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. J

    Fundamental Thermodynamic Relation and Helmholtz Energy

    I'm confused about the condition for spontaneity for the Helmholtz energy. My textbook (McQuarrie, "Physical Chemistry") derives the conditions as follows. We start with the combined law of thermodynamics: dU = δq + δw ≤ TdS – PdV since δq/T ≤ dS dU – TdS + PdV ≤ 0 For a process at...
  2. P

    Where did they come from and when?

    correct me if I am wrong but there are fundamental fields that are responsible for the forces and properties. A) how do these fields (em, higgs ect.) influence matter? B) where did they come from and when? Thanks
  3. 1

    Proving/deriving first fundamental theorem of calc

    I know this is going to be atrociously bad, but I like to try to prove things. \frac{f(x+\Delta x) - f(x)}{ \Delta x} = \frac{\Delta y}{\Delta x} => f(x+\Delta x) - f(x) = \frac{\Delta y}{\Delta x} \Delta x => f(x+\Delta x) = \frac{\Delta y}{\Delta x} \Delta x + f(x) Now...
  4. H

    Exploring the Fundamental Postulates of QM: Are They Truly Ad-Hoc and Strange?

    I find the fundamental postulates of QM very ad-hoc and strange. Compare them to the fundamental postulates of special relativity, special relativity naturally arises out of classical electromagnetism and the equivalence of all inertial frames, while QM seems to come out of nowhere.
  5. U

    The Lie bracket of fundamental vector fields

    Homework Statement The Lie bracket of the fundamental vector fields of two Lie algebra elements is the fundamental vector field of the Lie bracket of the two elements: [\sigma(X),\sigma(Y)]=\sigma([X,Y]) Homework Equations Let \mathcal{G} a Lie algebra, the fundamental vector field of an...
  6. F

    Equation linking all fundamental constants

    Folks, I remember once seeing an equation that linked all the important fundamental constants - electron charge/alpha, gravitational constant, planks constant, speed of light, electron mass etc - in a dimensionless relationship. But for the life of me I cannot find it -google doesn't know...
  7. A

    MHB Fundamental Theorem of Calculus Questions

    Last one for the night! These are the questions: This is my work: I think question 5 is correct (I hope), but I'm not entirely sure about question 6. Any help would be appreciated!
  8. T

    Is it worth it to persue fundamental Theoretical Physics? no strings

    Here's my battle plan: 1: Learn (the principles) of verified models for fundamental physics. QFT, GR, Standard Model, etc. 2: Try to think of what I don't know and what I'd like to experimentally test.I think I'd prefer to get a regular job and research physics in my free time, rather than be...
  9. B

    Tension and Fundamental frequency

    The fundamental frequency of a uniform wire with an AC current of constant magnitude was found at various different tensions and a graph of (T1/2,l) was plotted (l was the length between the two nodes of the wire when the fundamental frequency was found). How would the data on the graph...
  10. C

    Estimate the fundamental frequency of resonance sound induced by blowi

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  11. A

    Confusion about gravity and the fundamental forces

    I understand that there are 4 fundamental "forces" of nature that we've identified; gravity, electromagnetism, and the two nuclear forces. However, I have been under the impression that gravity isn't a "force" in the same way that the others are. From what I figured, a force is something that...
  12. S

    Write down orthonormal bases for the four fundamental subspaces [ ]

    Write down orthonormal bases for the four fundamental subspaces [...]" Homework Statement Problem: Write down orthonormal bases for the four fundamental subspaces of A = matrix([1,2],[3,6]]). (1 and 2 are on the first row whereas 3 and 6 are on the second row.) Solution: A =...
  13. V

    Network Analysis: Question on fundamental cutset matrix of a graph

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  14. S

    Finding a Fundamental Matrix for the System

    Homework Statement Find a fundamental matrix for the system x'(t)=Ax(t); where 5 -3 -2 A = 8 -5 -4 -4 3 3 The second part of the question is to find eAt Homework Equations I know that you have to find the 3 eigenvectors and then the 'general solution' without...
  15. E

    3 Fundamental questions must be answered.

    There is much talk and armwaving about evolution being a fact. However until these three questions can be answered it is not yet time to drop the word 'theory' and replace it with 'fact'. Anyone like to put me right? 1 Science has not been able to provide a mechanism that will enable...
  16. H

    Amplitude of harmonics higher than fundamental frequency

    Dears, I have an vacuum pump creating the vacuum approximately 10^-8 mbar. The rotor consists blades and it's placed in bearings. One side is ceramic bearing and the other one is maglev type(magnetic). I measure noise and vibrations of the pump. Significant peak of both units is naturally at...
  17. S

    Sound waves: How do we know it is the fundamental harmonic?

