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. S

    Precise definition of the limit of a sequence

    In the definition, 1) why must you find a n_0 \in N such that \forall N \geq n_0? You might as well say find a n_0 \in R such that \forall N > n_0. Just a matter of simplicity? 2) Why must |x_n - a| < \epsilon hold? I think |x_n - a| \leq \epsilon is fine as well, given that it must hold...
  2. A

    Definition of friction coefficients in a t-pipe

    Hiah, I've got a question concerning a t-pipe configuration and the corresponding friction coefficient values because there are two different friction coefficients stated in literature. Let's assume we have a simple t-pipe where the main passage is larger than the side branch. The friction...
  3. mnb96

    Question on definition of Lie groups

    Hello, I have a doubt on the definition of Lie groups that I would like to clarify. Let's have the set of functions G=\{ f:R^2 \rightarrow R^2 \; | \; < f(x),f(y)>=<x,y> \: \forall x,y \in R^2 \}, that is the set of all linear functions ℝ2→ℝ2 that preserve the inner product. Let's associate the...
  4. G

    The definition of multiplication

    Warning: Semantics battle may ensue, tread lightly So I was wondering the other day, the repeated-addition definition of multiplication only works for integers, for example you cannot use this to calculate the square of e or pi. So is there a rigorous definition for multiplication that is...
  5. M

    Displacement Operator definition

    Hi, Could someone explain how the following two definitions of the displacement operator are equal? The first is the standard one 1) e^{\alpha a^{\dagger}-\alpha*a} But what about this one? This is from a Fock state decomposition of a coherent state. 2) e^{\frac{-|\alpha^{2}|}{2}}e^{\alpha...
  6. srfriggen

    Quick definition question: Dihedral group

    A dihedral group of an n-gon denoted by Dn, whose corresponding group is called the Dihedral group of order 2n? What I gather from that is a square has 8 symmetries, an octagon has 16, a hexagon 12, etc?
  7. L

    Dirac's Quantum Mechanics - the definition of the time evolution operator

    Dirac's "Quantum Mechanics" - the definition of the time evolution operator I'm reading Dirac's "Principles Of Quantum Mechanics" to learn more about the formal side of the subject. I have a question about the way he defines the time evolution operator in the book. Either there's a mistake or...
  8. W

    Having trouble understanding/accepting the definition of work. Thought exercise

    One concept in physics that has never set well with me is the way work and energy are defined. According to all the physics sources I've looked at, work is defined as: W = \vec{F} \cdot \vec{d} (for a constant force over a distance) However, intuitively the notion of taking the dot...
  9. V

    Gauge Theory: Simplified Explanation & Examples for Physics

    Could someone explain what a gauge theory is, both in general and how it applies to physics? Please try to keep definitions relatively simple, even though the topic is exceedingly complicated. Examples are also greatly appreciated. Thanks!
  10. V

    Definition of a Tensor: Learn Physics Basics

    I am still a physics novice and am learning new things everyday. I've been looking at tensors recently and I'm finding that I can't really understand what they are. Could someone explain in relatively simple words what the definition of a tensor is and why they are so significant? Also, what is...
  11. G

    Confused with directional derivative definition of vectors

    Hi, been reading about GR and am quite confused about the new definition of vectors. My main problem with this is that the text uses partial derivatives as the vector basis, I understand this is related to directional derivatives but cannot see the direct mathematical relation. Secondly, how...
  12. Chestermiller

    Definition of Frame of Reference

    In studying SR, I've been subscribing to a particular definition of a Frame of Reference that makes sense to me. Recently, I've been made aware by another PF member that there may be other, broader, definitions that are valid and that people use. I would like to know more about these broader...
  13. QuestForInsight

    MHB What is an Abelian group and why is it useful in mathematics?

    Let $\mathbb{G}$ be a set with a map $(\xi, ~ \eta) \mapsto f(\xi, ~\eta)$ from $\mathbb{G}\times\mathbb{G}$ into $\mathbb{G}$. For every pair $(\xi, ~ \eta)$ in $\mathbb{G}$ let $f(\xi, ~\eta) = f(\eta, ~ \xi)$. Suppose there are elements $\omega$ and $\xi'$ in $\mathbb{G}$ such that for every...
  14. 1

    Physics textbook, replace sine with its definition (?)

    physics textbook, replace sine with its definition (?) What on Earth do they mean? "That will introduce either sin(theta) or cos(theta). Reduce the resulting two variables, x and theta, to one, x, by replacing the trigonometric function with an expression (its definition) involving x and y."...
  15. V

    What is the Definition of an Ordered Pair in Set Theory?

    I just started studying set theory, and I've seen this definition for an ordered pair (a,b) = {{a}, {a,b}} However, I don't understand how this definition makes sense. Could someone explain this definition to me? Maybe use a concrete example too?
  16. S

    Intuitive Definition of Electric Potential

    Hello all, For a few months, I've been (off and on) trying to come up with a more intuitive definition for Electric Potential (or Voltage, if you prefer), as all I can seem to find are mathematical equations. I believe I have finally come up with a satisfactory result, and I merely wanted to...
  17. D

    Understanding the Lorentz Group: What does O(1,3) mean?

