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. T

    Precise Definition of a Limit at Negative Infinity

    I'm working through some problems from Stewart's Calulus, 6ed. and am having some difficulty with certain limit proofs. In particular, there is no definition provided for limits of the form: $$ \lim_{x \to - \infty} f(x) = L $$ One of the exercises is to come up with a formal definition...
  2. T

    What is the definition of p_Θ?

    Homework Statement What is the definition of p_Θ? Homework Equations I search all over the web using the keyword linear "momentum polar coordinates", "lagrangian in polar coordinates", "hamiltonian in polar coordinates", "momentum in 2D polar coordinates" for about half an hour. but i...
  3. N

    Understanding the definition of standing wave

    Homework Statement For standing wave, I have read that there are certain points on the wave that don't move at all, nodes. However, for a standing wave the shouldn't the resultant wave have no displacement because we now have two waves on the same rope with the same amplitude and frequency...
  4. C

    Definition of Convergence: Can n -> -infinity

    Homework Statement I've been given a question that makes use of 5^(n)*sin(pi*n!) The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity? So basically, I...
  5. S

    Confirm limit of this sequence by using definition of limit

    Homework Statement I'm referring to the question and solution for part (b) in the attached TheProblemAndSolution.jpg file. Homework Equations Definition of limit. The Attempt at a Solution Should the equation with the two things in brackets have absolute value bars instead of brackets...
  6. Math Amateur

    MHB Definition of Direct Product Using UMP - Including Infinite Case

    I am reading Chapter 2: Vector Spaces over \mathbb{Q}, \mathbb{R} \text{ and } \mathbb{C} of Anthony W. Knapp's book, Basic Algebra. I need some help with some issues regarding the general UMP-based definition of external and internal direct products ... ... On page 63, Knapp defines...
  7. JonnyMaddox

    What is the generalization of the tensor product definition for a three form?

    Hello, the tensor product definition of a two form is \alpha^{1} \wedge \beta^{1} := \alpha \otimes \beta - \beta \otimes \alpha \alpha \wedge \beta(v,w) = \beta(v)\alpha(w)-\beta(w)\alpha(v) But what is the definition in this sense for a three form?
  8. shanepitts

    A different definition of radiation flux

    Would anyone have a different explanation and maybe an example of radiation flux? Here is the Wikipedia definition "Radiation flux is a measure of the amount of radiation received by an object from a given source", which is quite clearly explained.
  9. Greg Bernhardt

    Exploring Torque: Definition, Equations, and Explanation

    Definition/Summary Torque is the moment of a force about a point. Torque is a vector (strictly, a pseudovector, like angular momentum and any other moment of a vector), and is measured in units of Newton metres (N m or N.m). By comparison, energy is a scalar, and is measured in units of...
  10. Greg Bernhardt

    Stress and Strain Tensor Basics: Definition, Equations & Moduli

    Definition/Summary Stress is force per area, and is a tensor. It is measured in pascals (Pa), with dimensions of mass per length per time squared (ML^{-1}T^{-2}). By comparison, load is force per length, and strain is a dimensionless ratio, stressed length per original length...
  11. Greg Bernhardt

    Rolling: Definition, Constraint, Mass, and More

    Definition/Summary "Rolling" means moving along a surface without sliding. The (instantaneous) point of contact is stationary relative to the surface. In other words: it is the instantaneous centre of rotation (if that surface is stationary). Friction at the point of contact of a...
  12. Greg Bernhardt

    Inertia: Definition, Equations, and Laws of Motion

    Definition/Summary Inertia is the phenomenon that a force is required to cause change of velocity. The amount of inertial mass of an object is measured by measuring how much force it takes to accelerate it. The symbol for inertial mass is m. Equations Extended explanation The...
  13. Greg Bernhardt

    Heat and Work: A Definition and Explanation

    Definition/Summary Heat is the non-mechanical exchange of internal energy, U, between a system and its surroundings as a result of a difference in temperature. By contrast, work, W, is the mechanical exchange of energy as a result of force applied across a moving surface (such as the face...
  14. Greg Bernhardt

    Group Theory: Definition, Equations, and Examples

    Definition/Summary A group is a set S with a binary operation S*S -> S that is associative, that has an identity element, and that has an inverse for every element, thus making it a monoid with inverses, or a semigroup with an identity and inverses. The number of elements of a group is...
  15. T

    Conductor band overlap definition question

    i search all over the web but i cannot find the definition. Although I find all the literature saying that metal have valence band and conduction band overlapped, (and I cannot find a metal example with valence band and conduction band not being overlapped), I wonder if there is a...
  16. C

    Rigorous Definition of Infinitesimal Projection Operator?

