Uniqueness Definition and 246 Threads

Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison. When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it is unlike the personality traits that are prevalent in that individual's culture. When the term uniqueness is used in relation to an object, it is often within the realm of product, with the term being a factor used to publicize or market the product in order to make it stand out from other products within the same category.The notion of American exceptionalism is premised on the uniqueness of the West, particularly its well-defined secularism.

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  1. Alpharup

    Is the Limit of a Function at a Point Always Unique?

    Spivak proves that limit of function f (x) as x approaches a is always unique. ie...If lim f (x) =l x-> a and lim f (x) =m x-> a Then l=m. This definition means that limit of function can't approach two different values. He takes definition of both the limits. He...
  2. davidbenari

    A question related to the method of images and uniqueness theorems

    My question is best illustrated by an example from a Griffiths book on E&M: "A point charge q is situated a distance ##a## from the center of a grounded conducting sphere of radius R (##a>R##). Find the potential outside the sphere... With the addition of a second charge you can simulate any...
  3. S

    How Is Uniqueness Defined for Familiar Mathematical Sets?

    The formal way to define many mathematical objects is careful not to assert the uniqueness of the object as part of the definition. For example, formally, we might define what it means for a number to have "an" additive inverse and then we prove additive inverses are unique as a theorem...
  4. C

    MHB Does Theorem 1 Guarantee a Unique Solution for Given Differential Equations?

    Hello, In my book on Differential Equations, There is a Theorem that states: "Consider the IVP $\d{y}{x}=f(x,y), y(x_0)=y_0$ If $f(x,y)$ and $\pd{f}{y}$ are continuous in some $a<x<b$, $c<y<d$ containing the point $(x_0,y_0)$, then the IVP has a unique solution $y=\phi(x)$ in some Interval...
  5. ognik

    MHB Uniqueness of Solutions for 2nd Order Linear Homogeneous ODEs

    Hi, please review my answer, I suspect I am missing some fine points... y(x) is a solution to a 2nd order, linear, homogeneous ODE. Also y(x0)=y0 and dy/dz=y'0 Show that y(x) is unique, in that no other solution passes through (x0, y0) with a slope of y'0. Expanding y(x) in a Taylor series, $...
  6. fricke

    Uniqueness of magnetic vector potential

    I able to prove magnetic field is uniquely determined but I am confused how to prove that magnetic vector potential is also unique. Can I say that magnetic vector potential is uniquely determined since magnetic field has unique solution? Thanks.
  7. D

    Proving uniqueness of a mathematical object

    As I understand it, the usual method for proving uniqueness of a mathematical object (for example the identity element of a group) is to use a proof by contradiction. Now, for example, if we have ##a## such that ##ax=b## and we want to prove this is unique, we start by assuming the contrary...
  8. N

    Why Is the Derivative Uniqueness Proof Important?

    Hello. In the proof of uniqueness of ( multi-variable ) derivative from Rudin, I am a little stuck on why the inequality holds. Rest of the proof after that is clear .
  9. ELB27

    Determining Uniqueness of Reduced Echelon Form

    Homework Statement Is the reduced echelon form of a matrix unique? Justify your conclusion. Namely, suppose that by performing some row operations (not necessarily following any algorithm) we end up with a reduced echelon matrix. Do we always end up with the same matrix, or can we get different...
  10. G

    Solid Mechanics - Uniqueness of Plane Stress State

    Homework Statement My textbook says that the state of plane stress at a point is uniquely represented by two normal stress components and one shear stress component acting on an element that has a specific orientation at the point. Also, the complementary property of shear says that all four...
  11. R

    Impose Uniqueness on Diagonalization of Inertia Tensor?

    Given an inertia tensor of a rigid body I, one can always find a rotation that diagonalizes I as I = RT I0 R (let's say none of the value of the inertia in I0 equal each other, though). R is not unique, however, as one can always rotate 180 degrees about a principal axis, or rearrange the...
  12. F

    MHB Does the Existence and Uniqueness Theorem Guarantee Solutions for dy/dx = 2xy²?

    Given \frac{dy}{dx} =2xy^2 and the point y(x_0)=y_0 what does the existence and uniqueness theorem (the basic one) say about the solutions? 1) 2xy^2 is continuous everywhere. Therefore a solution exists everywhere 2) \frac{\partial }{\partial y} (2xy^2) = 4xy which is continuous everywhere...
  13. M

    How does the Wronskian determine uniqueness of solution?

