What is Definition: Definition and 1000 Discussions

A definition is a statement of the meaning of a term (a word, phrase, or other set of symbols). Definitions can be classified into two large categories, intensional definitions (which try to give the sense of a term) and extensional definitions (which try to list the objects that a term describes). Another important category of definitions is the class of ostensive definitions, which convey the meaning of a term by pointing out examples. A term may have many different senses and multiple meanings, and thus require multiple definitions.In mathematics, a definition is used to give a precise meaning to a new term, by describing a condition which unambiguously qualifies what a mathematical term is and is not. Definitions and axioms form the basis on which all of modern mathematics is to be constructed.

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  1. K

    I Definition of magnetic moment in quantum mechanics

    * The general formula for the magnetic moment of a charge configuration is defined as ##\vec{\mu} = \frac{1}{2} \int \vec{r} \times \vec{J} \,d^3r##* For an electron it's said that the correct equation relating it's spin and magnetic moment is is ##\vec{\mu} =g\frac{q}{2m}\vec{S}## * It's...
  2. U

    A question on the definition of the curl of a vector

    The curl is defined using Cartersian coordinates as \begin{equation} \nabla\times A = \begin{vmatrix} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A_x & A_y & A_z \end{vmatrix}. \end{equation} However, what are the...
  3. D

    I What is the ambiguity in forming new functions?

    Why do we want to always deal with single valued functions? In the classical treatment a function is a rule which assigned to one number another number. In the modern sense, it is a rule which assigns to each element in a set called the domain an element (one element) in a set called the range...
  4. angweieng

    A What are Z+, Z-, N+, and N- in the Axial Charge Definition?

    Hi all, I was trying to understand an equation where: axial charge, ##G_A=g^P_A(Z_+-Z_-)+ g^N_A(N_+-N_-)## What is the meaning of ##Z_+## , ##Z_-##, ##N_+## and ##N_-##? From the article I read, axial charge will be zero when the nuclei has zero spin. What if I have Germanium 73 which has...
  5. K

    I Definition of order of a partial differential equation

    How is the order of a partial differential equation defined? This is said to be first order: ##\frac{d}{d t}\left(\frac{\partial L}{\partial s_{i}}\right)-\frac{\partial L}{\partial q_{i}}=0## And this second order :##\frac{d}{d t}\left(\frac{\partial L}{\partial...
  6. C

    I What's the definition of "periodic extension of a function"?

    I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...
  7. joneall

    A Simple definition of Lie group

    I'm writing some notes for myself (to read in my rapidly approaching declining years) and I'm wondering if this statement is correct. I"m not sure I am posting this question in the right place. "Summary: The matrix representations of isometric (distance-preserving) subgroups of the general...
  8. Leo Liu

    I Multiplication and addition definition of congruence classes

    The definitions of them seem like arbitrary choices or an abuse of notation. I wonder what the reasons behind the definitions are. Thanks. PS. My instructor said such defs simplify the process of solving modular equations.
  9. arcTomato

    I Is my calculation of the power spectrum correct?

    Hello PF. I am thinking about the power spectrum when observing X-rays. We are trying to obtain the power spectrum by applying a window function ##w(t)## to a light curve ##a(t)## and then Fourier transforming it. I have seen the following definition of power spectrum ##P(\omega)##. Suppose...
  10. J

    Rigorous mathematical definition of sampling

    Please could you help me find a rigorous mathematical definition of sampling as it is used in mathematical statistics? Let ##X:\Omega\rightarrow\mathbb{R}## be a random variable and ##X_{1},...,X_{n}## is a statistical sample. What is its mathematical meaning in probability theory? Are we...
  11. chikchok

    I Fermi energy definition and Fermi-Dirac distribution

    1)In my book , there is a definition of fermi energy as topmost filled level in the ground state of an N electron system. This definition holds only for absolute zero,right? If it is not absolute zero,fermi energy is the energy at which the probability of a state being occupied is 50 percent...
  12. mcastillo356

    B What kind of definition is this?

