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. M

    Jackson's Electrodynamics. Question on Capacitance definition

    In Jackson's book he defines the capacitance of a conductor, "...the total charge on the conductor when it is maintained at unit potential, all other conductors being held at zero potential." I'm trying to get a more concrete definition in my head rather than the standard definition of...
  2. P

    Complex Analysis - Branch Definition

    Homework Statement Hi everyone, This is more of a definition clarification than a question. I'm just wondering if a branch is the same thing as a branch line/branch cut? I've come across a question set that is asking me to find branches, but I can only find stuff on branch lines/cuts and...
  3. K

    What Are the Requirements for a Subset to Be Considered a Subspace?

    I'm having trouble conceptualizing exactly what a subspace is and how to identify subspaces from vector spaces. I know that the definition of a subspace is: A subset W of a vector space V over a field \textbf{F} is a subspace if W is also a vector space over \textbf{F} w/ the operations of...
  4. S

    MHB Using definition of Laplace transform in determining Laplace of a step function

    I have a question that has stumped me a bit, i am not sure how to use the definition to calculate it, i can use the tables, but i don't think that's what is needed. Using the definition of the Laplace transform, determine the Laplace transform of I can do it with the table but i am not sure...
  5. 1

    Can someone help me better understand the formal definition for ordered pairs?

    I mean the one saying that: (a,b) is defined to be the set: {{a},{a,b}} What exactly does this set definition of an ordered pair mean? Namely, how does it attribute the relevant "order" of terms to the concept of an ordered pair? Thanks!
  6. 6

    Definition on an n-torus, two approaches.

    This wasn't originally a homework problem as such, so sorry if its confusing, but I thought I would ask it here; Homework Statement Show that the the two methods of creating the n-torus are equivalent. 1) The n-torus as the quotient space obtained from ℝn by the relation x~y iff x-y \in...
  7. X

    Starnge Recursion algorithm definition

    I'm reading the book "Algorithms Design" and a recursion algorithm is defined as: T(n)\leqqT(n/q)+cn But in the Karatsuba’s Algorithm, the recurrence for this algorithm is T (n) = 3T (n/2) + O(n) = O(nlog2 3). The last equation is strange, since 3T(n/2) is bigger than the set. Why they define...
  8. W

    What is the exact definition of a function?

    In text (Spivak) it says that a function is a collection of pairs of numbers with the following property: if (a,b) & (a,c) are both in the collection, then b=c; in other words, the collection must not contain two different pairs with the same first element. Now in an other text (Kolmogorov) I...
  9. B

    Definition wind tunnel blockage ratio

    Hello, I’m designing a 2D wind tunnel model for my master thesis. It will be a profile equipped with a fixed hinged trailing edge flap. I’m going to measure at different angle of attacks and different flap settings at low speeds (about 70 to 100 m/s). The aim is to measure steady and unsteady...
  10. Somefantastik

    Separable space definition and applications

    I read in my metric spaces book that a separable space is that which has a countable, dense subset. This definition has no intuitive meaning to me. I'm able to show if a space is dense or not, and I think I can show a space is countable. But, I'm missing the "so what?!" I would like to...
  11. FOIWATER

    Issue w/ electric field definition

    "an electric field is the region of space surrounding electrically charged particles and time-varying magnetic fields" Why do the fields need to be time varying? Additionally, if light is a frequency of electromagnetic radiation, and EM is made of magnetic and electric fields occurring at...
  12. H

    Derivative of function only using definition?

    Homework Statement f(x) = \left\{ {\begin{array}{*{20}{c}} {{x^2}\sin \frac{1}{x}}&{x \ne 0}\\ 0&{x = 0} \end{array}} \right. Is it differentiable at x=0? If it is, what's its value?Homework Equations The Attempt at a Solution I've calculated the derivative function for x not equal zero: f'(x)...
  13. H

    Generic definition of derivative?

    Generic "definition" of derivative? Hi. This is a theoric doubt I have since I went to class today. The professor "redifined" the derivative at point a. He draw a curve (the function) and the tangent at point a. Then he draw another two lines in the same point. Well, then he said that the error...
  14. D

    Quaternion Kaehlerian manifold, definition

    Hello, I am reading the paper of S. Ishihara, Quaternion Kaehlerian manifolds, I need it for understanding of totally complex submanifolds in quaternion Kaehlerian manifolds. I am afraid that I don't understand well the definition of quaternion Kaehler manifold, that is my question is the...
  15. F

    Definition questions for linear algebra

    i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W what i null A...
  16. M

    What is the Definition of a Limit in Mathematics?

