What is Fundamental: Definition and 946 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. D

    Fundamental subspaces of A?

    When studying linear algebra when encountering a system Ax=b, I always read of the fundamental subspaces of A: N (the null space, all solutions x of Ax=0), the column or domain space of A: (the space spanned by the columns of A, or in other words, all possible b for Ax=b), the row space (the...
  2. D

    Fundamental theorem for line integrals

    Hi, I have a question. In my calculus book, I always see the fundamental theorem for line integrals used for line integrals of vector fields, where f=M(x,y)i + N(x,y)j is a vector field.The fundamental theorem tells me that if a vector field f is a gradient field for some function F, then f is...
  3. N

    Fundamental Frequency of Two Pipe Organs

    Homework Statement Two organ pipes, open at one end but closed at the other, are each 1.18 m long. One is now lengthened by 2.50 cm Homework Equations λ = nL/4 fn = nv/4L v = λF The Attempt at a Solution Here's what I tried First I tried finding the fundamental frequency...
  4. E

    Fundamental theorem of calculus

    \frac{d}{dx} \int_a^b f(x) dx=f(b) This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.
  5. R

    Fundamental Theorem of Calculus

    Homework Statement All this information is in the attached file. Homework Equations All this information is in the attached file. The Attempt at a Solution What I tried to do was take the anti-derivative of the first equation and plug in the number 5. I'm not sure if that was...
  6. L

    ∫ of (x^2) from 0 to 2 proof w/o using fundamental therom, but w/ Riemann Sums

    Homework Statement \int_0^2 x^2 \, dx using true definition involving Riemann Sums (w/o fundamental theorem). Homework Equations I don't know what the relevant equations may be, perhaps some type of lim\sum f(x)(x_{j}-x_{j-1}) The Attempt at a Solution No attempt. Just seeking the...
  7. R

    Few Fundamental Thermodynamics Questions

    Hi I had a few thermodynamics questions on a thread that was locked. The link is here and my questions are directed at the last post. https://www.physicsforums.com/showthread.php?t=225555 1. Does the author mean that all quasistatic irreversble processes are reversible or if for a...
  8. D

    Fundamental Physics Prize - $27 million awarded

    Who said is there any money in Physics? http://fundamentalphysicsprize.org/news.html New annual US$3 million Fundamental Physics Prize recognizes transformative advances in the field Inaugural nine recipients of the Prize receive US$27 million in aggregate, all of whom agree to form a...
  9. C

    Another quick question, Fundamental Theorem of Calculus

    I am guessing the fundamental theorem of calculus, isn't not valid, if the integrand f depends on x. Right? For example if he had: \int^{x}_{0} f(u) ( x-u) du. If one would make F(x) = \int^{x}_{0} g(u) du, with g(u) = f(u) ( x-u). Then F`(x) = g(x) = f(x) (x-x) = 0. But this is not...
  10. A

    Questions regarding the Fundamental Theorem of Calculus

    If you define a function g(x) = \int_a^x \! f(t) \, \mathrm{d} t then from what I currently understand, g(x) gives the value of the area under the curve y=f(t) When you differentiate both sides, g'(x) gives the rate of change of the area underneath y=f(t), however, I don't understand...
  11. M

    Calculating the frequency of the fundamental vibrational mode

    Can you calculate the frequency at which a bond vibrates when you know what frequencies of EM radiation it absorbs? Using carbon monoxide as an example. It has a stretching frequency at around 2100 cm-1. In electron volts, that would be around 0.3 eV. If I'm not mistaken, this is the energy it...
  12. sunrah

    Fundamental solution of wave equation

    Homework Statement show that E(x,t):= \frac{1}{2} \left\{ \begin{array}{ll} 1 & \mbox{if $|x|<t $};\\ 0 & \mbox{else}.\end{array} \right. is a fundamental solution of the wave equation. Homework Equations LE = E_{tt} - \Delta E = \delta The Attempt at a Solution firstly...
  13. T