    I have done a handful of problems related to sound waves in air columns and one thing I have noticed is that, unless told otherwise in the problem formulation, one always assumes that sound wave that is formed is always the fundamental harmonic and thus the length of the air column comprises a...
  18. Ackbach

    MHB Hilarious post on Fundamental Physics Prize

    I thought this post was hilarious, especially the paragraph right before the first Update.
  19. caffeinemachine

    MHB A fundamental fact about Linear Algebra

    Hello MHB, This is probably my first challenge problem which falls in the 'University Math' category. $V$ is a vector space over an infinite field $F$, prove that $V$ cannot be written as a set theoretic union of a finite number of proper subspaces.
  20. Fernando Revilla

    MHB Patrick's question at Yahoo Answers (First fundamental theorem of Calculus)

    Here is the question: Here is a link to the question: Integration by parts? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  21. Darth Frodo

    Proof of the Fundamental Theorem of Calculus.

    Hi all I'm currently working my way through proving the FToC by proving something that is a foundation for it. So I need to prove that; L(f,P_{1}) ≥ L(f,P) where P\subsetP_{1} i.e where P_{1} is a refinement of P. So, Let P_{1} = P \cup {c} where c \in [x_{k-1},x_{k}] Let L' = inf{x|x...
  22. M

    Fundamental Group of a Cayley Graph

    Suppose we have a group with presentation G = <A|R> i.e G is the quotient of the free group F(A) on A by the normal closure <<A>> of some subset A of F(A). Is it true that that fundamental group of the Cayley graph of G (with respect to the generating set A) will be isomorphic to the subgroup...
  23. A

    Using the V-K Thm to find fundamental grp of sphere union disk in R3

    Hi, I am trying to get my head around the Van Kampen Theorem, and how this could be applied to find the fundamental group of X = the union of the unit sphere S2 in R3 and the unit disk in x-y plane? I was thinking of splitting the sphere into 3 regions - two spherical caps each having open...
  24. G

    Fundamental Theorem of Line Integration

    Homework Statement Suppose that F is the inverse square force field below, where c is a constant. F(r) = c*r/(|r|)^3 r = x i + y j + z k (a) Find the work done by F in moving an object from a point P1 along a path to a point P2 in terms of the distances d1 and d2 from these points to the...
  25. D

    Fundamental polygon of a Mobius strip

    Hey I am having a little bit of difficulty. The classification theorem for 2 - manifolds tells me that every 2 -manifold has the following representation: 1) connect sum of n-tori 2) connect sum of n-projective planes 3) a sphere Now, using Massey's book there is a very algorithmic...
  26. K

    Finding fundamental equation of the ideal VanWaals in Hemoltz

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  27. micromass

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  28. E

    Least energy required for creation of a fundamental particle

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  29. J

    How Does the Fundamental Theorem of Calculus Apply to Piecewise Functions?

    Homework Statement Use the The Fundamental Theorm of Calculus to Find the following answers f(x) = 0 IF x < -4 2 IF -4<=x<0 -3 IF 0<=x<5 0 IF x>= 5 g(x) = Integral of f(t) dt Between 4 and x Determine the value of each of the following: (a) g(−6)=...
  30. E

    Creation of a fundamental particle

    Do I always have to use pair production for the fundamental particles... If I have to produce a fermion,I have to create an anti fermion at the same time? what about bosons? i mean what about those who doesn't have antiparticle? Or do I have to create 2 of them as they themselves are their own...
  31. R

    Fundamental Theorem Of Calculus problems help

    Fundamental Theorem Of Calculus problems help! Homework Statement A)))) Find the derivative of g(x)=∫[8x to 4x] (u+7)/(u-4) dx B))) Use part I of the Fundamental Theorem of Calculus to find the derivative of h(x) = ∫[sin(x) to -3] (cos(t^3)+t)dt C))) F(x) = ∫[ 1 to √3]...
  32. K

    A fundamental question on homeomorphism

    It is well known that there does NOT exist a homeomorphism between R^m and R^n if m>n. My question is whether it is possible to construct a homeomorphism between R^m (as a whole) and a subset of R^n (note that we also suppose that m>n)? Intuitively, it is impossible. Is my intuition right...
  33. L

    Use part 1 of the Fundamental Theorem of Calculus to find the derivative.

    1. h(x) = ∫-3 to sin(x) of (cos(t^3) + t)dt 2. Okay, I know that you are supposed to replace t with the upper limit, and then I think you multiply that term by the derivative of the upper limit. So I thought it would be: cos(sinx)^3 * cos(x) + sinxcosx But what even is cos(sinx)...
  34. C

    What are the fundamental properties of matter?

