate. The distance/time formula with which we are familiar is an expression of the number of meters laid end to end alongside...
  8. wolram

    Is chaos more fundamental than any scientific rule?

    Is kaos more fundamental than any scientific rule, is it kaos that prevents us from knowing our origin and futuer?
  9. C

    Why aren't all overtones integer multiples of the fundamental?

    When plucking a string on an instrument, are all the overtones heard produced by the string itself (assuming all other strings are muted)? Would plucking the string without muting the others make a significant different? Another thing, why aren't all overtones integer multiples of the...
  10. L

    Fundamental Frequencies and Standing Waves

    Evening, Ive been having some trouble with questions about fundamental frequencies and labelling nodes and anti nodes in an air column. for example :a tuning fork is placed above a glass tube and then the sound of the air column is adjusted by raising or lowering the glass tube in the...
  11. R

    Fundamental Frequency of the earth's crust

    Here's one for you. Is there a way of working out what the fundamental frequency of the Earth's crust is? Has anybody done this. And what would happen if we somehow matched this frequency, say by all dancing to a particularly banging dance remix of Electric Light Orchestra's "Mr Blue Sky" all at...
  12. J

    What are the fundamental units of nature?

    So far, all i can think of is: mass, time, length & charge (maybe?). Temperature isn't even a fundamental unit, since it is a measure of kinetic energy of a particle (1/2*m*vrms^2=3/2*k*T, i think that's right...), am i missing any? If that's all of them, then that's pretty amazing to me.
  13. B

    Solution to Exercise: Fundamental Matrix for System of ODEs

    Hi, Please can someone help me on how to do this exercise. Give a fundamental matrix for the system: {x'(t)=-y(t) {y'(t)=20x(t)-4y(t) the solution is like: {v1(t)=e2t*cos(4t)[1;4]+e2t*sin(4t)[-1;-2], v2(t)=e2t*cos(4t)[1;4]+e2t*sin(4t)[0;-4]} [1;4]...are colunm vectors. IT is just...
  14. H

    Why fundamental forces change their strength?

    Yeah, same as above, Why do fundamental physical forces change their strength with distance? Shouldn't the force be the same everywhere? Why or Why not? :-p (how do you know, and what did you get the answer from) Thanks,
  15. K

    What Are the Lengths of Open and Closed Organ Pipes Based on Their Frequencies?

    I do not understand what is meant in this question: An open organ pipe has a fundamental frequency of 430 Hz. A closed organ pipe has a fundamental frequency that is the second harmonic of the open organ pipe. What are the lengths of the two pipes? Does the bolded part mean that the...
  16. D

    Natural and Fundamental Frequencies

    How could you tell if you had missed one of the natural frequencies when you were finding natural frequencies above the fundamental? thnx a lot
  17. K

    Directed Integration and the Fundamental Theorem of Calculus

    I'm still new to much of this stuff, so I do not claim to be an expert. But I thought I'd still comment. I'm trying to better understand the geometric calculus of David Hestenes: http://modelingnts.la.asu.edu/ A fairly common form for the Fundamental Theorem of Calculus is: \int_S...
  18. M

    The Fundamental Theorems of Calculus

    Can someone break down these theorems for me please because my book is horrible at explaining them. The examples the book gives shows the initial question but then the answer and none of the steps in between even on the very simple questions. I'm confused. ~Kitty
  19. N

    Tully-Fisher, fundamental plane - what about LSBs, dwarfs, etc?

    In https://www.physicsforums.com/showthread.php?t=112890", Space Tiger mentioned the Tully-Fisher relationship for spirals, and the Faber-Jackson relationship, "part of the fundamental plane of elliptical galaxies". Does anyone know of a review article on the extent to which these apply, to...
  20. J

    Fundamental theorem of calculus

    (that's a 3 on the last integral) http://img131.imageshack.us/img131/2549/jesus1cj.png I need to find which of those are true, now I thought I and III were true for sure. But when I do II with an example f(x) = x^2 I get x^2 - 9, so it's not true right? (I and III are not choices given for the...
  21. E

    Finding Fundamental Period of x(t) with 2 Exponentials

    I can't get my answer match up with correct answer in the book: i need to find fundamental period of this signal: x(t) = 2 cos(10t + 1) - sin(4t -1) i used formula for cosine Acos(wt + a) = ... that gives two exponentials... so I got pi/10 instead of pi for answer... any help is appreciated
  22. D

    Question about how two fundamental constants are measured

    How is the permittivity of free space, \epsilon_0, and the magnetic permeability in vacuum, \mu_o, measured? Thanks
  23. T

    What is the second fundamental form?

