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. Math Amateur

    MHB What is Stoll's definition of the natural logarithm function?

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's definition of the natural logarithm function (page 234 -235) The relevant section of Stoll reads as follows: In this section we read: " ... ... To prove (a), consider the function L(ax), x \gt 0. By...
  2. P

    Proving fundamental set of solutions DE

    Homework Statement Assume that y1 and y2 are solutions of y'' + p(t)y' + q(t)y = 0 on an open interval I on which p,q are continuous. Assume also that y1 and y2 have a common point of inflection t0 in I. Prove that y1,y2 cannot be a fundamental set of solutions unless p(t0) = q(t0) = 0.The...
  3. G

    Fundamental and contravariant representations

    The invariant of SL(2,C) is proven to be invariant under the action of the group by the following \epsilon'_{\alpha\beta} = N_{\alpha}^{\rho}N_{\beta}^{\sigma}\epsilon_{\rho\sigma}=\epsilon_{\alpha\beta}detN=\epsilon_{\alpha\beta} The existence of an invariant of this form (with two indices...
  4. Math Amateur

    MHB Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2

    I am reading Manfred Stoll's book: Introduction to Real Analysis. I need help with Stoll's proof of The Fundamental Theorem of the Calculus - Stoll: Theorem 6.3.2 Stoll's statement of Theorem 6.3.2 and its proof reads as follows: In the above proof we read: Since \mathscr{L}( \mathscr{P} ...
  5. G

    Fundamental frequencies of square wave and sine wave

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  6. T

    The Fundamental Theorem of Calculus

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  7. K

    Solve for y(x) using the Fundamental Theorem of Calculus

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  8. DrPapper

    Classical Fundamental Principles of Classical Mechanics - Kai S. Lam

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  9. P

    Wave Problem -- Total amplitude of fundamental and first three harmonics

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  10. PcumP_Ravenclaw

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  11. J

    Why is the fundamental theorem of arithmetic special?

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  12. I

    Why the muon is considered a fundamental particle?

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  13. abram

    Fundamental kinematics question

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  14. samgrace

    Which areas of maths are from which fundamental areas

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  15. P

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  16. M

    Can all fundamental forces be fictitious force ?

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  17. C

    The Fundamental Theorem for Line Integrals

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  18. nicolauslamsiu

    Fundamental theorem of calculus

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  19. E

    Fundamental capacitor discharge question

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  20. T

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  21. avito009

    Why is the strong nuclear force the strongest of the four fundamental forces?

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  22. PcumP_Ravenclaw

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  23. S

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  24. kmm

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  25. P

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  26. C

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  27. K

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  28. S

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  29. atyy

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  30. T

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  31. S

    Fundamental note vs fundamental frequency of string

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  32. jegues

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  33. B

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  34. ugenetic

    B, H, MMF, which one is the fundamental "driving" force?

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  35. C

    Best books on fundamental Statistics?

    Hi awesome Physicsforum members, I’m looking for a good introdoctury book for learning Statistics (self-study) which succeeds in explaining the concepts very clearly . What I’m looking for in a statistics book My goal with the book is not to achieve some abstract mathematical skills, but...
  36. marcus

    Is Unitarity Limited by Real Clocks in Quantum Gravity?

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  37. H

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    To which type of Fundamental forces below the Pauli's exclusion belong? Strong nuclear forces Coulomb forces Weak nuclear forces Gravitational forces
  38. Q

    Deriving Coulomb's law and other fundamental equations from F=ma?

    I saw a joke crib sheet for a physics class that said "F=ma, derive the rest," but is it actually possible to derive at least electricity and magnetism questions from Newton's second law?
  39. F

    A question about fundamental constants in Multiverse idea

    Has there been a debate as to whether fundamental constants can be chosen freely for a theory or some constraints make it unlikely that to be the case. for example the dimentionful parameters don't have a specific value, only combinations give FSC, that means(to my mind) that there is unlikely...
  40. O

    Do Fundamental Particles Experience Heat?