    Hi, PF, I've got a quote, and a philosophical question Figure shown suggests that a function ##f(x)## can have local extreme values only at ##x## points of three special types: (i) Critical points of ##f## (points ##x## in ##\mathfrak{D}(f)## where ##f'(x)=0##) (ii) Singular points of ##f##...
  13. cianfa72

    I Clarification on Rindler coordinates definition

    Hi, starting from this post Basic introduction to gravitation as curved spacetime I would ask for a clarification about Rindler coordinates. From this wiki entry Rindler coordinates I understand that the following transformation (to take it simple drop ##y,z##) $$T = x\sinh{(\alpha t)} ...
  14. P

    I Definition of Rindler Horizon: A Global View

    [Moderator's note: Thread spun off from previous one due to closure of the previous thread.] I have been thinking about this off and on, and though late to the thread, want to propose another way of looking at this that can be presented both at B level or A level. I post here at B level, and...
  15. A

    I What Are the Origins of the Universe?

    I see much confusion about the "origin" of the universe. When discussing what caused the big bang, we need to describe the conditions of the universe before the big bang. For me, the big bang occurred in the universe. The universe existed prior to the big bang and provided the energy to fuel...
  16. V

    Confusing definition of electrical potential energy

    If the book had said that electrical potential energy is the negative of work done by electrical force on a charge, then the definition would be very clear and easy to understand. So, why should the book give this confusing definition instead.
  17. A

    I The definition of the spectra in quantum mechanics

    for undergraduate students how to explain this?
  18. dRic2

    GNUplot -- how to improve image definition?

    Hi, I am used to plot data with the GNUplot software. This time I need to add an image to my plot. I did with the following line 'my_image.png' binary filetype=png center=(3.25,-9.1) dx=0.007 dy=0.0125 w rgbimage notitle but the final resolution of the picture is pretty bad compared to the...
  19. ergospherical

    I Understanding Papapetrou's Spinning Test Particles in GR

    I'd appreciate some clarification of this passage in the paper Spinning test particles in general relativity by Papapetrou, The definition is easy enough to understand, but what's the motivation? ##X^{\alpha}## are the coordinates of points on the worldline whilst ##x^{\alpha}## are...
  20. vibha_ganji

    B Definition of Stellar Parallax from Alonso and Finn Volume 1

    Hi! I read this definition of Stellar Parallax "It is expressed quantitatively by one-half the angle subtended by the Earth's diameter E1E2 perpendicular to the line joining the star and the sun (see Fig. 2-10)." (Source Alonso and Finn: Volume 1). But, I was unable to understand how they...
  21. S

    B Question about stress - strain graph and definition of hardness

    This is from my notes: Point D is called ultimate tensile strength and defined as highest possible within this material. So it means that point D should be at the highest point of the graph (more like absolute maximum in math)? Because it seems that from the graph point D is not at maximum...
  22. Charles_Xu

    I The definition of volume inside black hole?

    What is the definition of volume inside a black hole? we know the grr element of Schwarzschild metric is negative inside event horizon, so how to define a volume inside event horizon? if there is no definition of volume, is there the definition of density?
  23. G

    What is the Definition of the Joule and How Does it Relate to Energy?

    Mentor's note: Moved from HW for a better fit Homework Statement:: A joule is defined as "the energy transferred to an object when a force of one Newton acts on that object in the direction of its motion through a distance of one meter." Explain this definition. Relevant Equations:: 1J = 1N.m...
  24. Dario56

    Definition of Absolute Electrode Potential

    Hey guys, I have two questions: 1) I thought absolute electrode potential is galvani potential difference at the interface. However, it is given by this equation in John Bockris - Modern Electrochemistry: $$ E(abs) = ^M\Delta^S\phi - \mu_e^M/F $$ First term is galvani potential difference on...
  25. Poetria

    Domain of multivariable function - definition

    E.g. all real numbers could be a domain but not necessarily, etc. Am I right?
  26. nomadreid

    I Shouldn't this definition of a metric include a square root?

