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. gracy

    Definition of potential energy

    In my book definition of potential energy difference between two points as the work required to be done by an external force in moving without accelerating charge q from one point to another in electric field of any arbitrary charge configuration.I want to know why without accelerating ?
  2. S

    Definition of Divisibility

    Question: If x | y, (is true), then x ≤ y and x ≠ 0. For instance, if x > y, then there are no integer solutions to equation kx = y and thus, x does not divide y. Is this a correct proposition?
  3. nomadreid

    The definition of a field supposes simultaneity?

    A field is defined as a distribution throughout all space (or at least a portion of it). But all of (a portion of) space means all of space at a particular moment, no? But that sounds as if it assumes simultaneity. But I thought simultaneity was a meaningless concept, according to relativity...
  4. B

    Definition of dx: What is its Domain & Formalization?

    Homework Statement http://imgur.com/goozE9f Homework Equations ##(dx_i)_p i= 1,2,3## 3. The Attempt at a Solution [/B] I'm reading Manfredo and Do Carmo's Differential Forms and Applications. This is the very first page Would you mind explaining me what is meant by dx, as highlighted in the...
  5. H

    G-Force Definition: Acc & Weightlessness

    I am having considerable difficulty in finding a precise definition of g-force. My understanding is that it is the acceleration an object would undergo due to just the non-gravitational and non-electromagnetic forces acting on the object. A corollary of this is that an object experiences zero...
  6. S

    Definition clarification for Fourier transform

    I have been studying Fourier transforms lately. Specifically, I have been studying the form of the formula that uses the square root of 2π in the definition. Now here is the problem: In some sources, I see the forward and inverse transforms defined as such: F(k) = [1/(√2π)] ∫∞-∞ f(x)eikx dx...
  7. T

    How to prove the double integral definition of logarithm?

    Where does this definition come from: $$\ln n = \int_{0}^{\infty} \int_{1}^{n} e^{-xt} dx dt$$ Thank you very much.
  8. N

    What version of the definition of derivative

    If my understanding is correct the definition of a derivative is lim h->0 (f(x+h)-f(x))/h However, I've also seen this used: lim x->c (f(x)-f(x))/(x-c) are these both considered valid definition for the derivative or does the derivative have to tend towards zero? I am a bit confused because I...
  9. Math Amateur

    MHB Function Field over k --> k(x) - Rotman Definition and Proposition 3.29

    I am reading Joseph J. Rotman's book: A First Course in Abstract Algebra with Applications (Third Edition) ... I am currently focused on Section 3.3 Polynomials ... I need help with an aspect of the proof of Proposition 3.29 concerning elements of the function field k(x). Rotman's definition...
  10. J

    Definition for curvature of worldline in Minkowski space

    Hello, in this section of the wiki article on Rindler coordinates it is stated that the proper acceleration for an observer undergoing hyperbolic motion is just "the path curvature of the corresponding world line" and thus a nice analogy between the radii of a family of concentric circles and...
  11. Palak Verma

    What is the definition of net upward force?

    To keep the can immersed, an external downward force is needed to balance the net upward force(=F'b-W).
  12. B

    Better definition for complex number

    I was me asking why the complex numbers are defined how z = x + i y !? Is this definition the better definition or was chosen by chance? In mathematics, some things are defined by chance, for example: 0 is the multiplicative neutral element and your multiplicative inverse (0-) is the ∞. But, 1...
  13. Raman Choudhary

    Logic behind definition of Reparametrization

    What is the intuitive logic behind setting up the definition of reparametrization as being a bijective map and all that(the inverse map being smooth) and not alone that the reparametrisation must give us the same image curve.e.g if we see (t,t^2) as being describing the same curve as (t^3,t^6)...
  14. C

    Confusion about definition of compactness

    So the definition I have seen is: Given a topological space <S,F> it is compact if for any cover (union of open sets which is equal to S) there exists a finite subcover. By the definition of a topological space both S and the empty set must belong to the family of subsets F. Wouldn't <S, empty...
  15. G

    Canonical definition of Angular Momentum,

    Let's start with an arbitrary solid body rotating around a fixed axis of rotation with angular velocity ##\vec \omega## in the ## \hat z## direction. For simplicity, let's say the origin O is on the axis of rotation. Take a look at the picture I sketched in the next post. Tried my best to be...
  16. Roy Smith

    Is "Dark Matter" Really Matter? What is Matter?