    Say we have a particle, what are the most fundamental properties that distinguishes it from another kind of a particle. What is written in it's identity card? Spin, electric charge, rest mass, mean lifetime.. what else?
  35. T

    A question on degrees of maps of the fundamental group of the unit circle

    Hello, I'm reading a textbook and in the textbook we are discussing the fundamental group of the unit circle and having some difficulty making out what a degree of a map is and why when there is a homotopy between two continuous maps f,g from S^{1} to S^{1} why the deg(f)=deg(g) We have...
  36. T

    Showing the Fundamental Group of S^1 is isomorphic to the integers

    Hi, I am reading J.P. May's book on "A Concise Course in Algebraic Topology" and have approached the calculation where \pi_{1}(S^{1})\congZ He defines a loop f_{n} by e^{2\pi ins} I want to show that [f_{n}][f_{m}]=[f_{m+n}] I understand this as trying to find a homotopy between...
  37. 1

    Some fundamental questions on light matter interactions

    Hi there, I've a few (probably very basic) questions about photon-electron (light-matter) interactions. Here we go: 1) How can an electron "understand" that the frequency of incoming photon is equal to its excited state and so absorb that. Is there any "virtual" resonance between the...
  38. S

    Fundamental Group of the Torus-Figure 8

    So I'm revamping the question I had posted here, after a bit of work. I'm concerned with the homomorphism induced by the inclusion of the Figure 8 into the Torus, and why it is surjective. There seem to be a lot of semi-explanations, but I just wanted to see if the one I thought of makes...
  39. H

    Fundamental Shift and Scaling of Signals

    Homework Statement I'm confused on whether or not two functions would be equivalent or not. Let's say x(t) is a triangle with height 1, width 1. The hypotenuse of it is the function t (with a slope of 1). I'm told that x((t+2) / 4) then is making it 4 times as wide and left-shifting the...
  40. B

    The fundamental theorem of calculus(I think;) )

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  41. O

    Second fundamental theorem of calculus.

    Let f(x) be a non-stochastic mapping f: \mathbb{R} \to \mathbb{R}. The second fundamental theorem of calculus states that: \frac{d}{dx} \int_a^x f(s)ds = f(x). *QUESTION 1* Is the following true? \frac{d}{dx} \int_x^a f(s)ds = f(x). *QUESTION 2* Related to this, how can I...
  42. P

    Fundamental Theorem of Calculus

    Homework Statement Homework Equations The Attempt at a Solution I know this is not right. Could someone help me out here? Thanks
  43. V

    Second fundamental theorem of calculus viewed as a transform?

    You see this picture of the second fundamental theorem of calculus and you are taught in high school / early college calculus that the t is a dummy variable. However, couldn't you view this as some sort of transform? You convert a function f(t) into a function of f(x). Is this a valid way to...
  44. C

    Finding fundamental frequency with length and successive frequencies

    Homework Statement Standing waves on a 1.3 m long string that is fixed at both ends are seen as successive frequencies of 24Hz and 48 Hz. What is the fundamental frequency? Homework Equations fo = nv/4L The Attempt at a Solution Okay, so I don't really know if that is the right...
  45. T

    Question about the fundamental equation of thermodynamics

    In a PVT system, knowing the function U(S,V) (or S(U,V) ) allows us to know all the information about the system, so it's called the fundamental equation of thermodynamics (and it makes sense to me because it's derived from the 1st and 2nd law). But if we know the function P(V,T) for example we...
  46. marcus

    Fundamental Cosmology conference (Stockholm, November 2012)

    There's an interesting conference taking place this month at the NORDITA institute for theoretical physics in Stockholm. It brings together, from a number of different areas, researchers working on various models of what was happening around the start of expansion--replacing the "singularity"...
  47. R

    Nominations Welcomed for Fundamental Physics Prize

    Online nominations are now being accepted for the 2013 Fundamental Physics Prize. If you haven't heard of this new initiative, see http://www.fundamentalphysicsprize.org/news1 . The full information is included below. Thank you for your time, Rob Meyer Administrator ___ Online...
  48. W

    Linear Algebra Four fundamental subspaces small proof.

    Homework Statement Given A\in Mnxn and A = A2, show that C(A) +N(A) = ℝn. note: C(A) means the column space of A. N(A) means the null space of A Homework Equations These equations were proved in earlier parts of the problem... C(A) = {\vec{x}\in ℝn such that \vec{x} =...
  49. T

    Finding a fundamental set of solutions for a 2nd order differential equation

    Homework Statement 64y''+144y'=0 y1(0)=1 y'1(0)=0 and y2(0)=0 and y'2(0)=1 Homework Equations y1=c1*e^(r1*t) + c2*e^(r2*t)The Attempt at a Solution I start by finding the characteristic equation: 64r^2+144r=0 r1=-9/4 and r2=0 y1=c1e(r1*t) + c2e(r2*t) so I get y1=c1e^(-9/4 *t) + c2e^(0*t)...
  50. S

    Need help understanding Lagrange multipliers at a more fundamental level.

    I understand that for Lagrange multipliers, ∇f = λ∇g And that you can use this to solve for extreme values. I have a set of questions because I don't understand these on a basic level. 1. How do you determine whether it is a max, min, or saddle point, especially when you only get one...
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