    Heat is a product of excited energy states of the fundamental particles that make up atoms, correct? So do the particles, themselves, get "hot" - or is heat just experienced as radiation on the macroscopic scale? Do neutrons, for example, have a thermal property at all?
  41. M

    On Fundamental Closed String Solutions

    I am doing a project on string theory and my first task is to work out fundamental string solution from the I(10) string action in NS sector.I am following A.Dahbolkar, G.Gibbons, J.A.Harvey & F.Ruiz Ruiz, Nuclear Physics B340 (1990) 33—55. Why is it called fundamental string? I have not taken...
  42. J

    Fundamental theorem of calculus for double integral

    The popular fundamental theorem of calculus states that \int_{x_0}^{x_1} \frac{df}{dx}(x)dx = f(x_1)-f(x_0) But I never see this theorem for a dobule integral... The FTC for a univariate function, y'=f'(x), computes the area between f'(x) and the x-axis, delimited by (x0, x1), but given a...
  43. Wes Tausend

    Fundamental Valid Frame of Rest

    ... Three scientists are familiar with Special Relativity and it's rules. The three scientists decide to conduct a Special Relativity experiment. The plan is for two of them to get into two separate rocket ships and pass one another nearly head-on overhead while the third scientist...
  44. D

    Higgs field as the 5th fundamental force?

    Greetings, I am brand new to quantum physics. By that I mean I have no idea of the mathematics and know terribly little about the physics. In fact my best background comes from the book QED by Richard Feynman. Anyway, I do know that there are four fundamental forces which have force-carrying...
  45. C

    Fundamental groups and arcwise connected spaces.

    If a space X is arcwise connected, then for any two points p and q in X the fundamental groups ##\pi_1(X,p)## and ##\pi_1(X,q)## are isomorphic. This means that we can, up to isomorphisms, identify both groups with their equivalence class ##\pi_1(X)##. I started to think about the generality...
  46. Spinnor

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    If the B-mode sky plots could be Fourier transformed what would be a plot of the lowest order B-mode harmonic plotted on a sphere look like? I guess we need two functions of spherical coordinates, one function for amplitude at points on a sphere and one function for the orientation at the...
  47. P

    Fundamental theorem of Calculus

    Suppose a is a constant. If G(x) = \int_a^x \Big [ f(t) \int_t^x g(u) du \Big ] dt, what is G\,'(x)? My attempt, G\,'(x) = f(x) \int_x^x g(u) du = 0, and I am sure this is wrong.
  48. evinda

    MHB First fundamental theorem of Calculus

    Hello! :) I am looking at the theorem: "$f:[a,b] \to \mathbb{R}$ integrable We suppose the function $F:[a,b] \to \mathbb{R}$ with $F(x)=\int_a^x f$.If $x_0$ a point where $f$ is continuous $\Rightarrow F$ is integrable at $ x_0$ and $F'(x_0)=f(x_0)$". There is a remark that the theorem stands...
  49. L

    Fundamental theorem of calculus

    Homework Statement Let ##[a,b]## and ##[c,d]## be closed intervals in ##\mathbb{R}## and let ##f## be a continuous real valued function on ##\{(x,y)\in E^2 : x\in[a,b], \ y\in[c,d]\}.## We have that ##\int^d_c\left(\int^b_af(x,y)dx\right)dy## and ##\int^b_a\left(\int^d_cf(x,y)dy\right)dx##...
  50. D

    MHB Fundamental Matrix of \(\mathbf{A}\) - Find the Solution

    Given \[ \mathbf{A} = \begin{bmatrix} 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & -5 & -4 \end{bmatrix} \] find the fundamental matrix. If I had real eigenvectors, I could simply do \[ e^{\mathbf{A}t} = \mathbf{S}e^{\mathbf{D}t}\mathbf{S}^{-1}, \] but I have complex eigenvalues and eigenvectors. Also, if the...
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