    Hey, my name is Roy and I'm new to Physics Forums. I'm a retired medical and aerospace test engineer, now currently a freelance artist (kind of opposites right?) and I joined Physics Forums to hone my understanding of physics and ask the right questions in an ever expanding field of inquiry...
  17. F

    MHB Prove limit by definition

    What I've tried so far: $$ \begin{align*} \left |\frac {e^{x+y^2}-1-\sin \left ( x + \frac{y^2}{2} \right )}{x^2+y^2} \right | &\leq \frac {\left |e^{x+y^2}-1-\sin \left ( x + \frac{y^2}{2} \right ) \right |}{\left \| (x,y) \right \|^2} \\ &\leq \frac {\left |e^{x+y^2}-1 \right |+ \left...
  18. Math Amateur

    MHB Affine Algebraic Curves - Kunz - Definition 1.1

    I am reading Ernst Kunz book, "Introduction to Plane Algebraic Curves" I need help with some aspects of Kunz' Definition 1.1. The relevant text from Kunz' book is as follows:In the above text, Kunz writes the following: " ... ... If K_0 \subset K is a subring and \Gamma = \mathscr{V} (f) for...
  19. D

    Is there a definition of "hyperdimensional resonance"?

    I was talking to a friend recently and he said something to me that i just didn't understand. He is a physicist and i most certainly am not. I was telling him about similarities that i had observed between two things and was not sure if it was a coincidence or if there was a tangible link...
  20. Dong Aleta

    A question regarding the definition of acids and bases

    Hi! I just read in an inorganic chemistry book (by Whitten, et al.) that acids are defined as substances that produce H+ ions in dilute aqueous solutions, and bases are those that produce OH-. To me, this definition implies that a substance that has yet to produce an H+ or an OH- can already be...
  21. P

    Deriving the Definition of the Christoffel Symbols

    In Sean Carroll's Lecture Notes on General Relativity (Chapter 3, Page 60), in the chapter on Curvature, he derives the definition of the Christoffels Symbols by assuming the connection is metric compatible and torsion free. He then takes the covariant derivative of the metric and cycles through...
  22. S

    Finding derivative of x^x using limit definition

    I am trying to find the derivative of x^x using the limit definition and am unable to follow what I have read. Can someone help me understand why lim [(x+h)^h -1]/h as h ---> 0 = ln(x). This part of the derivatio
  23. L

    What is the physical definition of a state in quantum mechanics?

    Mathematically, a state in QM is a ray on the Hilbert space. But: 1) How would you define "state" from a physical point of view? I know a lot of examples but not a general definition. 2) Given a specific quantum system, to find all the states and so the Hilbert space, all I have to do is to...
  24. S

    Can someone verify this definition for Weyl tensor?

    I just want to make sure I have this right because when I go to different sites, it seems to look different every time. This is the Weyl tensor: Cabcd = Rabcd + (1/2) [- Racgbd + Radgbc + Rbcgad - Rbdgac + (1/3) (gacgbd - gadgbc)R] Is this correct?
  25. D

    What is the definition of Locality (in QFT)

    I am slightly unsure as to whether I have understood the notion of locality correctly. As far as I understand it locality is the statement that if two events occur simultaneously (i.e. at the same time) then no information can be shared between them (they are causally disconnected). Thus a...
  26. C

    Definition of the Path Integral

    Why is it that, in the definition of the path integral, we have the product of neighboring integrals of the form : ∫Φdx1...dxn when the whole idea is based on adding the contribution of neighboring paths. I need some help understanding why it is of the form ∫Φdx1...dxn and not the form...
  27. D

    What is the exact definition of equality?