    In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...
  27. K

    I Definition of Limit for vector fields

    Apostol defines limit for vector fields as > ##\quad \lim _{x \rightarrow a} f(x)=b \quad(\rm or\; f(x) \rightarrow b## as ##x \rightarrow a)## means that : ##\lim _{\|x-a\| \rightarrow 0}\|f(x)-b\|=0## Can't we say it's equivalent to ##\lim _{x \rightarrow a}(f(x)-b)=0##
  28. akashpandey

    I Why is the definition of work = force times displacement?

    Hello everyone, I was curious about how do we define a physical quantity and mathematical relation between them. For example: Work done is defined as product of force along the displacement and displacement i.e W=Fcos(theta)d. Now this definition is given in books straightforward but my...
  29. F

    The true definition of Poynting effect for simple shear

    Consider a simple shear x=kY, y=Y, z=Z. How is Poynting effect defined? If the normal stress along y is 0 but that along x isn't 0, is that also a sort of Poynting effect?
  30. harpazo

    What made 1970 a memorable year in music?

    I would like to know the meaning and/or background for the username you have chosen. Username stories are interesting. Do you agree?
  31. S

    I A strange definition for Hermitian operator

    In lecture notes at a university (I'd rather not say which university) the following definition for Hermitian is given: An operator is Hermitian if and only if it has real eigenvalues. I find it questionable because I thought that non-Hermitian operators can sometimes have real eigenvalues. We...
  32. W

    A Topological Definition of Holes

    Is it reasonable to define the n-dimensional holes of a topological space X as the non-zero Homology/Homotopy classes of X? Can we read these as obstructions to continuously shrinking a simple closed curve * to a point within the space? *I understand this is what we mean by a cycle.
  33. R

    I How can we define a limit approaching negative infinity?

    I have the following definition: $$ \lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$
  34. R

    MHB What is the definition of a negative infinity limit?

    I have the following definition: $$\lim_{x\to p^+}f(x)=+\infty\iff \forall\,\,\varepsilon>0,\,\exists\,\,\delta>0,\,\,\text{with}\,\,p+\delta< b: p< x < p+\delta \implies f(x) > \varepsilon$$ From this, how can I get the definition of $$\lim_{x\to p^-}=-\infty? $$
  35. CPW

    Acceleration in Polar Coords, Intuitive Definition video

    Summary:: I wish this video (and YouTube in general) was around when I took intermediate level mechanics as an undergraduate physics student: I wish this video (and YouTube in general) was around when I took intermediate level mechanics as an undergraduate physics student:
  36. T

    A Hypersurface Definition Confusion in General Relativity

    In my notes on general relativity, hypersurfaces are defined as in the image. What confuses me is that if f=constant, surely the partial differential is going to be zero? I'm not sure if I'm missing something, but surely the function can't be equal to a constant and its partial differential be...
  37. T

    Definition of a system in Boltzmann entropy

    Context Boltzmann first defined his entropy as S = k log(W). This seems to be pretty consistently taught. However, the exact definitions of S & W seem to vary slightly. Some say S is the entropy of a macrostate, while others describe it as the entropy for the system. Where the definition of...
  38. K

    I Questions about the definition of a metre (A. P. French's book)

    In his book, Newtonian Mechanics, while describing the standard unit for length, A. P. French writes (being American, he uses meter and not metre): My questions are twofold: The first one is about the physics. What does the text in orange really mean? There is no sidebar about "optical...
  39. U

    Clarification on electric quadrupole moment definition

    I have encountered two (?) definitions of the electric quadrupole moment. They are: $$Q_{ij}=\frac{1}{2}\int \rho(\vec{x}')x'_i x'_j\,\mathrm{d}^3x'$$ and $$Q_{ij}=\int (3x'_i x'_j-\delta_{ij}x'^2)\rho(\vec{x}')\,\mathrm{d}^3x'$$ I am trying to study radiation arising from the electric...
  40. H