    What is the fundamental premise of positive thinking

    What is the fundamental premise of "positive thinking" The only quantifiable effect of positive or negative thinking that I can think of would be how each mindset influences your actual behavior. But do positive thinkers believe that it also somehow produces effects via some unknown mechanism...
  14. M

    Fundamental Theorem of Calculus Part II

    1. Find the derivative of: ∫cos3(t) where a = 1/x and b = ∏/3 This was a part of a question on my first calc exam and I just wanted to know if I did it correctly. We can solve this using the Fundamental Theorem of Calculus, Part II The solution would be to simply plug in the values for a and...
  15. S

    Fundamental Solution to Laplace Equation

    Many texts in deriving the fundamental solution of the Laplace equation in three dimensions start by noting that the since the Laplacian has radial symmetry that Δu=δ(x)δ(y)δ(z) That all that needs to be considered is d^2u/dr^2 + 2/r du/dr = δ(r) For r > 0 the solution given is u= c1/r +...
  16. Math Amateur

    Algebraic Topology - Fundamental Group and the Homomorphism induced by h

    On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334) "Suppose that h: X \rightarrow Y is a continuous map that carries the point x_0 of X to the point y_0 of Y. We denote this fact by writing: h: ( X...
  17. T

    Fundamental flaws in human mathematics?

    Hi guys I'm wondering about something, currently in our mathematics our number system goes something like this, 1,2,3,4,5... etc all the way to 9, then the whole cycle is repeated when it reaches 10. I believe this method of counting seems to stem from the fact that we have 10 fingers and our...
  18. R

    MHB Linear ODE for a fundamental solution set

    Question: For the interval x > 0 and the function set S = { 3ln(x), ln2, ln(x), ln(5x)}, construct a linear ODE of the lowest order. My work: Taking the wronskian for this solution set, I get it as 0. Doesn't that mean that a linear ODE for this set cannot be found? I'm very confused here...
  19. A

    Problem involving fundamental principle

    Hello, I was unsure if I should post this or on the General Physics thread but since it's about Einstein's gravity, I'll post it here. All right, so the problem is as follows (from Surely You're Joking, Mr. Feynman!): I understand the question being asked. My question is, how you would...
  20. V

    The Fundamental Theorem of Algebra

    I just wanted to say first of all that I am not looking for any specific answers, just hoping someone could shed a light on the subjects at hand. Is the quadratic formula a specific example of some general root finding algorithm that solves for the n (or n-1?) roots of a nth degree...
  21. P

    Is gravity more fundamental than EM?

    If photons emit virtual gravitons (but gravitons don't emit virtual photons) why isn't gravity considered to be a more fundamental force?
  22. S

    Should luminous intensity be a fundamental unit?

    It seems to me that luminous intensity should really be put in terms of energy, not a special unit (which itself is based on some arbitrary specification of energy.) The other 5 units and Avogadro's number should be the only fundamental units.
  23. C

    What is entropy? on the most fundamental level.

    On the atomic level, what is entropy? how can I visualise it? Thanks
  24. S

    Second fundamental form of surface with diagonal metric

    Hello everyone, Let r(u_i) be a surface with i=1,2. Suppose that its first fundamental form is given as ds^2 = a^2(du_1)^2 + b^2(du_2)^2 which means that if r_1 = ∂r/∂u_1 and r_2= ∂r/∂u_2 are the tangent vectors they satisfy r_1.r_2 = 0 r_1.r_1 = a^2 r_2.r_2 =...
  25. P

    Evaluation fundamental physics

    There is an article in the field of relativity (not homework) that need evaluation, please advise institutes that might do that??
  26. E

    Block form of the generators of the fundamental representation of SU(2N)

    Homework Statement I am calculating the corrections to the beta functions of a quite general SU(N) gauge-yukawa theory coming from coupling an electro-weak (SU(2)xU(1)) sector similar to that of the Standard Model. To do this, I need to calculate...
  27. G

    Graphic Design & Science: Fundamental Particles Visualation

    Graphic Design & Science: Fundamental Particles Visualisation Hello my name is Brendon. I'm a graphic designer interested in science communication and currently studying a masters in graphic design. I was wondering if anybody would be able to help me with a few queries with regards to...
  28. T