    I've been reading Thomas Jordan's Linear Operators for Quantum Mechanics, and I am stalled out at the bottom of page 40. He has just defined the projection operator E(x) by E(x)(f(y)) = {f(y) if y≤x, or 0 if y>x.} Then he defines dE(x) as E(x)-E(x-ε) for ε>0 but smaller than the gap between...
  17. freddie_mclair

    Defining Cooling Power: What Does it Really Mean?

    Hi everybody! Well, these are just two basic questions that are bothering me. 1.When it is said that, for example, a Cryocooler has 1W of cooling power at 4.2K, what does it really mean? To me, the action of "cooling" depends on the material that it's being cooled down. So, for different...
  18. P

    Exploring the Definition of e Through Limits

    I am familiar with the fact that the number e can be defined several ways. One particularly interesting definition is the one based on limits, namely: e = limn \rightarrow ∞ (1 + \frac{1}{n})n My question is: wouldn't it be equally true to express e as the limit of the expression above as n...
  19. E

    A new limit definition of integral?

    I think i discovered a new way to define an integral, i don't know if it helps in any particular case, but its an idea worth posting i think. The idea is to define the height of the rectangles based on one single point of the function and then build up the next heights for the other rectangles...
  20. micromass

    MTW definition of differential froms made rigorous

    Hello guys! I've been trying to get some intuition for differential forms. I know the formal theory and I know how useful they are. But then I came across the following paper: https://dl.dropboxusercontent.com/u/828035/Mathematics/forms.pdf It describes an intuition for forms that is very...
  21. Math Amateur

    MHB Plane algebraic curves - basic definition of affine plane

    I am reading the book, "Introduction to Plane Algebraic Curves" by Ernst Kunz - which the author claims gives a basic introduction to the elements of algebraic geometry. The opening few paragraph of Kunz' text reads as follows:I am puzzled by Kunz statement: " \mathbb{A} (K) := K^2 denotes...
  22. B

    Finding delta in terms of epsilon-delta definition

    Homework Statement If f(x) = 3x+1 en assume δ > 0. Assume ε>0. Give a δ > 0 with the following property : |x-1|< δ => |f(x) - f(1)| < ε Homework Equations The Attempt at a Solution |f(x) - f(1)| < ε <=> |3x+1-(3*1+1)| < ε <=> |3x-3| < ε <=>...
  23. T

    Spectral Family Definition

    I've been reading Kreyszig's functional analysis book, and I'm a little confused why he defines the spectral family of a self-adjoint operator the way he does. For an operator ##T## he defines ##T_{\lambda} = T - \lambda I##. Then he defines ##T_{\lambda}^+ =...
  24. Runei

    Integral definition of factorial

    I'm watching V. Balakrishnan's video lectures on Classical Physics, and right now he's going through statistical mechanics. In that regards he's talking about Stirlings formula, and at one point, he wrote an integral definition of the factorial like the following n! =...
  25. A

    Waves, Bungee Jumping, Linear Density Definition

    Please look a picture. I think the book made a mistake. The answer should be 89.9N/m. Why? Because they calculated the linear density μ wrongly. They should have done $$\frac{75*10^{-3}kg}{1.8m}$$
  26. A

    Definition of compound and acidity impact

    Hi, I was studying the impact of electronegativity in a row of the periodic table on the acidity of compounds. I came upon a problem of definition: what exactly is a compound? Does it have to be electrically neutral AT ALL TIMES?
  27. L

    Defining Capacitance: Is Charge Stored per Unit Volt Accurate?