    Given an nth order DE, how (intuitively and/or mathematically) does computing the Wronskian to be nonzero for at least one point in the defined interval for the solution to the DE ensure the solution is unique and also a fundamental set of solutions? Also, is it true that if W = 0, it is 0 for...
  14. M

    Why Does ∂f/∂y Determine the Uniqueness of y in Differential Equations?

    Hi, I was just wondering why taking ∂f/∂y provides the interval on which y is unique (or not necessarily). Could someone possibly provide some mathematical intuition behind this and possibly a proof of some sort detailing why y is unique if ∂f/dy is continuous? Also, how exactly (if it can) is...
  15. M

    Why Is ∂f/∂y Used to Determine Uniqueness in Differential Equations?

    Hi, For differential equations, when trying to determine the uniqueness of an equation in the form dy/dx = q(y)p(x) (where p(x) and q(y) are any functions of x and y, respectively), is there any particular reason why dy/dx = f(x,y) = q(y)p(x) is then later differentiated with respect to y as...
  16. F

    MHB Application of existence and uniqueness theorem

    Given the differential equation y'=4x^3y^3 with initial condition y(1)=0determine what the existence and uniqueness theorem can conclude about the IVP. I know the Existence and Uniquness theorem has two parts 1)check to see if the function is differentiable and 2)check to see if \frac{\partial...
  17. ELB27

    Proof on a uniqueness theorem in electrostatics

    Homework Statement Prove that the field is uniquely determined when the charge density ##\rho## is given and either ##V## or the normal derivative ##\partial V/\partial n## is specified on each boundary surface. Do not assume the boundaries are conductors, or that V is constant over any given...
  18. M

    Understanding Uniqueness and Existence Theorems for ODE's

    How to understand Uniqueness and existence theorem for first order and second order ODE's intuitively?
  19. K

    Is the Uniqueness of IVP Solutions Always Binary?

    Are these statements correct, if not could you give me an example 1. If solution of IVP is non-unique then there are infinitely many solutions in short, if the solution to the IVP has at least 2 solutions then there are infinitely many solutions to this IVP 2.there are none IVP first...
  20. M

    A question about uniqueness of initial conditions

    Hİ. How can we sure that the initial conditions , say, for a second-order linear equation must be unique which is also the uniqueness of the solution.
  21. O

    MHB Proving Local Uniqueness of Autonomous System x'=f(x)

    i have an autonomous system x'=f(x) and teh function f is loc lip on its domain, if x and y are sol of the system defined on (alpha, beta) and x(s)= y(s) for some s in (alpha, beta) then x= y on (alpha, beta) is the solution to prove this problem similar to this one...
  22. C

    Where is the uniqueness of smooth structure for involutive distributions proved?

    I'm looking to prove the Global Frobenius theorem, however in order to do so I need to prove the following lemma: If ##D## is an involutive distribution and and ##\left\{N_\alpha\right\}## is collection of integral manifolds of ##D## with a point in common, then ##N = \cup_\alpha...
  23. K

    Uniqueness given specified surface charges and voltages

    Suppose we have a collection of conductors for which the voltage is specified on some conductors and the surface charge is specified on others. Is there a coherent way to specify this as a boundary value problem for the voltage (satisfying Laplace's, or in the presence of charge density...
  24. I

    Solutions to laplace's equation & uniqueness thrm #2 (Griffiths)

    In griffith's intro to electrodynamics (4rth edition), ch. 3, pg. 121. here is the second uniqueness thrm for the solutions to laplace's equation: the only part I'm confused about is, in the beginning where he says "in a volume V surrounded by conductors and containing a specified charge...
  25. O

    MHB Prove Local Uniqueness of DE Solutions on Interval

    if a function ls locally lip then considering this diff eq x'(t)= f(x(t) where now x and y are solutions of the DE on some interval J and x(s)=y(s) for some s in J. then how can I prove that there exists a positive number delta such that x=y on (s-delta, s+delta)∩ J
  26. H

    Proving the Uniqueness of Center of Momentum Frame

    Hello again, Two days ago, I started a thread asking about the same question more or less, and I was thinking that the matter was clear now in my mind, because I had made an error in my calculations... Before I begin, I want to admit that my English is not very good, and my exposition to...
  27. michael879

