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

    I Confused on definition of projection

    My textbook says: "if ## V = W_1 \oplus W_2 ##,, then a linear operator ## T ## on ##V ## is the projection on ##W_1## along ##W_2## if, whenever ## x = x_1 + x_2##, with ##x_1 \in W_1## and ##x_2 \in W_2##, we have ##T(x) = x_1##" It then goes on to say that "##T## is a projection if and only...
  2. P

    Thermodynamic definition of volume

    I'm studynig thermodymamics using the textbook 'Thermodynamics foundations and applications' (Beretta and Gyftopopulos). The definition of a system according to the authors consist in the specification of : -the costituents of the system ( atoms or molecules or prottons neutrons ... in...
  3. hackhard

    Why were momemtum, kinetic energy and work introduced?

    why were quantities like momentum, force , potential energy, kinetic energy,work ,etc needed to be introduced in physics? and why were they defined the way they are defined?. would it not be possible to explain nature without defining these quantities or by using alternate physical quantities ?
  4. M

    Definition of an operator in a vector space

    In the book that I read, an operator is defined to be a linear map which maps from a vector space into itself. For example, if ##T## is an operator in a vector space ##V##, then ##T:V\rightarrow V##. Now, what if I have an operator ##O## such that ##T:V\rightarrow U## where ##U## is a subspace...
  5. Z

    What's a firm/good definition of an "attractive" potential?

    What makes an attractive potential "attractive"? I've just started learning about 1D S.E.'s with delta potentials and see the phrase frequently but am having trouble discerning a good, clear definition of what that means.
  6. D

    Question on declaration and definition

    When I am creating an object via car audi audi=new car() what does the first line actually do? In non object oriented languages, some declaration like int mynumber to reserve a certain kind of bytes. But how does the compiler know what to reserve for the variable audi knowing it is of...
  7. mnb96

    Definition of chart for Lie groups

    Hello, I'm reading a book on Lie group theory, and before giving the definition of a Lie group G, the author defines the concept of chart as a pair (U(g), f) where: i) U(g) is a neighborhood of g∈G ii) f : U(g)→f(U(g))⊂ℝn is an invertible map such that f(U(g)) is an open subset of ℝn. My...
  8. M

    Gravitational Waves: Definition & General Overview

    What are gravitational waves in general, not just in the weak field/linearized theory?
  9. M

    Definition of Energy: Post-Relativity Questions

    This thread was triggered by @Anonymous Vegetable 's question re. nuclear fusion. In GR, energy density (in some coordinate system) is a parameter with physical implications- it is the (0,0) element in the stress energy tensor. This is in contrast with the situation in classical mechanics...
  10. R

    Current Definition: Conventional vs Electron Flow

    When the current is defined as being the conventional current then: i = dq/dt, i = integral of J*ds When the current is defined to be the electron flow: i = -dq/dt, i = - (integral of J*ds) Is this right?
  11. B

    Definition of a Cone: Does it Include Zero Vector?

    On this wikipedia page https://en.wikipedia.org/wiki/Cone_(linear_algebra) , "a subset ##C## of a real vector space ##V## is a cone if and only if ##\lambda x## belongs to ##C## for any ##x## in ##C## and any positive scalar ##\lambda## of ##V##." The book in this link...
  12. M

    Shor's algorithm, definition of modulo

    Hi guys, My question shouldn't take too long to be answered but I simply can't find anything using a google search. It's more of a problem from number theory rather than a physical one. I am referring to the Wikipedia article to Shor's algorithm and I still can't get my head around how the...
  13. Kilo Vectors

    Definition of a polynomial? and degree? integral and ration

    Hello What is the standard definition of a polynomial? according to the book I am using a polynomial is an algebraic expression which is integral and rational for all the terms. It gives no definition of integral or rational seperately, but I think integral means that the variables are to...
  14. S

    Definition of Image of a linear transformation

    Homework Statement The image of a linear transformation = columnspace of the matrix associated to the linear transformation. More specifically though, given the transformation from Rn to Rm: from subspace X to subspace Y, the image of a linear transformation is equal to the set of vectors in X...
  15. mnb96

    Definition of regular Lie group action

    Hello, in group theory a regular action on a G-set S is such that for every x,y∈S, there exists exactly one g such that g⋅x = y. I noticed however that in the theory of Lie groups the definition of regular action is quite different (see Definition 1.4.8 at this link). Is there a connection...
  16. S

    Understanding Isomorphisms for Linear Transformations

    Homework Statement I have a question about isomorphisms -- I'm not sure if this is the right forum to post this in though. A linear transformation is an isomorphism if the matrix associated to the transformation is invertable. This means that if the determinant of a transformation matrix = 0...
  17. A

    What is the definition of phase lag and phase difference?

    what is the definition of phase lag and phase difference in waves and how are these 2 related?please explain in simple words and with real life examples. Please explain phase,phase lag and phase difference from scratch.
  18. D

    Confusion about [itex]T[/itex] in the definition of entropy

    In the derivation of the Clausius inequality, T is the temperature of the reservoir at that point in the cycle, but in the definition of entropy it becomes the temperature of the system. This seems to work for a Carnot cycle, where the two are the same, but for other processes, such as an object...
  19. G

    MHB How do we see that these are mappings from the definition?

