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

    Definition of Magnetic Declination,Inclination

    I wish to know what exactly magnetic declination and inclination means. I used to feel that angle of declination was constant throughout the Earth equalling≈11.3°.But, it apparently isn't. Then, why is so much significance attached to 11.3° if it is not universally constant?
  2. M

    Definition of first law of thermodynamic?

    Homework Statement What is the definition for first law of thermodynamic? I was confused and which one is correct? (a) The CHANGE in internal energy of a closed system is equal to the heat that enters a system and the workdone on the system OR (b)...
  3. caffeinemachine

    MHB Graded poset definition trouble

    Graded poset on wiki: Graded poset - Wikipedia, the free encyclopedia Wikipedia defines a 'graded Poset' as a poset $P$ such that there exists a function $\rho:P\to \mathbb N$ such that $x< y\Rightarrow \rho(x)< \rho(y)$ and $\rho(b)=\rho(a)+1$ whenever $b$ covers $a$. Then if you go to the...
  4. C

    Stress-energy tensor definition

    I have seen two definitions with oposite signs (for one of the pressure terms in the formula) all over the web and books. I suspect it is related to the chosen metric signature, but I found no references to that. General Relativity An Introduction for Physicists from M. P. HOBSON...
  5. M

    Correct definition of surface tension

    Hi, I have a question, because I am confused with the definition of surface tension. In my book it is defined as follows: "Surface tension is the energy required to increase the surface area of a liquid by a unit amount" What do they mean with increasing the surface area, how do you do...
  6. Astrum

    What is the Intuitive Explanation for the Definition of Convergence?

    I'm a bit confused about how my book defines convergence. Definition: A sequence {an} convergences to l if for every ε > 0 there is a natural number N such that, for all natural numbers n, if n > N, then l a,-l l < ε note, l a,-l l = the absolute value Maybe someone could give me an...
  7. C

    Can the Kolmogorov definition of the conditional probability be proven

    It seems weird that such a relatively complex concept is simply given as a definition in most textbooks and then dismissed for further explanation other than using it intact or as a basis for further proofs.
  8. C

    Quantum Mechanics: Defining Dimensions

    How is a dimension defined in quantum mechanics?
  9. 7

    Integral and a derivative definition

    Hello, I just whant to know what mathematical rule alows me to do this? I mean i think it is u-substitution, but i am not sure how it is done here? It is weird to me as it seems that ##dt## just cancel out and limits are changed... $$ \int\limits_{0}^{t} \frac{dv}{dt} \cdot mv \gamma(v)\, dt...
  10. F

    Confusion regarding delta definition of limit

    I don't quite get the significance of the delta limit definition, if n>N and |sn−s|< ϵ , why does the limit converges does this simply means that there exist a number ε such that if n is great enough it will be greater than s by ε? But this doesn't make sense, because s is the value...
  11. phoenixthoth

    A statement equivalent to the definition of limits at infinity?

    I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition. So the question is this: are the following two statements equivalent? (1) \lim_{x\rightarrow\infty}f\left(x\right)=L (2) \exists c>0\exists...
  12. S

    Check this definition of a subset

    Homework Statement "We say a set T is a subset of a set S if every element of T also belongs to S( i.e T consists of some of the elements of S). We write T ⊆ S if T is a subset of S and T ⊄ S if not. For example, if S = {1, {2}, cat}, then {cat} ⊆ S, {{2}} ⊆ S, 2 ⊄ S. As another example, the...
  13. nomadreid

    Confusion on application of definition of degrees of freedom

    I am confused about the counting of degrees of freedom. Yes, I know that it is the number of vectors which are free to vary. But that definition gives way to different interpretations: (1) the number of data points minus the number of independent variables. This seems to be the basis of the...
  14. S

    What is the Definition of Limit in Real-Valued Spaces?

    So, it seems that in a real-valued setting, the limit and the derivative of a real-valued function is defined only if the domain is an open subset of Euclidean space. I'm a little confused as to why this is the case, and why we can't just define a limit and derivative on any subset of Euclidean...
  15. D

    Definition of an integral question

    definition of an integral question... if f(x) ≥ 0, how can you use the definition of an integral to prove that ∫(a,b)f(x)dx ≥ 0? This seems like it is an easy question, and seems like one of those things that seems obvious but hard to explain, and the only definition of an integral I've been...
  16. K

    Definition of Inertial Frame in GR: Math Explained

    How do we mathematically define a inertial frame in GR? Is it only a basis in some tangentspace or does it have to be induced by a coordinatechart? :/
  17. P

    Limit definition of derivative problem

    Homework Statement Using the definition of derivative find f'(x) for f(x) = x - sqrt(x) Homework Equations None. The Attempt at a Solution lim h --> 0 : ((x + h) - sqrt(x + h) - x + sqrt(x))/h 1 - (sqrt(x + h) - sqrt(x))/h Multiply by conjugate.. 1 - h/(h*(sqrt(x) +...
  18. C

    Is this a correct definition of one volt?

