What is Fundamental: Definition and 947 Discussions

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

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  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?

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  10. T

    Fundamental Theorem of Arithmetic

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  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

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  13. H

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  14. C

    Finding Fundamental Group: Step-by-Step Guide

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  15. L

    The Fundamental Theorem of Calculus, Part I

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  16. L

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

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  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

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    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

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  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

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  24. Pythagorean

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  25. Ranku

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  26. J

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  27. G

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  28. D

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  29. S

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  30. W

    Fundamental Theorem of Calculus concept

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  31. S

    Fundamental Matrix: Proving Vectors Solutions

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  32. R

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    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

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    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 (-∞, +∞)?

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  37. T

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  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

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  41. A

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    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

    Fundamental period of a signal

    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

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    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

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    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

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    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

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    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

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  50. facenian

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