    I've been grappling with the idea in my head as to how I would explain to someone exactly what equality between two mathematical objects actually means. This maybe a very stupid question, so apologies in advance, but if I'm honest I struggle to come up with an answer that doesn't involve using...
  28. person_random_normal

    Current Density: Definition & Measurement

    What is current density
  29. orion

    Learn Submanifold Definition in Differential Geometry

    I am trying to learn differential geometry on my own, and I'm finally getting serious about learning the subject. I am using several books which I supplement with material I find on the internet. I have found some excellent lecture notes on differential geometry written by Dmitri Zaisev. I like...
  30. M

    Basic questions on Nakahra's definition of topological space

    In chapter 2.3 in Nakahara's book, Geometry, Topology and Physics, the following definition of a topological space is given. Let X be any set and T=\{U_i | i \in I\} denote a certain collection of subsets of X. The pair (X,T) is a topological space if T satisfies the following requirements 1.)...
  31. C

    Precise definition of the limit (epsilon-delta)

    Sorry, I am really struggling with the precise definition of the limit. I have a specific question I'm trying to work out lim(x->2) (4x2+2)=18 skipping the introduction part any advice? I am just not sure how to get rid of the 2 value to re-arrange |(4x2+2)-18| to look like |x-2| |x-2|<delta...
  32. C

    Definition of derivative - infinitesimal approach, help :)

    Hi I'm reading Elementary calculus - an infinitesimal approach and just wan't to make sure my understanding of what dy, f'(x) and dx means is correct. I do understand the basic idea: You make the secant between 2 points on a graph approach one of the points and at this point you get the...
  33. C

    Using the precise definition of a limit (epsilon & delta)

    I'm trying to practise, precise definition of a limit (epsilon & delta) Just to check I'm along the right lines here's a previous question to the one I'm stuck on If epsilon > 0 then there is delta >0 ... All that introduction stuff, then Lim x-> 2 (3x-1) =5 Hence |x-2| < delta then |3x - 6|...
  34. H

    Definition of tensors - abstract and concrete

    I am well aware of an abstract definition of a general tensor as a map: \mathbf{T}:\overbrace{V\times\cdots\times V}^{n}\times\underbrace{V^{\star}\times \cdots\times V^{\star}}_{m}\longrightarrow\mathbb{R} I am happy with this definition, it makes a lot of sense to me. However, the physics...
  35. 24forChromium

    How Is Entropy Measured in Thermodynamics?

    When I learned the concept of specific heat capacity, I knew that 1J/(K*kg) means that it takes 1 Joule of energy to increase the temperature of a kilogram of matter by one Kelvin, but what does J/K, the unit of entropy, mean?
  36. T

    Definition question about atomic fluorescence

    Does atomic fluorescence involve: 1. spontaneous emission (or only), 2. stimulated emission (or only) 3. change in magnetic quantum number $$\Delta m \neq 0$$ ? Thank you. Rarely can I find a definition on the internet. My guess is that atomic fluorescence involves spontaneous emission only...
  37. AnkurGarg

    Vestigal organs definition clarification please

    Are temporary embryonic structures like gill clefts included under "Vestigeal organs"??
  38. M

    Epsilon-delta definition of limits

    Homework Statement Use the ε-δ definition of limits to prove that limx→2 x2 = 4. The Attempt at a Solution |x2 - 4| < ε 0 < |x - 2| < δ |x - 2| |x + 2| < ε And that's where I get stuck, can I divide both sides by |x + 2| to yield |x - 2| < ε/|x + 2| = δ In which case, where do I go from...
  39. Y