    Definition: If $S$ and $T$ are nonempty sets then a mapping from $S$ to $T$ is a subset, $M$, of $S \times T$ such that for every $s \in S$ there's a unique $t \in T$ such that the ordered pair $(s, t) \in M.$ Could someone please explain how these are mappings. The notation of the definition...
  20. W

    Definition of Compound Statement

    Hello, In my Real Analysis textbook (Schramm) they say that an example of compound statement would be "Either 1+1=2 or a pencil is a useful tool in neurosurgery." I was wondering why this isn't a non-statement since I don't see where the truth value of it would be. Thanks!
  21. Capisko

    What is the definition of the coefficient of linear and mass

    hi, i'm student in a university. I do not understand the definition of these concepts, and I would like to know the properties of these and it depends questions like They depend on the material used? the atomic number? thanks for answering capisko,
  22. J

    Implications of varying the definition of the derivative?

    I have been playing around with calculus for a while and I wondered what would it be like to make some changes to the definition of derivatives. I'd like to look at the original definition of derivatives in this way (everything is in lim Δx→0): F(x+Δx) - F(x) = F'(x) * Δx The Δx factor...
  23. T

    Def. of derivative and cosx=sin(Pi/2-x) to prove y'=-sinx

    A lot of web pages/books show how to use cosx=sin(Pi/2-x) and the chain rule to prove that the derivative of cosx=-sinx. My question is how to use this identity and the defintion of the derivative to prove the same thing. Or whether it is at all possible. Seeing that i get...
  24. C

    I have a question about the definition of a vector

    Why is it that a vector can be described in terms of a simple linear combination, like v = xi + yj + zk, where v is a vector, and i, j, and k are all unit vectors. It just seems a bit convenient that it's as simple as just adding the components this way. It seems like there should be a bit more...
  25. Z

    The definition of transverse amplitude in the process: ep→epπ

    When CLAS of Jlab describe the result of the process ep→epπ,the use the terminal "transverse amplitude" Ml±(W,Q2), El±(W,Q2),and "scalar(longitudinal) amplitudes"Sl±(W,Q2),they correspond to photons of the magnetic, electric, and Coulombic type.Are they observables? The index of...
  26. Alpharup

    Question on ε in epsilon-delta definition of limits.

    I am using Spivak calculus. The reason why epsilon-delta definition works is for every ε>0, we can find some δ>0 for which definition of limit holds. Spivak asserts yhat if we can find a δ>0 for every ε>0, then we can find some δ1 if ε equals ε/2. How is this statement possible? Since ε>0, then...
  27. Helios

    Phase Velocity Definition: A or B?

    As I'm seeing things, there are these choices for a definition of relativistic phase velocity u; a) u = (mc^2)/ p b) u = E / p Now I like choice a) because it leads to a correct-looking index of refraction via n = c / u. It leads to a correct-looking solution to the ray equation. However...
  28. W

    Formal Definition of Inner Join?

    Hi All, Hope this is not too simple/dumb. I think I have a good idea of what an inner join of two SQL tables T1, T2 along a common field F is, but , for an exercise, I am looking for a precise definition. I am looking for a definition of this sort: (please correct if necessary or let me know if...
  29. E

    Variables definition in Euler's introduction to analysis

    In his book, Euler gives the definition of a variable to be : "A variable quantity is an indeterminate or universal quantity, which includes within itself all completely determined values." What does he mean exactly in the last part of the sentence?
  30. H

    Vector Definition: Magnitude & Direction Effects

    Why we can not define a vector as a quantity which has magnitude and direction? Why we define the vectors according to behavior of its components in rotated coordinate-frames?
  31. Multiple_Authors

    Building a Definition for Heat - Comments

    Multiple_Authors submitted a new PF Insights post Building a Definition for Heat Continue reading the Original PF Insights Post.
  32. O

    Definite Integral by Definition

    Homework Statement Let A be the area of the region that lies under the graph of f(x) = 2x 2 + 5 between x = 0 and x = 4. Find an expression for A using n rectangles. Then evaluate this expression. Homework Equations Answer is 188/3 h= (4/n) The Attempt at a Solution [/B] The problem I am...
  33. rjbeery

    What is the definition of physical contact?

    We talk about local effects and surfaces being "in contact" with one another but do we have a vigorous definition for such a state? The objects reach a distance where their repelling EM charges resist and balance against a given force (such as gravitation)...but increased charges would make...
  34. W

    Relation between/among Tables/Entities: Definition and Condi

    Hi All, My apologies, I think I may have asked this question already, but I could not find it. Here it goes: I have seen the usage of the word 'relation', specifically a relation between tables but I have not seen a formal definition. From what I understand, tables X,Y are related to each other...
  35. B

    Definition of Principal Square Root?

