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. J

    Atmospheric eddies, waves etc: definitions

    Hi all. I'm reading up on types of atmospheric motion, and when discussing the meridional transport of various properties, the motion is generally split into three components: 1. Mean meridional circulation 2. Transient eddies 3. Stationary waves However, depending on what book...
  2. T

    Help understanding equivalent definitions for continuity

    I was hoping someone could help me understand the equivalence between the definitions for functions to be continuous between topological spaces, ie: For X and Y topological spaces, and f:X-->Y a function, my notes don't prove why these definitions are equivalent (possibly because I'm missing...
  3. C

    Elliptic function - different definitions

    Elliptic function -- different definitions Hi, I have recently discovered, that the definition of the complete elliptic integral of the first kind in Wolfram Mathematica (EllipticK[m]) is different from the usual (K(m)), given in Abramowitz-Stegun. Their domains are not the same. In...
  4. M

    Defining Emitter vs Observer for Schwarzschild Metric

    Let's say we are working with the Schwarzschild metric and we have an emitter of light falling into a Schwarzschild black hole. Suppose we define the quantity u=t- v where dv/dr= 1/(1-r_{s}/r) where r_s is the Schwarzschild radius. What is the u as observed by the emitter? I just need a...
  5. T

    Definitions and properties of limits (handwriting attached)

    Could someone look over this and see if I have any mistakes? I'm trying to show that ∫ y' dx = ∫ dy through definitions. http://imgur.com/6zCHYo5 Thanks!
  6. T

    Definitions and properties of limits (handwriting attached)

    I'm not entirely sure on the properties of limits, but this seems to work. Could someone look over this for me? http://imgur.com/6zCHYo5
  7. E

    Continuum Mechanics deformation definitions

    Homework Statement What do you understand by the following terms; (i) principal stretch (ii) an anisotropic material (iii) a dilatant deformation, (iv) a Lagrangian description of a deformation, and (v) a pure deformation. Homework Equations Am just trying to find descriptions for...
  8. C

    Confused about definitions in optics

    The width of a pulse is typically given in the time domain, correct? The effective width is the term to describe this. What is the spectral width in the frequency domain? How do you calculate spectral widths and effective widths? Thanks for your time.
  9. N

    Rigorous definitions in general relativity

    Hello (a) The universe U is a topological space whose elements are called events and as each event has a neighborhood homeomorphic to R^4. (b) A local coordinate system is a homeomorphism between an open subset of U and a bounded subset of R. (c) A world line segment is a continuous...
  10. B

    Movitation For Definitions In Physics

    Hello, I noticed in my physics textbook that we define certain relationships to be true. I can see how this is considerably helpful in deriving other relationships from these definitions; for instance, take position: we define these quantities to be so, and from it we can define other...
  11. N

    Rigorous definitions in quantum mechanics

    Hello It is question of specifying mathematical definitions which are cummunes in several theories. In classical physics, in special relativity, in quantum theories (wave functions and state vectors) and in general relativity, we can assert : (a) The universe U is a topological space...
  12. C

    Asymptotic notation informal definitions

    I was wondering if the O notation definition could be exchanged with the Ω notation and o could be exchanged with the ω notation. I ask this because of this: 2n² O(n²) means that 2n² <= c*n² which it is true for c=3 and n>=1 for example Instead, it would be like this: c*2n² <= n² which...
  13. S

    First order logic : definitions

    Hi all, Just a few question about FOL logic. What is the difference between terms and atoms, I read lot's of differents definitions, then when I think that I've understood, I find an exemple where both are used without any difference (for ordering by instance). An another question is : What...
  14. STEMucator

    Understanding Point Types in Real Analysis

    Just a few definitions I would like to verify so I'm not studying the wrong stuff. Interior Point : A point Q \in S \subseteq ℝ^n is an interior point of S if \forall \delta > 0, \exists N_{\delta}(Q) \subseteq S. The interior of S consists of all interior points and is denoted S˚ Boundary...
  15. Negi Magi

    Does anyone have an document which contains all definitions of physics laws

    I really need it now! Someone helps me
  16. K

    Definitions for Magnetic flux and Faraday's Law

    I was wondering if my understanding on Magnetic flux and Faraday's law was correct or not, is this okay? Magnetic flux is the number of lines in a magnetic field (Φ) Magnetic field strength (B) is the number of magnetic field lines over a given area B = Φ/A Other names for magnetic field...
  17. M

    Why are definitions for the studentized residual so confusing?