    Heat in a Rod Fundamental Solution

    Homework Statement The Attempt at a Solution So I know that I must have boundary conditions u(0,t) = 0 and ux(L,t) = 0. My textbook recommends reducing the given boundary conditions to homogeneous ones by subtracting the steady state solution. But, I thought these were already...
  29. R

    Proof of the fundamental theorem of calculus

    Homework Statement This is supposed to be a proof of the fundamental theorem of calculus. I'm not really sure what that proves, but to me at least it does not prove that the area under a curve is the antiderivative of the function and then inserting the upper x value and...
  30. S

    Fundamental- solving a second order differential equation

    I've completed my Engineering but doing a self study course in Dynamics of Structures and have got a very fundamnetal question concerning solution of differential equation and hope someone will be able to help me. Sorry if its too fundamnetal and stupid! Let us say we have to solve a...
  31. A

    Something weird with the fundamental theorem of calculus

    Suppose I know my function G is infinitely differentiable on the closed interval [a,b] and that all derivatives of G (including G itself) vanish at b. For any z in [a,b], I have by the FTC that \int_z^b G'(w) dw = G(b) - G(z). Or, switching limits, \int_b^z G'(w) dw = G(z) - G(b). One...
  32. C

    Codification of fundamental principals of physics

    There are a good number of very fundamental principles that underlie physics (conservation of information, principal of parsimony, time invariance, symmetry, etc.) that don't seem to be listed in modern physics textbooks. Are they codified somewhere?
  33. I

    Fundamental Theorem of Calculus

    Homework Statement F(x) = ∫ cos (1+t^2)^-1) from 0 to 2x - x^2 Determine whether F has maximum or minimum value Homework Equations The Attempt at a Solution I tried finding F'(x) = Dx (∫ cos (1+t^2)^-1) from 0 to 2x - x^2) = (2-2x)cos[(1+(2x-x^2))^-1] What do I do...
  34. M

    Fundamental Particles - Meson Decay Question

    Homework Statement The pion-zero meson has its mass quoted as 135.0MeV/c2. It decays into two gamma rays: ∏0 → γ + γ b) Assuming the ∏0meson to be initially at rest, calculate the energy and hence the wavelength of the two gamma rays. Homework Equations E= hc/λ The Attempt at a...
  35. N

    When is the fundamental thermodynamic relation true?

    dE = TdS - PdV, or equivalently \Delta E = \int T \mathrm d S - \int P \mathrm d V In general this is said to be derivable in the reversible case, however since S and V are state variables, it's also true for the irreversible case. But it can't be true for any irreversible case, since the...
  36. A

    Fundamental thermodynamic relation confusion.

    dE = dQ + dW = dQrev + dWrev = dQirev + dWirev. We have for an reversible process, dQrev = TdS and dWrev = -PdV. So; dE = TdS - PdV So this relation is for all changes (irreversible or reversible) since dS and dV are state functions. What doesn't make sense to me is the next part when...
  37. phys-if

    How Does the Equation F→∏ A l Z Relate to Fundamental Physics?

    I came across this. I guess you could say. F→∏ A l Z ...or... F→∏ AlZ The third symbol is supposed to be upside down. Does that matter?(The ∏ symbol) And between the symbol A and Z the line is supposed to be a tall line. (Not an L or I) Does this make sense?
  38. S

    How do I get the 1st fundamental form on Grassmann Manifold

    Consider G(n,m), the set of all n-dimensional subsapce in ℝ^n+m. We define the principal angles between two subspaces recusively by the usual formula. When I see "Differential Geometry of Grassmann Manifolds by Wong", http://www.ncbi.nlm.nih.gov/pmc/articles/PMC335549/pdf/pnas00676-0108.pdf I...
  39. fluidistic

    Fundamental equation and state equations of the ideal gas

    Homework Statement Find the fundamental equation of a monoatomic ideal gas in the Helmholtz potential representation, in the enthalpy representation, and in the Gibbs function representation. Assume the fundamental equation S= \frac{NS_0}{N_0} +NR \ln \left [ \left ( \frac {U}{U_0} \right )...
  40. Q

