Uniqueness Definition and 246 Threads

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...

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...

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...

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...

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, $...

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.

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...

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 .

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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.

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...

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...

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...

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...

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

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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}...

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...

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...

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...

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...

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...

Can anyone tell me What is UNIQUENESS THEOREM in electrostatics?

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...

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.

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...

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...

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...

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...

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.