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. J

    B Need a better definition of anticlockwise

    Dictionary says it is the opposite of clockwise. But if a clock is transparent and you look at it from behind, the same direction that was anticlockwise before is now clockwise. So it seems to depend on the point of view. But there are some electromagnetic equations that depend on clockwise...
  2. L

    A Definitions of Cylinder Sets and Cylinder Set Measure

    I'm trying to learn about Abstract Wiener Spaces and Gaussian Measures in a general context. For that I'm reading the paper Abstract Wiener Spaces by Leonard Gross, which seems to be where these things were first presented. Now, I'm having a hard time to grasp the idea/motivation behind the...
  3. J

    I What Are Some Examples of Symbolic Manipulation Not Included in Calculus?

    Not satisfied with the following definition of calculus. What is a better definition? More detailed? 1a : a method of computation or calculation in a special notation (as of logic or symbolic logic) b : the mathematical methods comprising differential and integral calculus —often used with the...
  4. DaTario

    A Thermal State in Relativity Theory: Can It Happen?

    Hi All, Considering a set of many many small hard balls which start colliding inside a box. The velocities of these balls being mostly greater than c/2. Is it possible, in this case, to speak of convergence to a thermal state in the same sense of ordinary thermodynamics (i.e., using...
  5. cianfa72

    I About the definition of a Manifold

    Hi, I'm a bit confused about the locally euclidean request involved in the definition of manifold (e.g. manifold ): every point in ##X## has an open neighbourhood homeomorphic to the Euclidean space ##E^n##. As far as I know the definition of homeomorphism requires to specify a topology for...
  6. W

    Engineering Identifying the purpose of a circuit from the VHDL definition file

    I have attempted to sketch the timing and got the following graph From reading the VHDL code my understanding is io each rising edge clock tick the system will carry out one of the following (choosing based on priority): reset if Clear_L is high load in a value if Clear_D is high UP count if...
  7. andrewkirk

    Insights Defining Large-Scale Isotropy: A Formal Definition

    Continue reading...
  8. J

    Definition of Heat and the First Law of Thermodynamics (discrepancy?)

    Zemansky defines Heat as : When a closed system whose surroundings are at a different temperature and on which diathermic work may be done undergoes a process, then the energy transferred by non mechanical means, equal to the difference between the change in internal energy and the diathermic...
  9. nomadreid

    I Faulty definition of r.e. set in Wiki?

    In https://en.wikipedia.org/wiki/Recursively_enumerable_set, in the introductory section one reads "...a set S of natural numbers is called recursively enumerable... if: ...There is an algorithm that enumerates the members of S." and in a later section it says "According to the...
  10. L

    I Definition of a symmetry transformations in quantum mechanics

    By the Wigner theorem, symmetries transformations are implemented by operators ##\hat{U}## that are unitary or antiunitary. This is what is written in most books. But I have read somewhere that, to ##\hat{U}## represent a symmetrie, it's necessary that ##\hat{U}^{\dagger} \hat{H} \hat{U} =...
  11. L

    I Definition of the potential energy operator

    In quantum mechanics, I can write the hamiltonian as ##\hat{H} = \hat{p}^{2}/2m + \hat{V}##. I am confusing with the definition of the operator ##\hat{V}##, who represents the potential energy. If the potential energy depend only on the position, is it correct write ##\hat{V} = V(\hat{x})##...
  12. A

    I What is the definition of the entropy of the Universe?

    My understanding is that to define the entropy of a system what you have to do is as follows: Define the boundaries of your system. Define a set of "microstates" of the system. Define a partition of microstates of the system where each element of the partition is measurable and known as a...
  13. L

    I Is the identity of a group unique?

    In general, the textbooks says that, if the set ##G## is a group, so to every element ##g \in G## there is other element ##g^{-1} \in G## such that ##g g^{-1} = g^{-1}g = e##, where ##e## is the identity of the group. But I am reading a book where this propriete is write only as ##g^{-1} g =...
  14. A

    The Definition Of Chemical Elements

    I don't get it - the definition of a chemical element is: "Cannot be chemically interconverted or broken down into simpler substances" -- But isn't every atom of an element made up of further sub-atomic particles, and atoms can be then broken apart, etc...? Also, how did people tell the...
  15. K

    B Definition of coordinate system

    In light of the modern definition of what is a coordinate system, namely it's a pair (U, f) with U a region of a m-dimensional manifold, and f a bijection from U to ##\mathbb R^m##, can we say that the polar coordinates on ##\mathbb R^2## are a coordinate system? I was thinking about this and...
  16. Math Amateur

    I Multivariable Differentiation .... McInerney Definition 3.1.1 ....

    I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1 and...
  17. Math Amateur

    MHB Multivariable Differentiation .... McInerney Definition 3.1.1

    I am reading Andrew McInerney's book: First Steps in Diofferential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.1: The Derivative and Linear Approximation ... I am trying to fully understand Definition 3.1.1...
  18. babaliaris

    I How was the speed of light calculated before the redefinition of the meter?