    Fundamental Theorem of Calc., Inc./Dec., and concavity

    Homework Statement I am having extreme trouble with the following problems: http://i.minus.com/iYs6ix6otGtLV.png Homework Equations For 26: If the first derivative is positive, then the function is increasing. If the first derivative is negative, then the function is...
  41. F

    Vertex of Fundamental Domains & Elliptic Points

    Dear Folks: Suppose \Gamma is a discrete subgroup of SL2(R), which acts on the upper half complex plane as Mobius transformation. F is its fundamental domain. If z is a vertex of F which does not lie on the extended real line ( that is R\bigcup\infty ) ,then must x be an elliptic point...
  42. fluidistic

    Carnot cycle, fundamental equation of ideal gas

    Homework Statement Assuming that the auxiliary system in the Carnot cycle is a monoatomic ideal gas whose fundamental equation is S=\frac{NS_0}{N_0} +NR \ln \left [ \left ( \frac{U}{U_0} \right ) ^{3/2} \left ( \frac{V}{V_0} \right ) \left ( \frac{N}{N_0} \right ) ^{-5/2} \right ]. 1)Find...
  43. B

    Misunderstanding in Fundamental Concept

    Okay, say a car is moving at a constant velocity and crashes into a wall. Now, observation would clearly illustrate to us that a destructive force was imparted onto the wall and the car. But how can that be if there there is no force, because there was a constant velocity? I must have some...
  44. P

    Find the fundamental group of a Riemann Surface

    Homework Statement χ is the Riemann Surface defined by P(w, z) = 0, where P is a complex polynomial of two variables of degree 2 in w and of degree 4 in z, with no mixed products. Find the fundamental group of χ.Homework Equations A variation of the Riemann-Hurwitz Formula states that if χ is...
  45. K

    Rieman Integral: The Fundamental Theorem of Calculus

    Homework Statement Let I := [a,b] and let f: I→ℝ be continuous on I. Also let J := [c,d] and let u: J→ℝ be differentiable on J and satisfy u(J) contained in I. Show that if G: J→ℝ is defined by G(x) :=∫u(x)af for x in J, then G'(x) = (f o u)(x)u'(x) for all x in J. 2. The attempt...
  46. 1

    How bad is this statement regarding the Fundamental Theorem for Line Integrals?

    State the Fundamental Theorem: Let F be a vector field. If there exists a function f such that F = grad f, then \int_{C} F \cdot dr = f(Q) - f(P) where P and Q are endpoints of curve C. _________________________________ I didn't receive any credit for this answer. Admittedly...
  47. A

    Fundamental Theorem for Line Integrals

    Vector field F(bar)= <6x+2y,2x+5y> fx(x,y)= 6x+2y fy(x,y)= 2x+5y f(x,y)= 3x^2+2xy+g(y) fy(x,y)=2x+g'(y) 2x+g'(y)= 2x+5y g'(y)= 5y g(y)= 5/2*y^2 f(x,y)=3x^2+2xy+(5/2)y^2 Then find the \int F(bar)*dr(bar) along curve C t^2i+t^3j, 0<t<1 I'm stuck on finding the last part for the F(bar)...
  48. F

    Help understanding the First Fundamental Theorem of Calculus

    The first fundamental theorem of calculus begins by defining a function like this: http://i.imgur.com/aWXql.png (sorry was not sure how to write this legibly in this post so I just uploaded on imgur) I kind of have a hard time wrapping my mind aruond this. How do you chose a? I...
  49. D

    I have a fundamental mis-understanding about gravity.

    I have a layman understanding about how the curvature of space describes the motions of planets and other large celestial bodies. What I don't understand is how curved space makes say; an apple fall to the ground. Any help appreciated. Thanks.
  50. C

    Medical Is leisure time and recreation a fundamental human need?

    Hey! Do we really need recreation or leisure time to function as humans? Could one get away with it and just work all of his life to whatever purpose he/she is set to go for? How does one distinguish between work and leisure time? Was there such a distinction in early humans? Or did one such...
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