    Hello all, This is very simple however I would like to understand why this is true. According to the definition of a limit, if we have limit of f(x) as x approaches infinity = a then for every ε>0 there exists a real number M such that if x>M then the absolute value of f(x)-a < ε. This...
  17. L

    Measure theoretic definition of conditional expecation

    I've been looking at the measure theoretic definition of a conditional expectation and it doesn't make too much sense to me. Consider the definition given here: https://en.wikipedia.org/wiki/Conditional_expectation#Formal_definition It says for a probability space (\Omega,\mathcal{A},P), and...
  18. S

    What is the Definition of a Manifold and How Does it Relate to Topology?

    i see the definition of differential manifolds in some book for example, NAKAHARA but what is the definition of manifold in general! and what the definition of topological manifold.
  19. M

    Gradient Flow - Definition and Sources

    Hi all, I am struggling to find any elementary material on the "gradient flow of a functional" concept. From introductions in advanced papers I seem to have understood that, assigned a functional F (u), the gradient flow is charactwerized by an equation of the type Du / Dt = P u , where P...
  20. M

    Proof of a limit involving definition of differentiability

    Homework Statement let the function f:ℝ→ℝ be differentiable at x=0. Prove that lim x→0 [f(x2)-f(0)] ______________ =0 x Homework Equations The Attempt at a Solution I am kind of lost on this one, I have tried manipulating the definition of a...
  21. S

    Napier's Constant Limit Definition

    Hi all ! I am terribly sorry if this was answered before but i couldn't find the post. So that's the deal. We all know that while x→∞ (1+1/x)^x → e But I am deeply telling myself that 1/x goes to 0 while x goes to infinity. 1+0 = 1 and we have 1^∞ which is undefined. But...
  22. S

    Is the Wigner D matrix definition applicable to spherical harmonic rotations?

    Homework Statement I'm not sure if this is the appropriate board, but quantum mechanics people surely know about spherical harmonics. I need to implement the Wigner D matrix to do spherical harmonic rotations. I am looking at...
  23. A

    Mathematical Definition of Energy

    What is the general definition of energy? I already know that it means ability to perform work and that Work = ∫Force d(displacement) = Δ Kinetic Energy = -Δ Potential Energy ( in a conservative field "a closed path integral of the force = 0"), Σ Kinetic-Potential = constant, ∫Kinetic-Potential...
  24. J

    The definition of 'reducible' in Hungerford's Algebra text

    He starts using the term 'reducible', as it came out of nowhere, from the page 162 of the text. I know, roughly, what kind of thing he mean by this 'reducible' obejct. (That is that an element is factored into two elements that are not units.) And this should not be a problem if this term is...
  25. M

    Definition of specific heat by via entropy

    In his Statistical Physics book, Landau introduces the specific heat as the quantity of heat which must be gained in order to raise the temperature of a body one by unit. I don't understand, how he directly jumps to the conclusion that that has to be (let's just say, for constant volume): C_V...
  26. S

    Landau: Explaining the Definition of "Number of States with Energy

    Question about Landau: Definition of "Number of states with energy" in an interval Hey! I am currently reading Landau's Statistical Physics Part 1, and in Paragraph 7 ("Entropy") I am struggling with a definition. Right before Equation (7.1) he gives the "required number of states with...
  27. A

    Definition of the residual spectrum

    If A is a bounded operator on a Hilbert space H, isn't the following true of the residual spectrum \sigma_r(A): \lambda \in \sigma_r(A) iff (\forall \psi \in H, \psi \neq 0)((\lambda - A) \psi \neq 0) iff \ker (\lambda - A) = \{0\} iff \lambda - A is injective? So isn't the condition that...
  28. B

    Ohda: Definition of Order in Baby Rudin

    Hi, All: Just curious: Rudin defines order in his "Baby Rudin" book ; an order relation < in a set S, as a relation* satisfying, for any x,y,z on S: 1) Either x<y , y<x , or y=x 2)If x<y and y<z , then x<z , i.e., transitivity. Just curious: why is Rudin only considering only...
  29. J

    Definition of The first law of motion

    Text book definition is "In the absence of forces, ("body") at rest will stay at rest, and a body moving at a constant velocity in a straight line continues doing so indefinitely". My thinking. Moon is orbiting Earth in a circular path, and not in a straight line. Still that motion follows...
  30. D

    Function Definition Without a Single Word

    Is this correct? A function ψ:A --> B is the set: ψ = { (x, y) | \forallx\inA\existsy\inB\ni(((x, y) \in ψ) \wedge ((x, z)\in ψ) \Rightarrow y = z)} Thanks.
  31. S

    What is the definition of EMF?

    Actually my sir asked me the definition of EMF so I just tell him that "Suppose a resistance(R) is connected across the terminals of a battery.A potential difference is developed across its ends.Current(or positive charge) flows from higher potential to lower potential across the resistance by...
  32. J

    About definition of 'Bounded above' and 'Least Upper Bound Property'

    The definition of 'Bounded above' states that: If E⊂S and S is an ordered set, there exists a β∈S such that x≤β for all x∈E. Then E is bounded above. The 'Least Upper Bound Property' states that: If E⊂S, S be an ordered set, E≠Φ (empty set) and E is bounded above, then supE (Least Upper...
  33. W

    Engineering Is My Circuit Linear? A Definition and Guidelines for Identifying Linearity

    Homework Statement In the Lectures, we are told that techniques like homogeneity and superposition work only for linear circuits, but in Chapter 3 of the Textbook (which is the only place I can find one) I see a definition of linearity as "A circuit is linear if and only if Homework...
  34. T

    Deriving bar instability mode- formal definition of instability?

