Uniqueness Definition and 229 Threads

Let's have a theory involving Dirac field \psi. This theory is decribed by some Lagrangian density \mathcal{L}(\psi,\partial_\mu\psi). Taking \psi as the canonical dynamical variable, its conjugate momentum is defined as \pi=\frac{\partial\mathcal{L}}{\partial(\partial_0\psi)} Than the...

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...

Homework Statement Determine if the vector x is unique. x = (-15, -3, 0) + x_3 (10, 0, 1) Note: The scalars should be vertically placed instead of horizontal. The Attempt at a Solution Seeing that there is a free variable, I said that x is not unique, but my teacher marked it wrong. Why...

I am familiar with the existence and uniqueness of solutions to the system \dot{x} = f(x) requiring f(x) to be Lipschitz continuous, but I am wondering what the conditions are for the system \dot{q}(x) = f(x) . It seems like I could make the same argument for there existing a...

[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...

I want to know whether the wave function of particle is unique? If not, could we find a ψ to rationalize the equation Pψ=Aψ, in which P is the momentum operator and A is a constant. Thank you!

Could you please explain the theory intuitively and provide a proof to it. I understand how to apply it but i want to understand the logic behind it.

I just don't understand the idea behind it. I hate it when they throw these theories at us without proofs or elaborate explanations and just ask us to accept and applym mthem. Anyone care to enlighten me?

is there a converse of uniqueness theorem for circuits have for charged conductors. or atleast is there such a thing in case of circuit analysis ..

Here's a problem from one of the last (or previous of last) year, which bothers me ssssssoooooooo much. I've been working on this like a day or so, and haven't progressed very far. So, I'd be very glad if someone can give me a push on this. \left\{ \begin{array}{ccc} e ^ x - e ^ y & = & \ln(1 +...

Homework Statement Prove the uniqueness of Laplace's equation Note that V(x,y,z) = X(x) Y(y) Z(z)) Homework Equations \frac{d^2 V}{dx^2} + \frac{d^2 V}{dy^2}+ \frac{d^2 V}{dz^2} = 0 The Attempt at a Solution Suppose V is a solution of Lapalce's equation then let V1 also be a...

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...

if u have 3 primes: x,y,z then prove its sum m=x+y+z is unique ? Thank you

I'm trying to uniquely determine a complex function given pairs of real valued functions derived from it. For example, if you have its real and imaginary parts, or phase and the magnitude, the function is uniquely determined from them. But what if you have the magnitude of the function and...

Need help with ODE and "Existence and Uniqueness Thm" I'm currently helping my neice study for her exams and going though last years test there was this one question that I wasn't sure about. ------------- Considering the initial value problem, \frac{dy}{dx} = f(x,y) Where f(x,y) = (1+x)...

If a polynomial p(x)=a_0+a_1x+a_2x^2+ \ldots +a_{n-1}x^{n-1} is zero for more than n-1 x-values, then a_0=a_1= \ldots =0. Use this result to prove that there is at most one polynomial of degree n-1 or less whose graph passes through n points in the plane with distinct x-coordinates. Let p(x) be...

Short question: Can anyone provide me with a nice synopsis of how to go about proving the "existence" of some object as often requested in math questions such as, "prove that X really exists and is unique"? In other owrds, in general, when presented with an "existence" question, is there a nice...

Using this convention A \overleftrightarrow{\partial }_{\mu} B =:A \overrightarrow{\partial}_{\mu} B - A \overleftarrow{\partial}_{\mu} B one can write the QED Lagrangian density simply as \mathcal{L}_{QED} =\frac{i}{2} \bar{\Psi}_{\alpha} \left(\gamma^{\mu}\right)^{\alpha}{}_{\beta}...

I think I've got this one, I'd just like someone to check my work Negate the statement (\exists! x \in S) P(x) Since (\exists ! x \in S) P(x) \Longleftrightarrow \{(\exists x \in S) (P(x) \} \wedge \{(\forall x,y \in S) [P(x) \wedge P(y) \longrightarrow x = y \} The negation would be...

Dear all, If a differential equation is Lipschitz continuous, then the solution is unique. But what about the implication in the other direction? I know that uniqueness does not imply Lipschitz continuity. But is there a counterexample? A differential equation that is not L-continuous, still...

Can anybody help me with the proof of Existence and Uniqueness Theorems.

I often hear that fingerprints are unique, and you will share yours with no one. How much truth is there to this? I'll accept that there is very little chance you will find someone with identical finger prints, but I'm not entirely convinced that no two people will ever have identical ones. On...

Hello guys. I've been looking at uniqueness requirements for the differential equation: \frac{dy}{dx}=f(x,y);\qquad y(a)=b And the extension of this to higher-ordered equations. I'd like to understand the sufficient and necessary conditions for uniqueness. Most proofs require that...

What is the difference in the "uniqueness" of the representations of Cartesian coordinates and in polar coordinates? :confused: Also, what is the non-uniqueness?

This is a numerical analysis question, and I am trying to prove that the p(0), p'(0), p(1), p'(1) define a unique cubic polynomial, p. More precisely, given four real numbers, p00, p01, p10, p11, there is one and only one polynomial, p, of degree at most 3 such that p(0) = p00, p'(0) = p01...

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...

In my class notes, I have two theorems which don't quite seem to fit together. Maybe you can help me out. Thm 1 If p(x) in F[x] splits in K, then E=F(a1,...,an) is the splitting field of p(x) in K (the a_i's are the roots of p(x)). Thm 2 If p(x) in F[x], then the splitting field of p(x) is...

Good evening, I am a first year engineer here and a first time poster also. I had a problem that has been bugging me for the last few days; after much head-scratching and tree-killing, I may have solved it. I am, however, not sure at all if all my assumptions along the way are correct. So...

I recall an prof of mine (from a grad EM course) telling me that once the divergence and curl are of a vector field is specified then the vector field is determined up to an additive constant. Helmhotz's Theorem states that any vector field whose divergence and curl vanish at infinity can be...