    Uniqueness of Maxwell's equations

    Hi all, I'm trying to derive for myself the uniqueness proof for Maxwell's equations, but I'm a little stuck at the end. I've managed to prove the following: \dfrac{A^\mu}{\partial{t}}\nabla{A^\mu}|_S = \dfrac{A^\mu}{\partial{t}}|_{t_0} = \nabla{A^\mu}|_{t_0} =0 \Rightarrow...
  28. B

    Existence and Uniqueness Theorem

    Hello Everyone. I have a question. Suppose I have a differential equation for which I want to find the values at which \displaystyle f(x,y) and \displaystyle \frac{\partial f}{\partial y} are discontinuous, that I might know the points at which more than one solution exists. Suppose that...
  29. evinda

    MHB Is the Lower Triangular Matrix in the Cholesky Decomposition Unique?

    Hello! :D I want to show that the lower triangular matrix L,which has the identity $A=LL^{T}$ ,where A is a positive-definite and symmetric matrix,is unique. That's what I have done so far: Suppose that there are two matrices,with that identity. Then, $$A=LL^{T}=MM^{T} \Rightarrow...
  30. J

    Feynman propagator and particle uniqueness

    In his layman's guide to QED Feynman defines a particle propagator as a function that gives you the amplitude that a particle, that was initially at spacetime event ##x##, will be found at spacetime event ##y##. But does this definition assume that the particle is unique so that if you find...
  31. P

    Proving Uniqueness in Continuous Functions with Positive Values

    Homework Statement Suppose that k(t) is a continuous function with positive values. Show that for any t (or at least for any t not too large), there is a unique τ so that τ =∫ (k(η)dη,0,t); conversely any such τ corresponds to a unique t. Provide a brief explanation on why there is such a...
  32. M

    Uniqueness of the solution with certain boundary conditions

    Hey! Speaking electrodynamics, I can't seem to get mathematically or even physically convinced that the solution with Dirichlet or Neumann boundary conditions is UNIQUE. Can someone explain it? Thanks.
  33. Y

    Proving uniqueness of inverse by identity (Groups)

    1. Which of the following is a group? To find the identity element, which in these problems is an ordered pair (e1, e2) of real numbers, solve the equation (a,b)*(e1, e2)=(a,b) for e1 and e2. 2. (a,b)*(c,d)=(ac-bd,ad+bc), on the set ℝxℝ with the origin deleted. 3. The question...
  34. A

    Is Uniqueness Necessary in Mathematical Measures?

    My book has a theorem of the uniqueness of the Lebesgue measure. But my question is: Is it necessarily a good thing that something in mathematics is unique and seems to indicate that this is very important. But my question is? Would the theory of measures fail if there existed another measure...
  35. E

    Proof of Uniqueness of Non-Identity Commuting Element in D_2n

    Homework Statement If ##n = 2k## is even and ##n \ge 4##, show that ##z = r^k## is an element of order 2 which commutes with all elements of ##D_{2n}##. Show also that ##z## is the only nonidentity element of ##D_{2n}## which commutes with all elements of ##D_{2n}##. Homework Equations...
  36. A

    What Are the Conditions for Uniqueness in Solving This PDE?

    Consider the PDE $$ U_{xy}+\frac{2}{x+y}\left(U_{x}-U_{y}\right)=0 $$ with the boundary conditions $$ U(x_{0},y)=k(x_{0}-y)^{3}\\ U(x,y_{0})=k(x-y_{0})^{3} $$ where $k$ is a constant given by $k=U_{0}(x_{0}-y_{0})^{3}$. $x_{0}$, $y_{0}$ and $U(x_{0},y_{0})=U_{0}$ are known. The solution...
  37. B

    Why is the matrix representation of a linear map unique?