    I cannot find a consistent definition of the studentized residual and the RMSEP, because I've noticed that various websites, lecture notes and software packages mix up 1 or 2 definitions along the way to the point that a "compound" definition ends up very different between one reference source...
  18. Femme_physics

    Geometric Tolerances: Standards & Definitions

    Geometric Tolerances - Do standards for defining "general geometric tolerances" exist For example, if I want to define non-geometric tolerances for the whole part, I just write what type of IT it is. For instance, IT8. And then the manufacturer just looks at the chart to know the tolerances he...
  19. dextercioby

    2 definitions and a theorem relating them

    Homework Statement Let E be a vector space and p,q 2 norms on it. By definition p,q induce the same topology on E, iff they assign the same neighborhood basis to the 0 vector. *QUESTION: What does the bolded part mean ? Does this mean that, if whatever A included in the system (=basis?)...
  20. D

    Are proofs needed for definitions? Conditional Probability

    My probability class has me wondering about pure math questions now. We started with the axioms and are slowly building up the theory. Everything was fine but then a definition of Conditional Probability P[A|B] = \frac{P[AB]}{P[B]} appeared and it's just not sitting right with me. I know that...
  21. M

    Different definitions of acids/bases

    I am having some difficulty in understanding the reason for the various types of acids/bases, of which i refer to bronsted-lowry, arrehenius, and lewis acids/bases. A bronsted acid donates an H+ and a bronsted base accepts a H+. However, a lewis accept acceps an e- pair and a lewis base donates...
  22. G

    I've been staring at these definitions for a while now

    what is the nature of the relationship, if any, between the two? if parsimony is simplicity in explanation and reductionism is simplicity of mechanism, then is reductionism externally projected parsimony?
  23. N

    Exploring Casimir Operators and Rest Reference Conditions in Particle Physics

    Hi guys! There is something I would like to get your help with... I am looking at the equation: W^{\mu}=-\frac{1}{2} \varepsilon^{\mu\nu\lambda\sigma}M_{\nu\lambda}p_{\sigma} Which is, if I understand correctly,a Casimir Operator. Now, I wish to look at a particle in its rest reference...
  24. A

    Comparing definitions of groups, rings, modules, monoid rings

    Hi, I wanted to see what people think about my current viewpoint on recognizing structures in abstract algebra. You count the number of sets, and the number of operations for each set. You can also think about action by scalar or basis vectors. So monoids groups and rings have one set...
  25. A

    Competing definitions of the Fourier transform

    Just began a serious study of the Fourier transform with a couple of books. One of them defines the Fourier transform on \mathbb R as \hat f(\xi) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\xi x}dx. Another defines it as \hat f(\xi) = \int_{-\infty}^\infty f(x)...
  26. A

    Help with definitions of dioxide and oxide

    Here is the definition of oxide I found in a dictionary: A compound of one oxygen atom combined with another element. And here is the definition of dioxide that I found in a dictionary: An oxide containing two atoms of oxygen in each molecule; binoxide. Now my question/confusion is...
  27. W

    What are the basic terms in semiconductor physics and how are they defined?

    hey guys i want to know the definition of these terms & what are they but please i don't want big words because i won't understand i want to understand it simply: 1-Fermi level 2-Density of states 3-Carrier concentration & the graphs (in the attachment) 4-Mass action law 5-Charge...
  28. L

    Statistical Definitions and Statement: X, S^2, μ, σ^2, True or False

    Homework Statement Let X1,…,Xn denote a random sample from a population with mean μ and variance σ^2. Assume that both μ and σ^2 are finite but unknown. Let X denote the sample mean and S^2 denote the sample variance. Are the following statements true or false? A-There is no difference...
  29. W

    Apparent Discrepancy Between Two Definitions of Newton's First Law

    So I was reading through my textbook (specifically, Physics for Scientists and Engineers, Eighth Edition, Volume 1 by Raymond A. Serway and John W. Jewett Jr.) and I noticed that, in one of the "Pitfall Prevention" sections (which are usually quite helpful - not this time, evidently), it says...
  30. R

    Follow-up and pipeline: definitions

    Hi I am looking for the actual meaning of some technical/procedure words in astronomy, like: 1) Follow-up 2) Pipeline As far I know, the follow up is the observation of an object or a field for a given amount of time, isn't it? Then, if it is, what's the difference with "standard" surveys...
  31. U

    Differences between basic definitions for compressors and turbines

    While making a comparative study between turbines and compressors, I noticed some differences between the way they are studied...pleasehelp me understand why these differences exist... 1. Efficiency for a turbine = (actual work/ideal work) whereas for compressors, it is (ideal work/actual...
  32. K

    Equivalence of definitions for regular representations

    There seem to be two definitions for a regular representation of a group, with respect to a field k. In particular, one definition is that the regular representation is just left multiplication on the group algebra kG, while the other is defined on the set of all functions f: G \to k . I do not...
  33. I

    Questions on inductive definitions in a proof

    Hi I was trying to solve the following problem from Kenneth Ross's Elementary Analysis book. here is the problem. Let S be a bounded nonempty subset of \mathbb{R} and suppose that \mbox{sup }S\notin S. Prove that there is a non decreasing sequence (s_n) of points in S such that \lim...
  34. D

    Force is ill-defined? contradictory definitions?