    I am totally confused about the Lorentz Group at the moment. According to wikipedia, the Lorentz group can be defined as the General Orthogonal Lie Group##O(1,3)##. However, the definition of the GO Lie Group that I know only works when there is a single number inside the bracket, not 2, e.g...
  18. G

    Hii have a few questions about entropy:why does the definition

    hi i have a few questions about entropy: why does the definition of entropy stress the fact that the heat exchange by the system is reversible(dS=dQ_rev/T)? am i right, that processes are only reversible iff ΔS=0 and therefore e.g. isothermic, isobaric and isochoric processes even of...
  19. G

    Convolution of a Gaussian with itself from the definition

    Homework Statement Find the convolution of g(x) = e^{-πx^{2}} with itself from -∞ to ∞ using the definition of convolution, not the Fourier Transform. The Attempt at a Solution See my attachment. My professor said that you have to use integration by parts, but I keep getting stuck...
  20. B

    Definition Of Natural Logarithm

    Hello, The definition is ln(x) = \int_1^x\frac{1}{t}dt I have read several sources regarding this, but what I can't seem to find is why it was defined this way. What is the justification for defining it this way, and how was ln (x) found to be the same as the that particular integral?
  21. K

    Definition of voltage (potential difference)?

    I understand the definition to be the amount of work done in moving a charge from one point to another divided by the charge. If you have a standard 1.5 volt battery, the charge should move easily from one end of the circuit to the other because the positive terminal attracts the electrons...
  22. L

    Formal definition of derivative: trig vs non trig functions

    for derivative sinx = cosx, by setting up into formal definition formula limΔx->0 \frac{f(x+Δx)-f(x)}{Δx} this formal definition of derivative is formulated from the cartesian coordinate system where the horizontal is x and verticle is y. But sinx is a trig function and trig functions...
  23. C

    Calculus, Definition of limit, Concept

    Hi comrades. According to spivak, the defition of limit goes as follows: " For every ε > 0, there is some δ > 0, such that, for every x, if 0 < |x-a| < δ, then |f(x) - l |< ε. " After some exercices, I came across with a doubt. Say that I could prove that | f(x) - l |< 5ε, for some...
  24. M

    Tangent Space Definition (Munkres Analysis on Manifolds)

    Hi all, I'm quite confused concerning the definition of tangent vectors and tangent spaces as presented in Munkres's Analysis on Manifolds. Here is the book's definition: Given ##\textbf{x} \in \mathbb{R}^n##, we define a tangent vector to ##\mathbb{R}^n## at ##\textbf{x}## to be a pair...
  25. Q

    What is the Definition of a Relation in Mathematics?

    It is kinda strange. There is no agreement on the definition of a relation. Some books says it is a set of ordered pairs. Other books says it is a subset of a cartesian product. How nice if everything can be agreed down to a few axioms like Euclid's elements. What is your favourite...
  26. C

    Using the Definition of the Derivative, Find the Derivative

    Homework Statement Using the definition of the derivative, find the derivative of g(t) = 1 / sqrt(t).Homework Equations I was told I could solve it by rationalizing it. I asked a question on Yahoo! Answers and saw someone work it out step by step, but I don't understand any of why they did what...
  27. V

    What is the definition of a coefficient

    What is the definition of a "coefficient" How would you define what a coefficient is in the context of differential equations? How do they influence the graph of a DE (variable and constant)? Thank you in advance
  28. V

    Definition of vector addition, Cartesian product?

    I'm reading through a multivariable calculus book and it starts off with some linear algebra. It defines vector addition as V \times V \rightarrow V. My text describes V as a set and describes the above process as a mapping. I believe the \times may represent a Cartesian product. Could someone...
  29. P

    Concept Of Linear Momentum and definition of Force

    I understood that Newton has introduced a concept called "Force" which is basically a cause for an effect i.e. if an object is in a state of rest and if applied a "force" then the object moves (change in velocity, ∴ accelerates) also if an object moves with a constant velocity and is disturbed...
  30. A

    1 is by definition 0.999999999 9?

    Hi there, I have a question regarding this statement: My question is whether we can say so... Thank you very much!
  31. A

    What is the definition of the spin of a particle?

    Hi, I am scratching the surface of information regarding particle physics. I have a basic understanding of standard model. What I am not quite understanding is what 'spin' is. I know that all fermions have a spin of 1/2, but what exactly is spin? Thanks
  32. A

    What is the Definition of C1-Close Curve in Whitney Topology?