    I am doing an undergraduate project on bars and I am trying to derive the bar instability mode given by Mo et al. It says "whether or not a disk is globally stable depends on the global properties of the disk... it is not possible to write down a universal dispersion relation or stability...
  35. T

    Parametric definition for a complex integral

    I have been working in complex functions and this is a new animal I came across. Let γ be a piecewise smooth curve from -1 to 1, and let A=∫γa(x2-y2) + 2bxy dz B=∫γ2axy - b(x2-y2) dz Prove A + Bi = (2/3)(a-bi) In the past anything like this defined γ and I would find a parametric...
  36. J

    The major problem of 5.1 Definition in Baby Rudin

    I will prove the following statement is true to show the flaw of 5.1 Definition in Baby Rudin. If in any case I'm wrong, please correct me. Thanks. Statment: Suppose f is a function defined on [a,a] with a \in \mathbb{R}. Then it is impossible to apply 5.1 Definition in Baby Rudin for this...
  37. ShayanJ

    An ambiguity in the definition of tensors

    One of the definitions of the tensors says that they are multidimensional arrays of numbers which transform in a certain form under coordinate transformations.No restriction is considered on the coordinate systems involved.So I thought they should transform as such not only under rotations but...
  38. M

    Definition of the derivative to find the derivative of x^(1/3)

    Homework Statement Use the definition of the derivative to find the derivative of x^(1/3) Homework Equations The Attempt at a Solution [(x+h)^(1/3) - x^(1/3)]/h I do not know where to go from here. If it were a square root I could conjugate.
  39. N

    Question on Definition of Fourier Transform

    I have a question, specifically to physics people, on their definition of the Fourier Transform (and its inverse by proxy). I'm an EE and math person, so I've done a lot of analysis of (real/complex) and work with (signal processing) the transform. In a physics class I'm taking, the professor...
  40. Roodles01

    Solving Sec Homework: Differentiate Ln(cos(5x)) wrt x

    Homework Statement Having started to differentiate Ln(cos(5x)) wrt x I checked the answer with WolframAlpha & got a different method & answer too. Homework Equations differentiate ln(cos(5x)) The Attempt at a Solution I used the Chain rule d(ln(cos(5x)))/dx = d(ln(u))/du *...
  41. M

    Help with Spivak's treatment of epsilon-N sequence definition

    I have just started my first real analysis course and we are using Spivak's Calculus. We have just started rigorous epsilon-N proofs of sequence convergence. I was trying to do some exercises from the textbook (chapter 22) but there doesn't seem to be any mention of epsilon-N in the solutions...
  42. B

    Definition of the Gibbs free energy

    In the equation \Delta G^{\ominus} = \Delta H^{\ominus} - T\Delta S^{\ominus} does the temperature refer to the temperature of the system, or the temperature of the surroundings? BiP
  43. W

    Proper definition of world lines in Galilean and Minkowskian spacetime

    I posted several questions on Galilean and Minkowskian spacetime on this forum lately, but I just don't seem to be able to get a real grip on things. I noticed that the core of my problems mostly arise from the definition of world lines. Therefore I tried formulating a definition of them in both...
  44. mnb96

    Question about derivative definition

    Hello, considering the definition of derivative, what would the following quantity be equal to? \lim_{\delta \to 0} \frac{f(x+g(\delta))-f(x)}{g(\delta)} In this case g(\delta) is a monotonic increasing function such that g(0)=0. For example we might have g(\delta)=\delta^3
  45. J

    Definition of a Topological Space

    Just a small (and, really, quite useless) little nugget here: In the definition of a topological space, we require that arbitrary unions and finite intersections of open sets are open. We also need that the whole space and the empty set are also open sets. However, this last condition is...
  46. Useful nucleus

    Are all open sets compact in the discrete topology?

    A subset K of a metric space X is said to be compact if every open cover of K contains a finite subcover. Does not this imply that every open set is compact. Because let F is open, then F= F \bigcup ∅. Since F and ∅ are open , we obtained a finite subcover of F. Am I missing something here?
  47. J

    Why/How does the definition of implication in mathematics work?

    I understand that we just have to fill the last two raws in the truth table with any value, and that we randomly chose True, and that the value True makes matters easier sometimes (I don't know an example of that, but I read that somewhere). But the question is, since mathematics is tied to...
  48. M

    Use the definition of E[g(Y)] to derive E[Y^2]

    Let Y ~ Bi(m,q). Use the definition of E[g(Y)] to derive E[Y^2]. Hint: Write Y^2 as Y(Y-1) + Y. You do not have to re-derive E[Y]. Not sure where to start with this; my initial reaction was to use the moment generating function?
  49. M

    Mathematically Precise Definition of Unit

    Does anyone know exactly what kind of mathematical object a unit (like meters, coulombs, etc.) is? Or what kind of algebraic structure units are elements of?
  50. S

    Natural definition for variance?

    Homework Statement I'm trying to find a natural definition for V[X|Y], but I'm not sure what a natural definition is. Any help?
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