    Would it be wrong to describe Capacitance as being the charge stored per unit volt? I have found on the internet that the definition of a Farad is the charge needed to cause one unit volt. Which means that the definition of Capacitance should surely be the same because the internet definition...
  28. M

    Exploring the Limit Definition of a Tough Derivative

    hey pf! can you help me with this $$\lim_{h \to 0} \frac{f(x+3h^2) - f(x-h^2)}{2h^2}$$ i know the definition and have tried several substitutions, but no help. anyone have any ideas?
  29. Math Amateur

    MHB Definition of epimorphism and equivalence to 'surjectivity'

    I am reading Paolo Aluffi's book Algebra: Chapter 0 which takes a (moderately) category theory oriented and infused approach to algebra. I am studying chapter 1: Set Theory and Categories and need help with formulating a definition of an epimorphism and with then proving it to be surjective...
  30. T

    Finding Limits Using Formal Definition

    Homework Statement Use formal definition of limits Find L = lim x→ c f(x). Then find a number δ > 0 such for all x f(x) = 3 - 2x c = 3 ε = 0.02 The Attempt at a Solution limx→3 3 -2x limx→3 3 - limx→3 2x 3 - 2(3) = -3 L = -3 I am not sure how to find delta
  31. R

    Work definition in thermodynamics

    Hello, I have been self-learning Thermodynamics and I am having a bit of trouble with calculating the work in different circumstances. Along the lectures we have come up with three different equations for work 1) W = pΔV 2) W = nRTln(V2/V1) 3) W = CvΔT So my questions are: 1) which...
  32. Math Amateur

    MHB Category Theory - Definition of Hom(X,X) in Awodey

    I am reading Steve Awodey's book, "Category Theory" (Second Edition). In Chapter 1 within a small section on monoids, Awodey defines Hom_{Sets} (X,X) as follows: " ... ... for any set X, the set of functions from X to X, written as Hom_{Sets} (X,X) is a monoid under the operation of...
  33. Hardik Batra

    Electric potential energy definition doubt?

    Electric potential energy of a charge q at a point in an electric field due to any charge configuration as the work done by the external force in bringing the charge q from infinity to that point without any acceleration.. In this definition, why charge is moved without any acceleration?
  34. S

    Could it be impossible to find derivative by basic definition?

    (I am sorry, totally forgot about solving quadratic equations, close the topic please) Homework Statement Finding derivative of the equation like F(x) = 5x / (1+x^2) by definition (ƒ(a+h) - f (a))/h is easy (point 2;2), but I got really stuck in finding a way to solve it by basic...
  35. P

    Question about limit definition of partial derivative

    I've seen it written two different ways: $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x + h, y) - f(x,y)}{h}$$ and $$\frac{\partial f}{\partial x} = \lim\limits_{h \rightarrow 0} \frac{f(x_0 + h, y_0) - f(x_0,y_0)}{h}$$ where the latter evaluates the function at...
  36. F

    Possible Errors in Writing Christoffel Symbols with a Symmetric Metric

    Homework Statement ¿Why \Gamma^{k}_{i j} = (1/2) g^{k p} (g_{i p ,j}+ g_{j p ,i}- g_{i j , p}) can't be writed like \Gamma^{k}_{i j} = (1/2) g^{k p} (2 g_{i p ,j}+g_{i j , p}) if i can say that the metric is symmetric? Homework Equations That is the relevant equation The Attempt at a...
  37. G

    S-Matrix definition and terminology

    A lot of textbooks give the definition of an S-matrix element as: \langle \beta_{out}| \alpha_{in}\rangle = \langle \beta_{in}| S| \alpha_{in} \rangle=\langle \beta_{out}| S| \alpha_{out} \rangle=S_{\beta \alpha} and that S|\alpha_{out} \rangle =|\alpha_{in} \rangle I don't...
  38. K

    How Do You Calculate Inverse Functions and Their Properties?