    MHB Plotting the definition region of a function

    Hello all I have this function with 2 variables: \[f(x,y)=\frac{\sqrt{x}+\sqrt{9-x^{2}+y^{2}}}{ln(\frac{1}{4}x^{2}+\frac{1}{16}y^{2}-1)}\] and I wish to plot the definition region (where the function is defined). I did the calculation manually, and asked MAPLE to plot my inequalities, and I...
  40. M

    Definition of Elementary Particle

    Just wondering if there's a precise definition of what it means to be an elementary particle. I had assumed it was related to not being able to convert it into multiple "smaller" things, but then a photon is called elementary when it can be converted into smaller energy positrons and electrons.
  41. S

    MHB Negating Definition of Function: A=>B

    We know that we define a function from the set A to the set B ,denoted by f: A=>B iff: 1) f is a subset of AxB 2) For every a belonging to A ,there exists a unique b belonging to B,such that (a,b) belongs to f In trying now to negate the above definition i got stuck ,particularly in negating...
  42. M

    Definition of charge of a free electron

    Hi there, I have a question about the definition of a charge of a free electron. Let's suppose that QED is the true theory of the interactions of charged particles. Presumably the charge on an (effectively) free electron, then, is the charge on an electron in which the electromagnetic...
  43. D

    Understanding the Tensor Product of Two One-Forms in Differential Geometry

    I'm relatively new to differential geometry and would like to check that this is the correct definition for the tensor product of (for simplicity) two one-forms \alpha,\;\beta\;\;\in V^{\ast} : (\alpha\otimes\beta)(\mathbf{v},\mathbf{w})=\alpha (\mathbf{v})\beta (\mathbf{w}) where...
  44. D

    Why does lowering the slew rate lower the electrical noise?

    From my research I am showing that if an input signal becomes too slow (ie: a low slew rate) then the noise can cause multiple state changes. But I am being told that if the slew rate is low then it will get rid of unwanted noise. I read my results from this: (I lost the link but I did copy...
  45. W

    Electrical mobility definition confusion

    Hi! According to this http://en.wikipedia.org/wiki/Electrical_mobility, the definition of electrical mobility ##\mu## is: ##\vec{v} = \mu \vec{E}##. But since electrical mobility is always positive, this means that the velocity is always parallel to the E-field regardless of charge. How can...
  46. W

    Understanding the Epsilon Delta Definition of a Limit

    Hi I'm new to limits and calculus in general. Our professor told us there existed some rigorous proof for a limit, but it was "beyond the scope of the course". All we needed to know about a limit was that (1)$$\lim_{x\to a} f(x)$$ is true iff when x approaches a from both directions p(x)...
  47. S

    Accuracy Defined: What Does 10% Mean?

    I want to ask a strange question. When the gun has 10% accuracy, what does 10% mean? I mean it doesn't really tell me anything. How large is the area the bullets spread if the gun has 10% accuracy? I have already attempted to find articles in google, and I still didn't get the answer about the...
  48. M

    The definition of mass of an electron (after the renorm group)

    Hi there, I have a question about the rest mass of an electron. As we all know, the charge of an electron is a function of the energy at which the system is probed. When defining the charge, we typically use as our reference scale the charge measured in Thompson scattering at the orders of...
  49. P

    The epsilon-delta definition

    I'm trying to wrap my head around the epsilon-delta definition. "Let ##f## be a function defined on an interval that contains ##a##, except possibly at ##a##. We say that: $$\lim_{x →a} f(x) = L$$ If for every number ##\epsilon > 0## there is some number ##\delta > 0## such that: ##|f(x) - L| <...
  50. ShayanJ

    Problem with definition of tensor

    In textbooks, a tensor is usually defined in terms of its transformation properties. But this definition actually seems vague when it comes to checking a set of quantities to see whether they form a tensor or not. Imagine I have four functions and want to see whether they form a 2d 2nd rank...
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