    One volt is when: The difference in (electrical potential energy per unit charge (q)) between two places equals one. Where electrical potential energy equals EPE at distance R from charge Q = (1/4piEpsilonNought) * Q/R Is this Correct? Thanks!
  19. G

    Is this a good definition of entanglement

    I don't like the way Wiki describes entanglement. Here is my own definition. Tell me if it is in essence correct. If I have left out an important detail please let me know objects are entangled if and only if by changing the property of one object one instantaneously changes the property of...
  20. D

    MHB Definition of H in pulley diagram

    Why can $H = \left(\ell - \frac{b}{\sin\theta}\right)$ where $\ell$ is the length of the rope. (everything is frictionless.) http://img690.imageshack.us/img690/579/pulley.png
  21. M

    Classical definition of probability & kolmogorovs axioms

    I've seen in some probability theory books that the classical definition of probability is a probability measure, it seems fairly trivial but what is the proof for this? Wikipedia gives a very brief one using cardinality of sets. Is there any other way?
  22. mef51

    Checking Linear Independence. Using Wronskian vs. Using Definition

    Homework Statement Is the set $$ \{cos(x), cos(2x)\} $$ linearly independent?Homework Equations Definition: Linear Independence A set of functions is linearly dependent on a ≤ x ≤ b if there exists constants not all zero such that a linear combination of the functions in the set are equal to...
  23. J

    What is the definition of a 'true vector'?

    In my class notes my professor defined a true vector as a vector which does not depend on origin placement. Once he defined it he went on with an example of how a vectors magnitude is conserved in two different coordinate systems. So my question is what is the definition of a true vector? Is...
  24. A

    MHB Proofs on growth rates of functions theorems using definition of a limit

    Hello, I am working through some proofs from the following document: Function Definitions Under Calculation of Big - Oh, some theorems are provided that classify the growth rates of functions in relation to one depending on what the limit is as the input approaches infinity. One proof is...
  25. TrickyDicky

    Manifold Definition in Topology

    In general terms a manifold can be defined simply as a topological space locally resembling Euclidean space with the resemblance meaning homeomorphic to Euclidean space, plus a couple of point set axioms that avoid certain "patological" manifolds and that some authors reserve for the definition...
  26. A

    What is the connection between sine and cosine and geometry?

    Ordinarily in mathematics, when you want to define a function, it is without reference to geometry. For instance the mapping f:ℝ→ℝ x→x2 And though I don't know much about mathematics I assume you somehow proof that the function is well defined for all numbers, check if the derivative exists and...
  27. M

    Union Definition: P(AB) and P(AB)c Explained

    Homework Statement Can you guys explain to me what the following mean. We are working on probability and unions, and these came up on the homework and need to know what these mean in order to solve the problem. Thanks P(AB) P(AB)c Where c is the compliment. Also i want to...
  28. H

    Very basic derivative definition question f(x)=x

    Homework Statement Find the derivative of f(x) = x using the definition of a derivative. (when Δx → 0) Homework Equations (x + Δx) - x -------------- Δx The Attempt at a Solution I know the answer is 1. I graphed the function of f(x)=x and confirmed this, however the...
  29. J

    Definition of Fourier transform

    Hi All, Usually the Fourier transform is defined as the one in the Wiki page here (http://en.wikipedia.org/wiki/Fourier_transform), see the definition. My question is can I define Fourier transform as \intf(x)e^{2\pi ix \varsigma}dx instead, i.e., with the minus sign removed, as the...
  30. elfmotat

    Finding Constant \alpha_M in SET Definition

    So I was looking through Wald when I noticed his definition of the stress-energy for an arbitrary matter field: T_{ab}=-\frac{\alpha_M}{8\pi} \frac{1}{ \sqrt{-g}} \frac{\delta S_M}{\delta g^{ab}} where S_M is the action for the particular type of matter field being considered, and \alpha_M...
  31. T

    Definition of the boundary map for chain complexes

    I've been poking around, learning a little about homology theory. I had a question about the boundary operator. Namely, how it's defined. There's two definitions I've seen floating around. The first is at: http://en.wikipedia.org/wiki/Simplicial_homology The second, at...
  32. J

    When y= a constant, how do you find the interval of definition?

    I used the linear equation method to solve a D.E. and got y=3/4 at the end. I'm asked to find the interval of definition but I don't know how to do that when Y is just a constant :/
  33. pellman

    What is the calculational definition of the 'big bang'?

    You sometimes see statements regarding so-many-minutes after the big bang. Or 10^-23 seconds after the big bang. But this exactly is the event this is measuring from? How is it defined? I presume it has some definition from within the framework of general relativity.
  34. Y

    Verify definition of circular polarization in Balanis.