    Homework Statement Let ## C= \{ x \in R : x \geq 1 \} ## and ## D = R^+ ## For each f defined below, determine ## f(C), f^{-1}(C), f^{-1}(D), f^{-1} (\{1\}) ## a.) ## f: R -> R ## is defined by ## f(x) =x^2## I have problems with the definitions The Attempt at a Solution a.) ## f(C)= { 1 ...
  39. carllacan

    Alternative definition of constants

    Hi. First off, sorry about the title, its not very descriptive but I had no clue on how to sum my question. I'm reading Sakurais' Modern Quantum Mechanics. In the discussion of the K operators (p47) he compares it to the classical momentum operator, states that K = p/(some constant) , and...
  40. TheFerruccio

    Confusion over the definition of a Green's function

    This is how I learned about Green's functions: For the 1-D problem with the linear operator L and the inner product, (\cdot,\cdot), Lu(x) = f(x) \rightarrow u=(f(x),G(\xi,x)) if the Green's function G is defined such that L^*G(\xi,x) = \delta(\xi-x) I understand how to arrive at this...
  41. evinda

    MHB How can I show with the definition that f is continuous?

    Hello! (Smile) I am given this exercise: $$f(x)=\left\{\begin{matrix} \frac{e^x-1}{x} &, x \neq 0 \\ 1& ,x=0 \end{matrix}\right. , x \in [0,1]$$ Show that $f$ is integrable in $[0,1]$,knowing that if $f:[a,b] \to \mathbb{R}$, $f$ continuous,then $f$ is integrable in $[a,b]$. So,I have to...
  42. gfd43tg

    Standard Gibbs energy change definition

    Hello I am working on deriving the expression relating the equilibrium constant K to the change in Gibbs energy. This part seems to be followed okay, but here I am not following why the change in Gibbs energy of reaction is defined this way. I can see why K is defined in a way because...
  43. D

    Expanding the definition of inertial coordiante systems.

    Einstein has a thought experiment with two trains which he uses to prove linear motion without acceleration is inertial. Inertial means there is no physical test which will prove which train is moving and which is stationary, no coordinate system is preferred and that coordinate system are...
  44. S

    Question about the definition of df

    I guess I have several definitions of df flying at me, and I am having trouble getting a continuous definition. So in basic Calculus, we are taught df = f'(x)dx, and it's taught as sort of a linear approximation of the change of f for small values dx, whch makes sense with the definition of the...
  45. J

    What is the definition of max and min for multiple numbers?

    I found in the wiki a definition for the max of 2 numbers: https://en.wikipedia.org/wiki/Ramp_function But is definition is only for 2 numbers, how would be the definition for 3 numbers? Also, which is the definition of minimum function?
  46. V

    Formal definition of set operations

    Are set operations on a set ##X## defined as operations on ##2^X##? In other words a binary operation on ##X## is an operation ##\omega:2^X\times{}2^X\rightarrow{}2^x##? Surely the basic set operations could be defined that way, but then some weird non-standard operation like...
  47. D

    Understanding Euler's Number: Its Significance & Definition

    Can anyone give me a good definition of Euler's number and its significance. I see it everywhere, it's prolific in science and engineering.
  48. G

    Which is the Correct Answer for a Separable Differential Equation?

    Homework Statement A separable differential equation is a first-order differential equation that can be algebraically manipulated to look like: a. f(x)dx +f(y)dx b. f(y)dy = g(x)dx c. f(x)dx = f(y)dy d. g(y)dx = f(x)dx e. both f(y)dy=g(x)dx and f(x)dx = f(y)dy Homework Equations...
  49. E

    Ampere's Law: A Clear Definition

    Could someone give me a word definition of Ampere's Law as I have googled some definitions and they all seem a bit confusing. Thanks :)
  50. J

    Explicit definition for antiderivative?

    Accidentally I wrote in the wolfram f(x) = f(1/x) the the wolfram give me the solution for this equation (f(x) = Abs(log(x))). Hummmm, nice! Thus I thought: given the definition of derivative, ##f'(x) = \frac{f(x+dx)-f(x)}{dx}##, is possible to isolate f(x) in this equation? If yes, how? I...
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