    Determining the growth of two functions using Big-Oh definition

    My attempt involved using the big-Oh notation, I think this should work but I am not sure how to go about it. The two functions are g(n) = 6^n/n^5 and h(n) = (ln n)^84. I thought that I could use the inequality 6^n < ln(n)^84 and 6^n/|n^5| = |g(n)| < 6^n and put those inequalities together...
  41. Eclair_de_XII

    B Sequential Compactness of Sets: A Definition

    Definition: A set is sequentially compact if all sequences contained in the set contain a subsequence that converges to a point in the set. Let ##N\in \mathbb{N}## and suppose that ##m\geq N##. Let ##x\in K_m##. Since ##K_m\subset K_{m-1}\subset \ldots \subset K_N##, it follows that ##x## is an...
  42. Arman777

    I Definition of Tensor Identity Simplification

    Is there a simplifcation of ##g^{\alpha \delta}g_{\beta \gamma} ## or what is equal to ?
  43. A

    Can a supernatural phenomenon by definition exist?

    Or would any apparent supernatural phenomenon ultimately be boiled down to natural laws that we just don't understand yet? By definition, a supernatural phenomenon doesn't obey natural laws, or a certain subset of natural laws. Like say for instance, we observer a large star orbiting a small...
  44. S

    I Subset definition: universal quantifier over which set?

    This is a somewhat trivial question, but I never managed to learn much logic back in the day...so: The definition of a subset can be written as: ## A \subset B \Leftrightarrow \forall x (x \in A \Rightarrow x \in B) ## However, over which set is ##\forall## supposed to quantify? It seems to...
  45. B

    The definition of generalised momentum

    Why, in lagrangian mechanics, do we calculate: ##\frac{d}{dt}\frac{\partial T}{\partial \dot{q}}## to get the (generalised) momentum change in time instead of ##\frac{d T}{dq}##? (T - kinetic energy; q - generalised coordinate; p - generalised momentum; for simplicity I assumed that no external...
  46. F

    [Complex analysis] Contradiction in the definition of a branch

    I find the following definition in my complex analysis book : Definition : ## F(z)## is said to be a branch of a multiple-valued function ##f(z)## in a domain ##D## if ##F(z)## is single-valued and continuous in ##D## and has the property that, for each ##z## in ##D##, the value ##F(z)## is one...
  47. E

    Definition of energy level width

    First, it is easy to see that n=4 after the collision because: E_1=-13.6\frac{1^2}{1^2}eV=-13.6eV E_4=-13.6\frac{1^2}{4^2}eV=-0.85eV E_5=-13.6\frac{z^2}{5^2}eV=-0.544eVBut, I never saw a definition for the width of an energy level. I tried to use something I saw online that said it was...
  48. A

    Domain of definition of the function f(x,y)=x^y

    Good day as said in the title i need the domain of definition of of the function f(x,y)=x^y for me as x^y=expontial (y*ln(x)) so x>0 but the solution said more than that I really don't understand why we consider the case (0,y) in which while should be different from 0, because I will never...
  49. L

    MHB Definition of onto function

    Which of the following statements is equivalent to saying that a function f:A→B is onto? There are 8 options, select all that are correct. x f(A)=B x In the arrow diagram representing f, every point in B has an arrow pointing at it. x ∀y∈B ∃x∈A such that f(x)=y x f−1(B)=A...
  50. SamRoss

    B What is the Definition of a Candela?

    I'm trying to wrap my head around the definition of a candela: "The candela is the luminous intensity, in a given direction, of a source that emits monochromatic radiation of frequency 540×10^12 hertz and that has a radiant intensity in that direction of 1/683 watt per steradian." I'm aware...
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