What is Definitions: Definition and 273 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. Math Amateur

    I K-algebra/associative algebra - equivalence of definitions

    I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows: I have to say I do not find that this definition gives me a good intuitive idea...
  2. Math Amateur

    MHB Equivalence of Definitions of Algebras in D&F and Cooperstein | Peter

    I am reading Dummit and Foote's book "Abstract Algebra" (3rd Edition) and am focused on Chapter 15: Commutative Rings and Algebraic Geometry ... ... On page 657 D&F give a definition of a k-algebra ... as follows:I have to say I do not find that this definition gives me a good intuitive idea...
  3. JulienB

    I General questions about integrals (definitions)

    Hi everybody! I'm currently studying integrals, and I would like to clarify a few definitions, especially about the criterions of convergence/divergence of an integral. Basically if that's okay for you guys I'm going to list and number a few statements and I'd like to know if they are true or...
  4. G

    Divergence and transversal extension integral definitions

    Hi. I am reading a paper about gaussian beams and the author says that gaussian beams have simultaneously minimal divergence and minimal transversal extension. In order to prove it, the author states that \mathrm{divergenece} \propto \int_{-\infty}^{+\infty} \frac{d\,k_{x}}{2\pi}...
  5. P

    Helmotz decomposition definitions.

    I' m studing the hodge helmotz decomposition of a flow Field, and i have Found different definitions. I'm Not sure to have assigned the rigth meaning to the terms of the decomposition. Look At The picture( i don't write here cose there are several equations).
  6. H

    Definitions of parity conservation

    Definition 1: The expectation value of the observable related to the parity operator ##\hat{P}## is constant over time. That is, \frac{d}{dt}\langle P\rangle=0 \int\Psi^*(r)\ \hat{P}\ \Psi(r)\ dr=constant \begin{align}\int\Psi^*(r)\ \Psi(-r)\ dr=constant\end{align} Definition 2: If the...
  7. N

    Understanding 'Observe' in Physics: Definitions and Explanations

    When I read any physics articles online I always end up receiving the wrong message. especially in terms such as 'observe'. what does this term mean when its used to describe exeperiments such as the double slit
  8. N

    What are the definitions and properties of Riemann sums?

    Just want to see if I actually understand what these all mean. Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...
  9. R

    Understanding Projectile Motion: Definitions and Concepts

    QUESTION REMOVED
  10. bcrowell

    Alternative definitions of geodesic

    I teach both physics and math at a community college, and I've volunteered to give a short talk for students at our weekly math colloquium that has to do with curvature and non-curvature singularities in relativity. This is a tall order, given that I can't even assume that all the students will...
  11. Math Amateur

    MHB Algebras - k-algebra and R-algebra - reconciling two definitions ....

    I am trying to get a full understanding of the notion of an algebra ... I have thus consulted two books - Cohn: "Introduction to Ring Theory and Dummit and Foote: Abstract Algebra. Cohn defines a k-algebra (or linear algebra) as follows: If we want to amend the above definition to an R-Algebra...
  12. newjerseyrunner

    Angle and trig definitions in curved space

    I was going to ask a question about whether or not pi was constant or changed with curved space. I found the answer on here that it does indeed change. Then I started thinking about the ramifications of that. sine waves are dependent on pi, so they should change too. Does sin(theta) =...
  13. Ahmad Kishki

    Logical structure of definitions

    Is the claim that all definitions are biconditionals, true? Is the converse statement true as well, that all biconditionals are defintions?
  14. M

    "Proving" that definitions "work"

    So I have a friend who wants to become an engineer who is overtly obsessed with his mathematical foundations at the moment. He has confessed recently that he didn't understand the definition of the derivative, and asked me to elaborate. And so I did. However, what he asked next kind of...
  15. topsquark

    MHB What Are Some Alternative Sources for Help with Lie Algebra Definitions?

    First off: I am not complaining about any of the members here or elsewhere. I do not post questions expecting help. When I post a question I would like an answer but I do not require it. Getting an answer depends on who is around and who is willing to help when I post the question. Still...
  16. Delta2

    Humidity definitions and measurement devices

    I know there are three definitions of humidity. Absolute , relative and specific. Can you tell me if on a typical commercial device (such as the one i have in a clock i have) that measures humidity as percentage is it the relative humidity that it measures? Can the percentage become greater...
  17. S

    Confused by different definitions of position

    Hello, I am self-studying an introductory mechanics textbook and while I feel I understand the material there pretty well, I came across across an online definition of position which seems at odds with the explanations and definitions in my book. The definition that confused me is: ""Position...
  18. nomadreid

    Are the two definitions of "bit" compatible?

