What is Uniqueness theorem: Definition and 42 Discussions
In mathematics, a uniqueness theorem is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include:
Alexandrov's uniqueness theorem of three-dimensional polyhedra
Black hole uniqueness theorem
Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.
Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients.
Division theorem, the uniqueness of quotient and remainder under Euclidean division.
Fundamental theorem of arithmetic, the uniqueness of prime factorization.
Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients.
Picard–Lindelöf theorem, the uniqueness of solutions to first-order differential equations.
Thompson uniqueness theorem in finite group theory
Uniqueness theorem for Poisson's equation
Electromagnetism uniqueness theorem for the solution of Maxwell's equation
Uniqueness case in finite group theoryA theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition).
Often in potential calculus problems, the uniqueness theorem of the solution of the Poisson problem with Dirichlet and Neumann boundary conditions is improperly "invoked," without bothering too much about making such an application rigorous, i.e., showing that indeed the problem we are solving...
Hi Pfs,
Please read this paper (equation 4):
https://ncatlab.org/nla b/files/RedeiCCRRepUniqueness.pdf
It is written: Surprise! P is a projector (has to be proved)...
where can we read the proof?
Does the second uniqueness theorem just say that if there is an electric field that satisfies Gauss's law for a surface surrounding each conductor + a surface enclosing all the conductors, it is indeed the true electric field, and no other electric field will satisfy those conditions?
in this example in Griffiths' electrodynamics, he says the following :(Figure 3.7 shows
a simple electrostatic configuration, consisting of four conductors with charges
±Q, situated so that the plusses are near the minuses. It all looks very comfort-
able. Now, what happens if we join them in...
There is a nice uniqueness theorem of electrostatics, which I have found only after googling hours, and deep inside some academic site, in the lecture notes of Dr Vadim Kaplunovsky:
Notice that the important thing here is that only the NET charges on the conductors are specified, not their...
I'm studying ODEs and have understood most of the results of the first chapter of my ODE book, this is still bothers me. Suppose
$$\begin{cases}
f \in \mathcal{C}(\mathbb{R}) \\
\dot{x} = f(x) \\
x(0) = 0 \\
f(0) = 0 \\
\end{cases}.
$$
Then,
$$
\lim_{\varepsilon \searrow...
1. For regions that contain charge density, does the 1st uniqueness theorem still apply?
2. For regions that contain charge density, does the 'no local extrema' implication of Laplace's equation still apply? I think not, since the relevant equation now is Poisson's equation. Furthermore...
We work with Maxwell's equations in the frequency domain.
Let's consider a bounded open domain ## V ## with boundary ## \partial V ##.
1. The equivalence theorem tells me that if the field sources in ## V ## are assigned and if the fields in the points of ## \partial V ## are assigned, then I...
I don't understand proof of uniqueness theorem for polynomial factorization, as described in Stewart's "Galois Theory", Theorem 3.16, p. 38.
"For any subfield K of C, factorization of polynomials over K into irreducible polynomials in unique up to constant factors and the order in which the...
Following my instructor's notes the statement of the Uniqueness Theorem(s) are as follows
"If ##\rho_{inside}## and ##\phi_{boundary}## (OR ##\frac{d \phi_{boundary}}{dn}## ) are known then ##\phi_{inside}## is uniquely determined"
A few paragraphs later the notes state
"For the field inside...
I am under the impression, there is no unique solutions to Einstein's field equations for a cosmological constant, or for higher dimensional spacetimes. Has anybody got a counter example for a solution including the cosmo constant to show there are multiple solutions, for example, i know of the...
Homework Statement
Boundary conditions are i) V=0 when y=0 ii) V=0 when y=a iii) V=V0(y) when x=0 iv) V=0 when x app infinity.
I understand and follow this problem (separating vars and eliminated constants) until the potential
is found to be V(x,y) = Ce^(-kx)*sin(ky)
Condition ii...
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, $...
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...
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...
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
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...
Can someone give me a qualitative example of the uniqueness theorem of a first order linear differential equation? I have read the definition, but I am not 100% positive of what it means in regards to an initial value problem.
Im confused about what a unique solution is when/if you change the...
Hello everyone!