    Attached is a scan of how Balanis define plane wave with circular polarization with Ey having a phase of +∏/2 respect to Ex component of the E field. I don't quite agree with the book. The second attachment is my derivation. The definition of CW or CCW is with respect to direction of...
  35. J

    Definition of a solution of a first order ODE

    Given an open connected subset D of the (t,x) plane and a function f\in C(D,\mathbb{R}), we say \varphi\in C^1(\text{proj}_1D,\mathbb{R}) is a solution of the first order differential equation x'=f(t,x) if and only if \forall t\in \text{proj}_1D,\quad (t,\varphi(t))\in D and \forall t\in I...
  36. P

    Definition of the wedge product on the exterior algebra of a vector space

    Hi, I am currently reading about differential forms in "Introduction to Smooth Manifolds" by J. M. Lee, and I was wondering exactly how you define the wedge product on the exterior algebra \Lambda^*(V) = \oplus_{k=0}^n\Lambda^k(V) of a vector space V. I understand how the wedge product is...
  37. P

    Charge carrier injection in heterostructures - help with concept definition

    Hi, I have this report to do on "Charge injection in heterostructures". I have been searching and reading but I still have some trouble with the basics, i.e. defining the concept. As far as I understood a heterostructure is a junction between two or more different semiconductors and the...
  38. S

    Integral through a path in 2D (or ND) What's the usual definition ?

    Integral through a path in 2D (or ND) What's the usual "definition"? [Bold letters are vectors. eg: r] We have a scalar function f(r) and a path g(x)=y. I see two ways to reason: (1) The little infinitesimals are summed with the change of x and on the change of y separately. (2) The little...
  39. S

    What is the Definition of No Slipping in Physical Terms?

    What does it really mean in physical term? Does it mean no friction? No loss of mechanical energy? Thanks!
  40. C

    Magnetic Moment Definition Verification/ Proof

    I saw the equation here http://en.wikipedia.org/wiki/Magnetic_moment#Current_loop_definition for the definition of the magnetic moment for a non-planar loop. Can someone tell me if there's a name for this equation m= \frac { I }{ 2 } \int { \overrightarrow { r } } \times d\overrightarrow { r }...
  41. M

    The definition of whole numbers

    I'm not a mathematician of any sort so excuse me if my question is stupid. I just realized that I could not define the set of whole numbers without referring back to them or to the operation of addition, which then itself can't be defined. How would you define whole numbers?
  42. V

    Definition of electric charge as rationalized charge

    definition of electric charge as "rationalized charge" Hi All, I wonder about the meaning of the term "rationalized" when saying "rationalized electron charge." Does this mean that the charge is given in natural units? Thank you very much! Best
  43. A

    Deriving the algebraic definition the dot product

    Is there a way of deriving the algebraic definition of the dot product from the geometric definition without using the law of cosines?
  44. A

    Limit definition and infinitely often

    Limit definition and "infinitely often" If we have a sequence of real numbers x_{n} converging to x, that means \forall \epsilon > 0, \exists N such that |x_n - x| < \epsilon, \forall n \geq N. So, can we say P (|x_n - x| < \epsilon \ i.o.) = 1 because for n \geq N, |x_n - x| < \epsilon...
  45. I

    Confusion with definition and notation of reciprocal.

    Hello everyone, I have some conceptual issues with aforementioned definitions. How is exactly multiplicative inverse defined? Say, for a rational, nonzero number a/b, its reciprocal is b/a. Is there a certain operation that transforms a/b to b/a? Also, the notation for multiplicative...
  46. G

    Definition of potential energy

    Why is change in potential energy is defined as PE1 - PE0 = -W I mean I could see it for example for gravity if we took PE0 to be zero at ground and we integerated -mgy(y^) we get -mg(y0 - y1) -> -mgh,but is their a proof somewhere where it shows it will be always negative work ? Thank you.
  47. D

    What is the relationship between arc voltage and current in ion sources?

    I could not figure out definition of arc voltage and arc current. Is that true arc voltage if the potential differences between dual electrode and the current is flow through one of these electrode? any help would be appreciated.
  48. L

    Intuition about definition of laplace transform

    why was laplace transform developed i have googled it and found that it is something about shaping a family of exponential and vector projections etc i couldn't get it. some simply said that it was used to make a linear differential equation to algebraic equation but i couldn't understand how...
  49. genphis

    Is the definition of space relative ?

    I have just finished reading stuart clark's book 'The Universe' and i find myself pondering the question of space its possible infinite size,shape, and its relation to our universe. a) if space did not exist before our universe's expansion. What are we expanding into and what are we pushing...
  50. S

    What does the N mean in a Cauchy sequence definition?

    What does the "N" mean in a Cauchy sequence definition? Hi everyone, I have a question regarding Cauchy sequences. I am trying to teach myself real analysis and would appreciate any clarification anyone has regarding my question. I believe I have an intuitive understanding of what a Cauchy...
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