    First, I am not sure that this is the right place for this post, because earlier Physics Forums broke up the "computing and technology" into several subheadings, such as information science as opposed to hardware, and so forth, that I don't see anymore. Anyway, so here is an elementary question...
  19. R

    Exploring Potential Energy in Oscillating Systems

    1. With respect to any oscillating system, what is the difference between ΔPE, PE, and PEaverage?Homework Equations ---- The Attempt at a Solution Hi all. I want to preface this by saying that we have been discussing the Lennard-Jones potential and particle theory in class recently, after...
  20. DavideGenoa

    Definitions of Lebesgue integral

    Dear friends,I know the definition, from A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа, of Lebesgue integral of measurable function ##f:X\to \mathbb{C}## on ##X,\mu(X)<\infty## as the limit ##\int_X...
  21. I

    MHB Definitions of Functions and Spaces

    Hi everyone, I am in second year university and am taking linear algebra this semester. Never having been a strong maths student, I am certainly struggling with some basic concepts and especially notation. I have tried searching on the web but have had difficulty in finding something which...
  22. Math Amateur

    MHB Definitions of Algebras in Cohn and in Golan

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Cohn's book, in Chapter 2: Linear Algebras and Artinian Rings, we find the definition of an algebra ... ... but in Jonathan Golan's book ["The Linear Algebra a Beginning Graduate Student...
  23. Math Amateur

    MHB Definitions of Algebras in Cohn and in Dummit and Foote

    I am reading "Introduction to Ring Theory" by P. M. Cohn (Springer Undergraduate Mathematics Series) In Chapter 2: Linear Algebras and Artinian Rings we find the definition of an algebra ... ... but in the chapter on module theory on page 342 of Dummit and Foote we find a different definition...
  24. B

    Multiple definitions for equilibrium

    During my studies I failed to understand thermonynamics and compared the whole subject to black magic. This frustrated me a lot. Years later I tried to restudy it by myself reading sources with less conventional approaches. I had finally come to believe I could make sense of it. But re-reading...
  25. 1

    Vectors - are the definitions truly equivalent?

    There are two "formal" definitions of vectors (and tensors in general) which I've learned. The first is what I consider the "better" definition, one I learned in linear algebra. We call a set X a vector space over a field F whenever that set has properly defined operations of scalar...
  26. A

    MHB Limit Definitions and Finding the Unknown Quick Question

    Hey guys, Here's another quick question this time from a problem set I'm having trouble with at the moment. Question: So, for a, I computed f(x) = x^5 This is because of the numberator's right side. If 2+h is raised to the 5, this must be the function. Moreover, a=2 because 2 is already on...
  27. A

    MHB Limit Definitions and Extreme Value Theorem Help Needed

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For 1a, I just took \lim_{{h}\to{0}} of the function using [ f(x+h)-f(x) / h ] and simplified. Ultimately, this gave me...
  28. T

    Prove the 3 definitions of entropy are equivalent (stat. mechanics)

    Homework Statement S(E,V) = kln(\Gamma(E) )\\ S(E,V) = kln(\omega(E) )\\ S(E,V) = kln(\Sigma(E) )\\ S entropy, k Boltzmann's constant. Prove these 3 are equivalent up to an additive constant. Homework Equations \Gamma(E) = \int_{E<H<E+\Delta}^{'}dpdq\\...
  29. V

    Confused about formal definitions of probability theory

    I think the first thing that is confusing me is the terminology. There are too many similar terms (e.g. probability measure, probability distribution, probability density function, probability mass function) What are the general concepts and what are the instances of those concepts? Like, are...
  30. J

    Problem with differential/integral definitions

    If work ##W = \Delta E = \int_{s} \vec{F} \cdot d\vec{s}##, so work can't be ##\frac{dW}{d\vec{s}} = \vec{F}## like is here: http://en.wikipedia.org/wiki/Work_%28physics%29#Path_dependence cause this implies that ##W = \int \vec{F} \cdot d\vec{s} = E##, but the work is the variation of...
  31. W

    Unifying Different Definitions of Adjoint Map

    Hi, this question seem to fall somewhere between Analysis and Algebra; I just choose this section; sorry if it is the wrong one. I would appreciate any suggestions, refs., etc. I'm basically trying to see if the different definitions of adjoint maps can be unified into a single...
  32. M

    Can a Differential Equation Have an Unrelated Variable?

    I'm not sure if this is particularly important, but so far through my studies I've only encountered DE with two related variables (e.g. ## \frac {dy}{dx} = 3x##). Now, given another function with an additional variable that is UNRELATED to the two other variables, can this still be considered...
  33. Entanglement

    The original definitions of Ohm, ampere, volt and coulomb

    What's the original definition of Ohm, ampere, volt and coulomb, And which unit was defined according to the other ??!
  34. J

    2 definitions for argument, why?