So today is was my first day of differential equations and I understood most of it until the very end. My professor started talking about partial derivatives which is Calc 3 at my university. He said Calc 3 wasn't required but was recommend for differential equations. He...
Hello, I'm trying to make a sort of "system theory approach" to dynamic Maxwell's equations for a linear, isotropic, time-invariant, spacely homogeneous medium.
The frequency-domain uniqueness theorem states that the solution to an interior electromagnetic problem is unique for a lossy...
Consider a solid conductor with a cavity inside. Place a charge well inside the cavity. The induced charge on the cavity wall and the compensating charge on the outer surface of the conductor will be distributed in a unique way. How does this follow from the Uniqueness Theorem of EM? David...
Hi all.
Suppose that U1 is the solution of the Laplace's equation for a given set of boundary conditions and U2 is the the solution for the same set plus one extra boundary condition. Thus U2 satisfies the Laplace's equation and the boundary conditions of the first problem, so it's a solution...
My book stated the following theorem: If the functions P(x) and Q(x) are continuous on the open interval I containing the point x0, then the initial value problem dy/dx + P(x)y = Q(x), y(x0)=y0 has a unique solution y(x) on I, given by the formula y=1/I(x)\intI(x)Q(x)dx where I(x) is the...
As far as I can understand it, Picard's Existence and Uniqueness of ODEs theorem relies on the fact that a the given function f(x,t) in the initial value problem dx/dt = f(x,t) x(t0) = x0 is Lipschitz continuous and bounded on a rectangular region of the plane that it's defined on. And the...
Homework Statement
Consider the IVP compromising the ODE.
dy/dx = sin(y)
subject to the initial condition y(X) = Y
Without solving the problem, decide if this initial value problem is guaranteed to have a unique solution. If it does, determine whether the existence of that solution is...
Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZX4GEpxPRgC&lpg=PP1&dq=lang%20complex%20analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis.
But I think part of the reason was that I kept concentrating on the second sentence...
sorry about my English
Homework Statement
In Purcell 3.7 (Problem) and Griffith there is a question,look at fig.
(we have 4 conductors with charges +Q,-Q,+Q,-Q (b),what will happen
if we connect them with tiny wires in pairs )
Griffith say that c distribution of charge ,can't be a...
There is one thing I don't understand about this and is that besides the Dirichlet and Neumann conditions there seems to be a third one which is important when the method of images is used and is never mentioned. The problem is that Newmann condition requires especification of \frac{\partial\phi...
I read a lot of books on the uniqueness theorem of Poisson equation,but all of them are confined to a bounded domain \Omega ,i.e.
"Dirichlet boundary condition: \varphi is well defined at all of the boundary surfaces.
Neumann boundary condition: \nabla\varphiis well defined at all of the...
Homework Statement
Show that this problem has a unique solution:
\frac {dy}{dx}=\frac{4x+2e^{y}}{2+2x^2}
given that y(0) = 0.
Homework Equations
Test for exactness: If (when rewritten into (2+2x^2)y' - 4x+2e^y = 0 ; which i hope is correct) My = Nx then there is an exact...
Homework Statement
I have a situation with a charge distribution for a system of static charges in a vacuum. It then asks to state the uniqueness theorem for such a system.
Homework Equations
The Attempt at a Solution
I know that the uniquessness theorem means that once you have...
it states that in a given volume V surrounded by conductors or for that matter infinity if the charge density \rho and the charge on each conductor is fixed then the electric field is uniquely determined in that volume V
Can someone use this find the field in certain situations.
For Example...
[SOLVED] Griffith's Second Uniqueness Theorem
Homework Statement
I am having trouble understanding the Second uniqueness theorem in Griffith's Electrodynamics book which states that
"In a volume V surrounded by conductors and containing a specified charge density rho, the electric field is...
Greating my friends,
I have just returned home today from heart surgery.
I still feeling the effects of the operation, because I'm affried the hospital send me home a bit to early. But I have to have these questions finished before tomorrow.
So therefore I would very much appreciate...
Consider the system of linear differential equations:
X' = AX where X is a column vector (of functions) and A is coefficient matrix. We could just as well consider a first order specific case: y'(x) = C(x)y
We know that the soltuion will be a subset of the vector space of continuous...