    Is the principle square root just the positive and negative roots of any number (as opposed to just the positive)? I've seen some confusing definitions of this term online and thought I'd double-check with knowledgeable math people here. Lastly, if it is just the + and - roots of any number...
  36. Avatrin

    Epsilon delta definition of limit

    I am struggling to properly understand the \varepsilon-\delta definition of limits. So, f(x) gets closer to L as x approaches a. That is okay. However, taking the leap from there to the \varepsilon-\delta definition is something I have never really been able to do. Why is the formulation we...
  37. E

    Why is dispersion important in wave propagation?

    In the propagation of non-monochromatic waves, the group velocity is defined as v_g = \displaystyle \frac{d \omega}{d k} It seems here that \omega is considered a function of k and not viceversa. But in the presence of a signal source, like an antenna in the case of electro-magnetic wave or a...
  38. J

    Definition of Energy in Friedmann equations?

    The first Friedmann equation for a flat Universe is given by: $$\bigg(\frac{\dot{a}(t)}{a(t)}\bigg)^2 = \frac{8 \pi G}{3} \rho(t)$$ The energy density ##\rho(t)## is given by: $$\rho(t) \propto \frac{E(t)}{a(t)^3}$$ where ##E(t)## is the energy of the cosmological fluid in a co-moving...
  39. Dishsoap

    Definition of "brief" for a personal statement?

    Here's the description of the personal statement for one university for which I'm applying for graduate school: Please upload a document briefly describing your past work in your proposed or allied fields of study, including non-course educational experiences, teaching, or other relevant...
  40. E

    Exploring the Role of Superconformality in Coupling Supergravity to Matter

    I have been trying for a while to read a precise definition of a Vector Multiplet (to whom ##N=2## Supergravity theories couple to in ##4D##) but was not lucky in finding a self-contained one. The best I got was that on https://en.wikipedia.org/wiki/Supermultiplet though it was on...
  41. L

    Einstein-Cartan Theory: Dynamical Definition of Spin Tensor

    Hi, this is my first message on thi forum :D I apologize in advance for my english. I'm doing my thesis work on the theory of relativity of Einstein-Cartan. I'm following the article of Hehl of 1976; it's title is "General relativity with spin and torsion: Foundations and prospects". I can't...
  42. S

    Definition of a specific kind of motion

    Homework Statement It is not really a homework question, but rather a translation problem. I searched everywhere, but I still cannot find a good translation into English of a term that is defined as: "In mechanics, the motion of a moving reference frame relative to another (primary) reference...
  43. W

    "Crow Flies Distance" On a Rectangular Grid -- Definition?

    Hi all, I am looking for a precise definition of " Crow Flies Distance" on a rectangular grid. I have not found a precise definition yet, but from what I have read, I think it would be the standard Euclidean distance ## d((x_1,y_1),(x_2,y_2)) = \sqrt {( (x_1-x_2)^2+(y_1-y_2)^2}) ## Is this...
  44. naiasetvolo

    The definition of the electric field?

    Hey guys, I need an explanation on the definition of the electric field. It was said in a post that " the definition of the electric field is defined in terms of how it is measured or tested". What do they mean by measured/tested?
  45. T

    Double check apparent power definition

    Hey just to make sure, can we say apparent power S = V^2 / Z for a complex impedance Z? And do we have to worry about conjugates at all? Cheers
  46. E

    What is the difference between standard and isotropic metrics?

    The metric $$ds^2=-R_1(r)dt^2+R_2(r)dr^2+R_3(r)r^2(d\theta^2+sin^2d\phi^2)$$ when changed to $$ds^2=-R_1(r)dt^2+R_2(r)(dr^2+r^2d\Omega^2)$$ upon setting ##R_2(r)=R_3(r)##, the later metric holds the name of isotropic metric. My question what is the difference between the first and the second...
  47. Math Amateur

    MHB Paul E Bland's "Direct Product of Modules" Definition - Category-Oriented

    I am reading Paul E. Bland's book: Rings and Their Modules and am currently focused on Section 2.1 Direct Products and Direct Sums ... ... I am trying to fully understand Bland's definition of a direct product ... and to understand the motivation for the definition ... and the implications of...
  48. Titan97

    Interpreting Curl in Vector Fields: ∇×v

    In a river, water flows faster in the middle and slower near the banks of the river and hence, if I placed a twig, it would rotate and hence, the vector field has non-zero Curl. Curl{v}=∇×v But I am finding it difficult to interpret the above expression geometrically. In scalar fields, the...
  49. PhysicsKid0123

    What Is Conformal Mapping in Complex Analysis?

    "Definition: A map ƒ: A ⊂ ℂ→ ℂ is called conformal at z0, if there exists an angle θ ∈[0,2Pi) and an r > 0 such that for any curve γ(t) that is differentiable at t=0, for which γ(t)∈ A and γ(0)= z0, and that satisfies γ ' ≠0, the curve σ(t) = ƒ(γ(t)) is differentiable at t=0 and, setting u =...
  50. Dimitri655

    What is the Definition of Dimension in Mathematics?

    Hey guys! After watching another awesome video of minutephysics: I couldn't help but wonder, what is a dimension? Thanks for your replies in advance,
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