Table of Contents
What is uniqueness theorem explain?
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: Fundamental theorem of arithmetic, the uniqueness of prime factorization.
What is uniqueness theorem in electromagnetic theory?
The electromagnetism uniqueness theorem states that providing boundary conditions for Maxwell’s equations uniquely fixes a solution for those equations. However, this theorem must not be misunderstood as that providing boundary conditions (or the field solution itself) uniquely fixes a source distribution.
What is the uniqueness theorem in differential equations?
Existence and uniqueness theorem is the tool which makes it possible for us to conclude that there exists only one solution to a first order differential equation which satisfies a given initial condition.
What is uniqueness theorem in elasticity?
For a finite material it is shown that a solution giving specified stresses over all boundaries is effectively unique, i.e. the stresses are uniquely determined throughout the material and the displacements are unique except for rigid body displacements, and that a solution giving specified displacements over all …
How do you use uniqueness theorem?
Existence and Uniqueness Theorem. The system Ax = b has a solution if and only if rank (A) = rank(A, b). The solution is unique if and only if A is invertible.
What is uniqueness theorem in complex analysis?
Uniqueness Theorem. Uniqueness Theorem: Let D ⊂ C be a domain and f , g : D → C is analytic. If there exists an infinite sequence {zn} ⊂ D, such that f (zn) = g(zn), ∀n ∈ N and zn → z0 ∈ D, f (z) = g(z) for all z ∈ D. Find all entire functions f such that f (r) = 0 for all r ∈ Q.
What is Picard’s Theorem?
Great Picard’s Theorem: If an analytic function f has an essential singularity at a point w, then on any punctured neighborhood of w, f(z) takes on all possible complex values, with at most a single exception, infinitely often.
Who was first to prove Taylor’s theorem?
Taylor’s theorem (without the remainder term) was devised by Taylor in 1712 and published in 1715, although Gregory had actually obtained this result nearly 40 years earlier. In fact, Gregory wrote to John Collins, secretary of the Royal Society, on February 15, 1671, to tell him of the result.
What is the importance of uniqueness theorems?
Theorems that tell us what types of boundary conditions give unique solutions to such equations are called uniqueness theorems . This is important because it tells us what is sufficient for inputting into SIMION in order for it to even be able to solve an electric field.
What is the uniqueness theorem in charge simulation?
The uniqueness theorem states that if we can find a solution that satisfies Laplace’s equation and the boundary condition V = V0 on Γ, this is the only solution. In the charge simulation method we seek equivalent (fictitious) charges near the surface of the conductor as illustrated in Figure 7.8.
What is the meaning of the word unique in math?
Lesson Summary. The word unique means one of a kind. In mathematics, when a theorem contains statements that use the word ‘unique,’ or that there is only one element that satisfies a certain condition, we call it a uniqueness theorem, and the proof of a uniqueness theorem is called a uniqueness proof.
Does the existence and uniqueness theorem apply to first-order linear ODEs?
First, let’s apply the Existence and Uniqueness Theorem to IVPs involving first-order linear ODEs. In the last section of this chapter we’ll sketch a proof of this important result. Because linear equations model many important physical situations, it’s important to know when these equations have unique solutions.