    In the wiki, I found this definition for the argument: http://en.wikipedia.org/wiki/List_of_trigonometric_identities#Exponential_definitions However, in other page of the wiki (http://en.wikipedia.org/wiki/Complex_conjugate#Use_as_a_variable), I found this definition for...
  35. A

    Clarifications regarding definitions of Taylor, Laurent etc. series

    I want to know the difference between various kinds of series like Taylor, Laurent and Asymptotic. I have some understanding but I want some clarifications. Here is what I understand:- 1) Taylor series is just f(0) + x.f'(0) + x2.f''(0)/2! + ... 2) Laurent series is applied when taylor series...
  36. Avatrin

    Memorizing Mathematical Definitions

    Hi Usually when learning math, understanding the theorems and ideas helps tremendously to remember math. I get that... I got through calculus, linear algebra and complex analysis easily. The problem for me started with three branches in mathematics: Real analysis, measure theory and abstract...
  37. C

    Open set (equivalent definitions?)

    I've seen open sets ##S## of a bigger set ##X## being defined as 1) for every ##x\in S## one can find an open disk ##D(x,\epsilon)## centered at ##x## of radius ##\epsilon## such that ##D## is entirely contained in ##S##. Where $$D(x,\epsilon)= \left\{y \in X: d(x,y) < \epsilon\right\}$$...
  38. W

    Factor (Quotient) Space definitions.

    I'm learning algebra by myself and this concept is confusing me. Please excuse me if I define anything wrong... I've never expressed myself in this language before. Lets say we have a group G and a group G' and there exists a homomorphism R: G → G' and for any element g \in G, the...
  39. O

    What can I consider 'fundamental definitions'?

    What are considered fundamental definitions? For context, a question was posed where the prof. provided parametric equations for the motion of a satellite in orbit and said that we can use the provided equations, fundamental definitions, and no other equations to solve the problems. What does...
  40. atyy

    Is ADM Energy Equivalent to Komar Mass in All Spacetimes?

    Can I check whether these are right? Here let's define the ADM mass as length of the ADM energy-momentum vector. In the Schwarzschild spacetime ADM energy = Schwarzschild mass parameter In a spacetime in which the ADM energy and the Komar mass are both defined ADM energy = Komar mass
  41. S

    Bit Specific Assembly Definitions

    I have an example given in the textbook that defines port A bits 5 as PA5 equ 0x40004080. Port A is 0x40004000, and the first 7 bits are for data. I do not see how bit 5 is 0x80. I figured that would be bit 7 and bits 5 would be 0x20. We use ARM assembly language for LM4F120H5QR microcontroller.
  42. N

    Standards Definitions are weird

    Hello, While riding home today, I started to wonder what was the definition of a second in the context of time. I looked it up and found out that the definition has changed somewhat over the years and it is now defined as (paraphrased) 9,192,631,770 periods of the cesium atom. A meter is...
  43. J

    A briefing of Topology's most important definitions and results?

    I just need to know the basic ideas of topology, and the most important results, because I'll have differential geometry the next semester. Does anyone have a good material for this? Or you can just say what to search for and I'll search it. Thank you :)
  44. stripes

    Equivalent definitions of convergence

    Homework Statement \mathbf{D1:}\forall\varepsilon>0,\exists K\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n\geq K\Longrightarrow|x_{n}-x|<\varepsilon \mathbf{D2:}\forall\rho>0,\exists M\epsilon\mathbb{N},\forall n\epsilon\mathbb{N},n>M\Longrightarrow|x_{n}-x|\leq\rho Show these two...
  45. D

    Showing exp(x) definitions are the same.

    Homework Statement e^{x}=\sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!} and e^{x}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}. I want to show that \sum\limits_{k=0}^{\infty}\frac{x^{k}}{k!}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^{n}. 2. The attempt at a...
  46. S

    Transmittance: Conflicting definitions?

    Not sure if this is in the right section, but I'm not sure where else it would fit. I'm currently researching a variety of optics-based topics, and I'm a bit confused by what appear to be some conflicting definitions of transmittance. I've seen the following: 1) It's the ratio of monochromatic...
  47. Alpharup

    The need of definitions in physics

    My high school physics textbook gives much emphasis on memorizing definitions.We have to memorize the definition of force, inertia, etc...We have to write the definitions as it is in the textbooks... For example, the definition of "force" in physics textbook is, "Force is defined as that...
  48. F

    Rank-2 tensor: multiple definitions

    Hi all! In a paper they say that a certain quantity is a rank-2 tensor because it transforms like a spin-2 object under rotations, that is: if the basis vectors undergo a rotation of angle \phi, then this quantity, say A, transforms like A\mapsto Ae^{i2\phi} As far as I knew, a rank-2...
  49. K

    Question on variable definitions in Paschens law? (breakdown voltage)

    Hello all, In the equation V= apd/Ln(pd)+b as described by Paschens law regarding breakdown voltage, I was wondering what the "L" and the "n" stood for. Example of what I'm looking for: p stands for pressure. Thanks.
  50. E

    Trouble with the definitions of equivalent and collinear.

    My teacher, textbook and the internet have differing definitions. First of all: Equivalent. My teacher says that two parallel vectors with the same magnitude are equivalent, but my textbook says that two vectors in the same direction are equivalent. ×--> <--× are these equivalent?And...
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