    Friedberg proves the following theorem: Let V and W be vector spaces over a common field F, and suppose that V is finite-dimensional with a basis \{ x_{1}...x_{n} \}. For any vectors y_{1}...y_{n} in W, there exists exactly one linear transformation T: V → W such that T(x_{i}) = y_{i}...
  38. R

    Uniqueness of eigenvectors and reliability

    Dear All, In general eigenvalue problem solutions we obtain the eigenvalues along with eigenvectors. Eigenvalues are unique for each individual problem but eigenvectors are not, since the case is like that how we can rely that solution based on the eigenvector is correct. Because if solution is...
  39. B

    Uniqueness of identity elements for rectangular matrices

    Let A be the set of n \times n matrices. Then the identity element of this set under matrix multiplication is the identity matrix and it is unique. The proof follows from the monoidal properties of multiplication of square matrices. But if the matrix is not square, the left and right...
  40. P

    Uniqueness - exact differential equation

    Hi folks! This one got me in doubts... Homework Statement Solve IVP (Initial Value Problem): (2xy+sin(x))dx+(x^{2}+1)dy=0, y(0)=2 Is the solution unique? Motivate why! Homework Equations Relevant equations for solving the exact equation... The Attempt at a Solution I can...
  41. M

    Uniqueness of value of Riemann Integral(proof)

    The proof is in the document. I highlighted the main points that I am questioning in the document. I am questioning the fact that A = B... (The following is in the document) |A-B|=ε where they define the value of ε to be a positive arbitrary real number (ε>0). And for A = B that means ε must...
  42. L

    How to prove uniqueness solution of the 3D wave

    Homework Statement The three dimensional wave equation: c∂^{2}u/∂t^2 = ∇^2 u boundary conditions : u(x,y,z,t) = F(x,y,z,t) on S initial conditions: u(x,y,z,0) = G(x,y,z) ∂u/∂t(x,y,z,0)=H(x,y,z) Homework Equations how to prove the uniqueness solution of the above equation...
  43. B

    Uniqueness Theorem in Electrostatics - Explanation

    Can anyone tell me What is UNIQUENESS THEOREM in electrostatics?
  44. B

    Existence and Uniqueness Theorem

    Suppose you have an ODE y' = F(x,y) that is undefined at x=c but defined and continuous everywhere else. Now suppose you have an IVP at the point (c,y(c)). Then is it impossible for there to be a solution to this IVP on any interval containing c, given that the derivative of the function, i.e...
  45. B

    Existence uniqueness wronskian

    Homework Statement y''-4y=12x Homework Equations I don't know The Attempt at a Solution http://imageshack.us/a/img7/944/20130207102820.jpg I'm not sure if I did this right, I'm putting this here to make sure. Please respond within 3 hours if you can because it will be due.
  46. B

    Lipschitz Condition, Uniqueness and Existence of ODE

    Homework Statement Find a solution of the IVP \frac{dy}{dt} = t(1-y2)\frac{1}{2} and y(0)=0 (*) other than y(t) = 1. Does this violate the uniqueness part of the Existence/Uniqueness Theorem. Explain. Homework Equations Initial Value Problem \frac{dy}{dt}=f(t,y) y(t0)=y0 has a...
  47. T

    Prove Uniqueness Theorem: |\phi(t) - \psi(t)| ≤ ∫0t

    I am having to justify the steps in a proof of the uniqueness theorem. I am supposed to show why the inequality follows from the initial equation. http://i.imgur.com/AxApogj.png \phi(t) - \psi(t) =∫0t 2s[\phi(t) - \psi(t)] ds |\phi(t) - \psi(t)| =|∫0t 2s[\phi(t) - \psi(t)] ds| \leq...
  48. L

    No problem, glad I could help!

    Homework Statement Using the delta-epsilon definition of limits, prove that of lim f(x) = l and lim f(x) =m, then l=mHomework Equations Delta-epsilon definiition of the limit of f(x), as x approaches a: For all e>0, there is a d s.t if for all x, |x-a|<d, then |f(x) -l|<e The Attempt at a...
  49. R

    Uniqueness Theorem, Concentric Equipotential Cylinders & Moving Charge

    Homework Statement Consider two electrodes 2 mm apart in vacuum connected by a short wire. An alpha particle of charge 2e is emitted by the left plate and travels directly towards the right plate with constant speed 106 m/s and stops in this plate. Make a quantitative graph of the current in...
  50. J

    What is the Uniqueness Theorem and its Application in Physics?

    Could someone give me an applied math example of the uniqueness theorem in the physical sciences (physics, chemistry, biology)? Because I am not sure of its application. I understand that there is an interval (x,y)~intial conditions.
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