    Correct me if I am wrong but as far as I know, force is generally defined in three ways ways: 1) F = \frac{d p}{d t} 2) F = m\dot v 3) F = ma This is all well in good usually...until the case arises when mass is variable. Then two contradictory cases arise: If we take definition 1...we...
  35. L

    Carrier Swing & Deviation Ratio: Clarifying the Definitions

    Can anyone help me to find out the definition of "carrier swing"&"deviation ratio"? The definitions which i found on surfing the net is too vague.Can someone help me with this?
  36. reddvoid

    Two different definitions for sinc ?

    i've seen in some texts they use sin(pi t)/pi t = sinc(t) and in some they've used just sin(t)/t = sinc(t) each gives different answer for example if i want to find FT of rect(t/tow) using former one gives sinc(w tow/2 pi) and if i use former one i get sinc (w tow / 2) so how...
  37. A

    Have the definitions of time and metric meter changed over years?

    i remember the French came up with metric meter by measuring the distance between equator and north pole and then divided by an integer to come up meter. it that still the defintion for meter? also, it seems now that a second is defined by the integer number of oscillation of atomic clock...
  38. G

    Definitions of integral over a bounded set.

    Hi! I want to learn a course of "general relativity". For this, I've realized that I have to master the differential geometry. So, I've chosen Lee's book called " introduction to smooth manifolds". In the appendix of the book, some required knowledege of integrations on an euclidean space...
  39. M

    LOGIC: A Request for Clarification of definitions

    With the study of logic, lots of words get thrown around that I don't really understand their complete meaning. With a deductive argument the conclusion is true if the premises are true, and an argument is valid if all the inferences (and the conclusion) follow logically from the axioms. These...
  40. M

    General Definitions of Impulse and Work

    In classical mechanics, if we consider the motion of a particle of mass m, then The mass m is constant The vector \vec{c} can be: \ldots or \vec{r} or \vec{v} or \vec{a} or \vec{j} or \ldots \vec{c}_1 = d\vec{c} / {dt} \vec{c}_2 = d^2 \, \vec{c} / {dt^2} Definition of...
  41. M

    Definitions of Momentum and Work

    In classical mechanics, if we consider the motion of a particle of mass m, then m=constant\vec{v}=d\vec{r}/dt\vec{a}=d\vec{v}/dt\vec{j}=d\vec{a}/dt\ldots Definition of Momentum (\vec{M}) \vec{M} \; = \int_a^b m \, \vec{a} \, dt \; = \int_a^b m \,d\vec{v} \; = \Delta \; m \, \vec{v} If...
  42. M

    Alternative definitions of energy?

    I had an interesting challenge earlier this year in physics class, and I got a good grade on my answer, but I'd like to see what other people think about this. Energy is defined in the dictionary as being the ability to do work, while work is defined as the application of energy (roughly...
  43. P

    Why so many definitions of an inertial frame?

    A Newtonian inertial frame is one where objects obey Newton's first law. Schutz (A first course in general relativity) says an inertial frame cannot be constructed in a gravitational field because it's then impossible to synchronize the frame's clocks? For the same reason an inertial frame...
  44. T

    Specific technical definitions of quantum terms

    I feel that a lot of the misunderstaing in quantum physics arises from the misinterpretations of terms used by physicists. I think that the forum should have a page where all the quantum terms are explained in a precise technical manner to avoid confusion. I will state pop culture's...
  45. jinksys

    Prove the definitions of Linear Transformations

    Homework Statement Show that 2.1.1 is equivalent to the totality of 2.1.2 and 2.1.3.Homework Equations The Attempt at a Solution aTx + bTy = aT(x) + bT(y) = T(ax) + T(by) = T(ax + by) ?
  46. P

    Source, path, reflection differences (definitions)

    What are the definitions of source difference, path difference and reflection difference. Thanks!
  47. A

    Sobolev Spaces different definitions

    Hi, I am studying PDEs and I am confused by the definition of Sobolev spaces as they are different in two books. I'll write the definitions and mention the points of difference which I see despite which I still can't see the difference in definitions. 1) PDEs by Lawrence Evans Let U be...
  48. N

    Conceptual definitions of vector topics

    (Is this thread in the right place?) A few questions on vectors: 1. I was wondering if anyone could explain in conceptual terms what the dot product and cross products represent. (I understand how these are calculated, but why are they important?) 2. Also, would it be accurate to describe a...
  49. S

    The FAQ on proofs should emphasize definitions

    I think the FAQ on proofs would be improved if it emphasized the use of defintions. It says that theorems and axioms are used in proofs, but many many textbook type proofs hinge on "parsing" definitions correctly. As alluded to in the FAQs related to "is .999.. = 1?", many difficulties that...
  50. J

    Confused by separate definitions of sets which are bounded above

    I have been consulting different sources of analysis notes. My confusion comes from these two definitions \begin{defn} Let S be a non-empty subset of $\mathbb{R}$. \begin{enumerate} \item $S$ is Bounded above $ \Longleftrightarrow\exists\,M > 0$ s.t. $\forall\, x\in S$, $x\leq M$...
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