    OK, this is more of a spot for an elaboration on a question I just posted in another thread. Not quite duplicating threads, I hope, I just wanted to have this not buried in another spot... So, the question is this: Let's say that we have a smooth manifold \mathcal{M}that may be viewed as a...
  24. S

    Is There a Fourth Fundamental Force Discovered in 1986?

    We know that there are three fundamental forces: 1. Gravitational Force 2. Strong Nuclear Force 3. Quark-Quark Force (unified by Weak Nuclear and Electromagnetic Force) I've heard about the fourth fundamental force which was found in 1986 and changed the relation: F = G*m*M/R*R to some...
  25. D

    The Fundamental Theorem of Calculus

    Id like to know if the following argument is valid. Take an arbitrary function f(x). f(x)dx can be thought of an infinitesimal area of a certain form (I emphasise this because I use it later in the argument) determined by the form of the function f(x). Let's denote its integral by Y. \int{...
  26. B

    Fundamental Questions: Voltage, Current & AC Signals

    I am confused with some fundamental things... Does voltage and current go in opposite direction when represented in a drawn circuit? Does current go the way opposite to the actual direction of electrons flow? In an AC signal, for example, a generator is placed at point B and it's...
  27. J

    What are the fundamental physics equations?

    Howdy all, I've begun to get started in learning about physics, but am still lightyears removed from any of the mathematical aspects of the science. Thus I am now looking to move from themes to numbers (and I'm kinda nervous :rolleyes: lol) Anyway, here's where I'm heading currently...
  28. E

    Fundamental frequency of a guitar string?

    Question: One of the 63.5-cm-long strings of an ordinary guitar is tuned to produce the note {\rm B_3} (frequency 245 Hz) when vibrating in its fundamental mode. 1. If the tension in this string is increased by 1.0%, what will be the new fundamental frequency of the string? The first part of...
  29. M

    Applying the fundamental theorem of calc to calc III, confused

    Hello everyone, i have the following problem I'm confused about! Can anyone guide me to what I'm suppose to do? I tried the following but it was wrong, he then told me I can just apply the theorem to |r'(u)| insteed of the stuff under the square root which would be more difficult. This is...
  30. C

    An extension of the Fundamental Theorem of Calculus

    In a book of Introduction to Probability I found this statement: " Let be F(x) = \int_{-\infty}^{x} f(t)dt. Then, by the Fundamental Theorem of Calculus, F'(x) = f(x)." With the minus infinity on the lower limit, it is this a valid aplication of the FTC? Thanks.
  31. Loren Booda

    Are all fundamental particles singular?

    Is there a fundamental particle (like a Planck black hole) that has a finite radius?
  32. W

    How Do You Calculate the Phase Difference Between Two Sinusoidal Waves?

    2 sinusiodal waves are defined y1=2sin(20x-32t) y2=2sin(25x-40t) what is the phase difference between this two waves at point x=5 and t=2s. all length in cm. I sketch the graph y/x and y/t and both have their max amplitude at x=3.2 , and t=3.125s. so what is the different in phase...
  33. A

    Mathematica Fundamental rules for physics mathematical derivations.?

    Hello. I am in confusion because I cannot understand which formula will be used in this and that derivation,because there are many formulas of physics.Then how we come to know that this specific formula must be used in this derivation It is a great confusion for me. Please solve my problem...
  34. mattmns

    Fundamental Theorem of Line Integrals

    Quick question, what is the approach to this problem? Keep in mind I am supposed to use the Fundamental Theorem of Line Integrals. \int_{C} 2ydx + 2xdy Where C is the line segment from (0,0) to (4,4). Unless I am missing something I need to make that into the form of \vec{F} \cdot...
  35. Loren Booda

    The most fundamental physics that we can observe directly with our senses?