    I'm reading a paper and have came across the term 'Cn-close' in the sense of a curve being C1-close to a circle for example, but can't find a definition of this term anywhere, and would be grateful if anyone could help.
  33. J

    A definition of a Potential Well

    Hello I read the following sentence when reading about ion traps: "By changing the trapping voltage we are changing the depth of the potential trapping well, therefore at the same axial position there is a corresponding increase in the potential well, which means that the ion will have to...
  34. W

    Heat Capacity and Entropy Textbook Definition - Quick Question

    Just a quick question of something I found in my textbook but can't get how they produced it. C_p =(∂Q/∂T)_p that is the definition of heat capacity at a constant pressure p. Q is heat and T is temperature. This equation is fine and I know how to derive it. Now it is the next line which...
  35. P

    Definition of a Planet (and other stuff)

    I searched for "definition" and "planet" but found no thread which matched this purpose; if one already exists then it is significantly old, but I will apologize anyways. It used to be that our solar system had Nine planets. Then some trans-Neptunian object (Eris) forced some astrological...
  36. V

    What Are Born Final State Particles in High Energy Physics?

    I've been doing a lot of readings on NLO calculations for high energy physics, and several papers I have read mention "Born final state" particles, "Born level" processes/trees/diagrams etc. None of them seem to define them, however, and my searches on the Google and in textbooks have been...
  37. H

    Epsilon Delta Definition of Limit

    So far, all I understand is that the definition proves that there's a value of f(x,y) as f(x,y) approaches (x0,y0) which is sufficiently close to but not exactly the value at f(x0,y0). I am probably completely off... but I just don't understand the purpose of proving this. I also don't...
  38. V

    Definition of Order of Operations

    I realize now that I took something for granted when I first learned it god knows when.. So I though of starting a discussion as to why were the order of operations defined the way they were? I mean, is there some kind of natural explanation as to why we should compute exponents first and...
  39. E

    Finite Definition of Languages problem

    Homework Statement A palindrome over an alphabet Ʃ is a string in Ʃ* that is spelled the same forward and backward. The set of palindromes over Ʃ can be defined recursively as follows: i) Basis: λ and a, for all a that are elements of Ʃ, are palindromes. ii) Recursive step: If w is a...
  40. P

    About the definition of discrete random variable

    About the definition of "discrete random variable" Hogg and Craig stated that a discrete random variable takes on at most a finite number of values in every finite interval (“Introduction to Mathematical Statistics”, McMillan 3rd Ed, 1970, page 22). This is in contrast with the assumption that...
  41. B

    Understanding Infinite Limits: Clarifying the Definition and Misconceptions

    I have attached this definition that my book provides. My question is does that part "for each M > 0 there exists δ > 0 such that f(x) > M, mean that whenever you M close to the limit, you can find a δ that will give M1 that is closer to the limit?
  42. B

    Definition of Integration (Reimann sums, etc.)

    So I was just working through Courant's calculus and am a bit confused as to where a few variables are pulled out of. Homework Statement Integration of f(x) = x We can see that a trapezoid is formed, so the relevant equation: 1/2(b-a)(b+a) is the value of this integral. To confirm that our...
  43. S

    Reimann Integral definition confusion.

    My lecture notes say: Let f:[a,b]->R be bounded. F is said to rienmann integrable if: L=\int_{a}^{b} f(x)=U where : L=Sup(L(f,P)) and U=Inf(U,(f,P)) but everywhere else(internet) there's a definition with epsilon. I have the epsilon stuff later under "riemann...
  44. P

    Can there be more than one definition of a GROUP?

    I'm reading a book about Group Theory (by Mario Livio: The Equation that Couldn't be Solved ). On page 46 he explains that four rules and one operation define a group: The rules are Closure, Associativity, the existence of an Identity Element and finally the existence of an Inverse. He cites...
  45. B

    Why is Delta Dependent on Epsilon in the Formal Definition of a Limit?

    I am reading about the formal definition of a limit, and its corresponding proof, and there is one thing that I don't quite understand, yet. It says that delta depends on epsilon, but what I wonder is why is it not the other way around. Indeed, why does delta have dependency on epsilon?
  46. S

    Linear independance and span (Definition)

    Would I be correct in saying that: If Span(S)≠0 then S is linearly independant. If Span(S)=0 then S is linearly dependant. With S being a subset of V.
  47. A

    A question about definition of a category

    I know it may sound idiotic to ask questions about definition of something, but I'm going to do that now. I've seen the definition of categories in several different contexts, in all of them categories consisted of objects like groups, rings, R-modules (in particular, vector spaces), topological...
  48. J

    Constants definition - turbulent vel. profile

    Homework Statement Hello! I need some help with a problem: Problem: Turbulent flow beteween parallel flat plates. It is defined: [ tex ] \tau = \mu \frac{d\bar{u}}{dy}-\rho\bar{u'v'} [ \tex ] The exercise gives that [ tex ] \tau = a y [ \tex ] and [ tex ] \rho\bar{u'v'} =...
  49. A

    Accumulation point definition.

    We say a point x in X (which is a topological space) is an accumulation point of A if and only if any open set containing x has a non-empty intersection with A-{x}. Well, I'm creating examples for myself to understand the definition. Suppose X={a,b,c,d,e} and define...
  50. I

    Epsilon-delta definition of limit

    While solving problems regarding Epsilon-delta definition of limit from my textbook i found that every answer was like ε= a×∂,where a was any constant.Is it necessary that ε should always be directly proportional to ∂ for limit to exist?? Cant they be inversely proportional? If they can please...
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