    I'm reading a book from the authors Halliday and Resnick and it says that The meter is the length of the path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second. This time interval was chosen so that the speed of light c is exactly c = 299 792 458 m/s. I...
  19. ibkev

    I Dot product definition: deriving component form

    ## \newcommand{\ihat}{\hat{\boldsymbol{\imath}}} \newcommand{\jhat}{\hat{\boldsymbol{\jmath}}} \newcommand{\khat}{\hat{\boldsymbol{k}}} ## Several times now I've seen the following technique for deriving the component form of the dot product. It always felt clean and simple until last night when...
  20. S

    A Definition of bi-local measurement by Masanes et al.

    Dear experts, I'm currently working my way through the paper Masanes, Galley, Müller, https://arxiv.org/abs/1811.11060. On page 3, they define what they call a bi-local measurement: If we have two systems a and b and we define an outcome probability function for some measurement f on system a...
  21. T

    I Strange Dot Product definition

    Hi i have seen in abook the dot product defined as follows: Dot(A,B)=(1/4)[Norm(A+B)^2-Norm(A-B)^2] how this definition connect with the common one: Dot(A,B)=Sum(ai*bi) Thanks!
  22. Wi_N

    Finding Boundaries of a Definition Area

    Homework Statement Problem is part of a double integral. but my boundries are: 1<=x^2 + y^2 <=9 so between 2 circles with r1=1 and r2=3 and x<=y and y<=sqrt(3x) the first boundry is obviously pi/4 and/or 3pi/4 the answer is pi/3 and i have no idea how u get that. u obviously have to...
  23. Antony Death

    B Is the current definition of gravity accurate?

    The current definition of Gravity is: The force of attraction between bodies as a result of their mass. Gravity affects both the space and time of the area surrounding a mass, diminishing with distance, so is the current accepted definition truly accurate? Do I have the correct definition and if...
  24. Robin04

    I Understanding the definition of a soliton

    I'm learning about solitons from a book called Solitons and Instantons by R. Rajaraman. He defines (page 14-15) a soliton as a solution to a (possibly non-linear) PDE where the energy density of the system is of the form ##\epsilon (x,t) = \sum_i \epsilon_0(x-a_i-u_i t-\delta_i)##, as ##t...
  25. J

    Definition of isolation and pulse response time for a 3-way power splitter

    I will use 3 way power splitter and power detector. 1. This is power splitter data sheet. In this data sheet, there are different isolation values. What does it mean?? Also in that point, I wonder definition of isolation at power splitter. 2. This is power detector data sheet. In this data...
  26. K

    I Definition of Cartesian Coordinate System

    I was asking myself what is the definition of a Cartesian Coordinate System. Can we say that it's a coordinate system such that - the basis vectors are the same ##\forall x \in R^n## - the basis vectors are orthonormal at each ##x \in R^n## So for instance, normalized polar coordinates do not...
  27. VuIcan

    I Interpreting The Definition of Tensors

    Hello, I've just been slightly unsure of something and would like to get secondary confirmation as I've just begun a book on tensor analysis. I would also preface this by saying my linear algebra is somewhat rusty. Suppose you have the inertia tensor in some unprimed coordinate system such that...
  28. ZuperPosition

    Abstract definition of electromagnetic fields on manifolds

    Hello, In the sources I have looked into (textbooks and articles on differential geometry), I have not found any abstract definition of the electromagnetic fields. It seems that at most the electric field is defined as $$\bf{E}(t,\bf{x}) = \frac{1}{4\pi \epsilon_0} \int \rho(t,\bf{x}')...
  29. SebastianRM

    I What is the 'formal' definition for Total Derivative?

    A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations. My question is, where does this equation comes from?
  30. Jonathan Stanley

    B Einstein's definition of synchronization

    At the time ##t _0##, a ray of light goes out ##A##, reflected at ##B## at time ## t_1##, and arrives back at ##A## at time ##t_2##. So Einstein provides: ##t_2-t_0 = (t_1-t_0) + (t_2 - t_1) = \frac{l _{rod}}{c - v _{rod} } + \frac{l _{rod}}{c + v _{rod} }## Where: Rod with ends A and B ##v...
  31. Y

    I Definition of 'dynamical equilibrium'?

    "Assuming that the system is in dynamical equilibrium – that the magnitude of the total potential energy is equal to twice the kinetic energy – a so-called “dynamical mass” can be derived..." https://astrobites.org/2012/03/16/what-defines-a-galaxy/ I am trying to understand what is meant by...
  32. C

    MHB Recursive definition and induction

    Hey. The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$. Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$. Now, I know that is correct intuitively but...
  33. Buzz Bloom

    Do biologists agree on a definition of "species"?

    I have read through the 2013 thread https://www.physicsforums.com/threads/definition-of-species-under-attack.688447/ . This was the only thread title I found when I searched for "species definition". Although the question about a definition for "species" was discussed, I found no clarity about...
  34. K

    A Definition of Tensor and.... Cotensor?