    What is the most fundamental physics that we can observe directly with our senses?
  36. J

    Confusion on just what fundamental means

    Confusion on just what "fundamental" means I just read the Tao of Physics and I'm slightly confused. Are quarks and leptons fundamental units of matter or are they temporal manifestations of energy with E=mc^2? Is is that all matter is comprised of bundles of energy and that quarks and...
  37. M

    Relativity vs. QMwhich is more fundamental

    I have my own views on the question of between the two, QM or Relativity, is one more fundamental. It seem that Rel. is based on fundamental principles, i.e. the principle of relativity. As I see it this could never be violated but some of the foundational principles of QM could turn out to be...
  38. A

    Fundamental Frequency of Helium-Filled Pipe

    A certain pipe produces a fundamental frequency f in air. If the pipe is filled with helium at the same temperature, what fundamental frequency does it produce? (Take the molar mass of air to be M_air, and the molar mass of helium to be M_He). The ratio γ of heat capacities for air (7/5) and...
  39. M

    The significance of the speed of light as a fundamental constant

    The Doctrine of Wave particle Duality with its theories of the disassociation of particles (i.e the ability of particles possessing both dimensions and mass to be in two places simultaneously) leads in a logical succession of ideas to the “Many Worlds Theory”. This theory could be considered to...
  40. G

    Which Anatomy and Physiology Textbooks Are Best for Premed Students?

    Over this semester I studied quite a few physiology and anatomy books ranging from the classics of Gray's anatomy, Last's anat, Guyton & Hall's medical physiology and also Lange's histology and Lange's physiology. A word of advice to any premed student buying or reading from these books...
  41. K

    Fundamental Vote: Is It or Isn't It?

    Let's vote
  42. H

    Fundamental Frequency and Musical Harmonics

    Ok, I was given a question in homework and was never told the right answer. If the Fundamental Frequency = 15Hz, the 3rd Harmonic = ? I estimated it at 45Hz because of an example The example was; Harm 1 = 100 x 1 = 100Hz Harm 2 = 100 x 2 = 200Hz Harm 3 = 100 x 3 = 300Hz If so...
  43. S

    Would i use the fundamental counting principal or is this a permutation

    thare are 8 candidates for three student seats how many different ways can the seats be chosen? With the FCP I get 24 If it is a permutation I get 336 which is right and why? Thanks for the help
  44. S

    How can I correctly identify the flow of current in a node in a circuit?

    Hi Guys, I'm a first year Electrical Engineering student and we're just covering the basics of circuit analysis. Currently we are going over Nodal analysis. I'm having some trouble correctly identifying the currents that are entering and leaving a node in a circuit. I can get most simple...
  45. S

    Van Kampen's theorem and fundamental groups

    I didn't see a topology forum, so I thought I'd post this question here. Can anyone give any pointers on using van Kampen's theorem? I understand the basic way it works, decompose a space X into open, path-connected sets, say U and V. Then pi1(U) * pi1(V) = pi1(X)/N, where N is a normal...
  46. K

    Harmonics are integer multiples of a fundamental frequency

    I understand that harmonics are integer multiples of a fundamental frequency. Also, that the relative strengths of the harmonics are what make the same note on different instruments sound different. Why are these other frequencies made? How many integer multiples are there? Why do our...
  47. R

    More fundamental and more difficult

    Which one is more fundamental or more difficult in solving real-life problems, professional engineering work or theoretical research?
  48. C

    Fundamental Theorem of Calculus

    Could you check whether I am doing these questions right: 1. \int_{0}^{4} (2+x) dx . So I use the Fundamental Theorem of Calculus F(b)-F(a) and receive: \frac{(x+2)^{2}}{2} = F(4) - F(0) = 16 2. \int_{-1}^{1} (4t^{3} - 2t) dt = t^{4} - t^{2} = F(b)-F(a) = 0 3. \int_{0}^{3}...
  49. Reshma

    Questions on the fundamental theorems

    I have these two queries regarding the fundamental theorems: 1. The fundamental theorem for divergences is given by: \int_v (\nabla.\vec v)d\tau = \oint_s \vec v.d\vec a According to this theorem, the integral of the divergence of a vector over a volume is equal to its surface...
  50. P

    What is the FTC saying about the derivative of the integral of a function?

    Use part I of the Fundamental Theorem of Calculus to find the derivative of \int_x^{3} sin(x^3) dx F'(x)=_________________ (answer goes here) i think i need to integrate the problem first, but it seems impossible. can someone help?
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