    Why are there (at least) two definitions of a tensor? For some people a tensor is a product of vectors and covectors, but for others it's a functional. While it's true that the two points of view are equivalent (there's an isomorphism) I find having to switch between them confusing, as a...
  35. A

    What do surface tension vectors mean in this quote?

    I was reading Fundamentals of Inket Printing and it said the following: "The surface tension in a liquid causes a force to act in the plane of the free surface perpendicularly to a free edge in that surface." Can someone explain to me what this means? What's the direction of the force? I have...
  36. Robin04

    I Understanding the definition of continuous functions

    Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...
  37. C

    MHB Webpage title: The Importance of Correct Order in the Definition of Limit

  38. bbbl67

    B So what is the new definition of the kilogram?

    So this article "Quantum leap for mass as science redefines the kilogramme" said that there is a new definition of the kilogram coming. But they neglected to mention what that new definition is exactly. All they said was that it's now based on Planck's Constant. So I worked my way backwards...
  39. M

    I Modern Definition of Mass: Does a Photon Have It?

    If a photon does the 4 things listed below, why do we say it doesn’t have mass? I guess I’m asking if there’s a clear definition of mass. follows geodesics in spacetime (classically: accelerated in a grav field), curves spacetime (classically: creates a gravitational field), has momentum /...
  40. C

    Definition of work done by torque

    I' m trying to derive the work done by a torque from W = ∫ F ⋅ ds and I' ve looked up the internet, it said: W = ∫ F ⋅ ds ( since ds = dθ × r ) ---------------------------------------- ( Line 1 ) it can be written as W = ∫ F ⋅ dθ x r this is a vector triple product , thus can also...
  41. M

    I Proving the limit of a sequence from the definition of limit

    Say that we are asked to prove, using the definition of limits, that the sequence ##\frac{4n^2+3}{n^2+n+2}## tends to ##4## as ##n## tends to infinity. The following is a screenshot of the solution I found in a YouTube video: (Note that in the definition above, "g" denotes the limit - in this...
  42. Mr Davis 97

    I Flipping the sign in the definition of derivative

    Is it true that if ##f## is differentiable at ##a## that ##f'(a) = \lim_{h\to a}\frac{f(a+h) - f(a)}{h} = \lim_{h\to a}\frac{f(a-h) - f(a)}{-h}##. That is, can the sign of ##h## be flipped. I've seen this a few times and it seems a bit dubious.
  43. binbagsss

    A Modular forms- definition of a cusp

    this is probably a stupid question but for the fundamental domain for SL2(Z), we say the cusp is only at infinity. Compare say to hecke subgroups which are congruence subgroups where we say the equivalence classes are given by the points where the fundamental domain intercepts the real axis as...
  44. Math Amateur

    MHB Remark on the Definition of Differentials .... Lafontaine page 5 ....

    I am reading the book "An Introduction to Differential Manifolds" (Springer) by Jacques Lafontaine ... I am currently focused on Chapter 1: Differential Calculus ... I need help in order to fully understand a remark by Lafontaine following his definition of differentials ... Lafontaine's...
  45. F

    I Get Relation from Stress-Energy Tensor Def.

    Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity : $${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$ with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...
  46. redtree

    I The definition of velocity in the de Broglie relation

    I apologize ahead of time for the simplicity of the question, but this has really been bothering me.Given the de Broglie relation, assuming natural units, where ##\hbar = 1##: \begin{equation} \begin{split} \vec{k} &= M \vec{v} \end{split} \end{equation}My question regards velocity and...
  47. Zeynel

    B The definition of “vector” in math and physics

    I'm learning APL and this is how a vector is defined https://tryapl.org: All data resides in arrays. An array is a rectangular collection of numbers, characters and arrays, arranged along zero or more axes. We can use more specific terms for some arrays, like a single number is a scalar, a list...
  48. Danny Boy

    A Quantum synchronization description used in a paper

    In the paper "Steady-state spin synchronization through the collective motion of trapped ions" it states the following: "Steady-state synchronization of atomic dipoles forms the foundation for ultra-stable optical lasers utilizing narrow-linewidth atoms coupled to a lossy cavity mode. The...
  49. Math Amateur

    MHB Proof of Apostol's Definition 3.2 and Theorem 3.3: Help Appreciated

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focuses on Chapter 3: Elements of Point Set Topology ... I need help regarding a remark of Apostol's made after Definition 3.2 and Theorem 3.3 ...Definition 3.2 and Theorem 3.3 read as follows: In a note at the...
  50. K

    A Intrinsic definition on a manifold

    I'm reading "The Geometry of Physics" by Frankel. Exercise 1.3(1) asks what would be wrong in defining ##||X||## in an ##M^n## by $$||X||^2 = \sum_j (X_U^j)^2$$ The only problem I can see is that that definition is not independent of the chosen coordinate systems and thus not intrinsic to...
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