Physics
Uniqueness Theorem
The Uniqueness Theorem states that the solution to a given problem is unique, provided that certain conditions are met. In physics, this theorem is often used to prove that the solution to a particular problem is the only possible solution. It is a fundamental principle in many areas of physics, including electromagnetism and fluid dynamics.
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3 Key excerpts on "Uniqueness Theorem"
eBook - PDF- Saurabh Kumar Mukerji, Ahmad Shahid Khan, Yatendra Pal Singh(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
31 3 Theorems, Revisited 3.1 Introduction The majority of field problems reduce either to the solution of Laplace’s equation or that of Poisson’s equation. Once a solution is found, it becomes a bone of contention whether the obtained solution is the only possible solu-tion. The answer to this question is given by the Uniqueness Theorem, which is considered as a litmus test in all such cases. This chapter begins with the Uniqueness Theorem and explores its applicability and compatibility for the solutions of Laplace’s and Poisson’s equations. The domain of the unique-ness theorem is further extended to encompass the vector magnetic poten-tial and Maxwell’s equations. This chapter also includes the discussion on Helmholtz’s theorem and the generalised Poynting theorem. A new concept of approximation theorems is also introduced and their usefulness for the Laplace equation, vector magnetic potential and eddy current equation is explored. 3.2 Uniqueness Theorem The Uniqueness Theorem spells that if a solution exists for a given equation under certain specified boundary conditions it is the only possible solution and may be referred to as unique. In the following subsections this theorem is discussed vis-a-vis the Laplace and Poisson equations, vector magnetic potential and Maxwell’s equations. 3.2.1 Uniqueness Theorem for Laplace and Poisson Equations In general, there are infinite solutions for the Laplace and Poisson equations. However, under certain boundary conditions unique solutions for these equa-tions can be found. The Uniqueness Theorem 1 reviewed here describes these conditions that are equally valid for both Laplace and Poisson equations. 32 Electromagnetics for Electrical Machines Let V 1 and V 2 be any two solutions for the Laplace equation for potential distribution in a given volume v .
eBook - PDF- P. C. Deshmukh(Author)
- 2019(Publication Date)
- Cambridge University Press(Publisher)
These theorems provide the logical backdrop of the succinct relationships provided by the Maxwell’s equations. Before we get to Maxwell’s equations, we shall therefore first establish these two important theorems. In this chapter, Maxwell’s equations are presented in the framework of the aforementioned two theorems, instead of the historical context of the empirical laws of electricity and magnetism developed separately prior to Maxwell’s work. The empirical laws of importance were formulated by Coulomb, Biot, Savart, Oersted, Ampere, and by Faraday and Lenz. We shall show, after introducing the four equations of Maxwell, that all the earlier empirical laws automatically fall under the canopy of the Maxwell’s equations. Theorem 1 (Uniqueness Theorem) Given (i) the divergence and (ii) the curl of a vector field in a simply connected region, i.e., a domain in which one can shrink every closed path continuously into a point without plunging out of that domain, and (iii) if the vector field’s normal components at the 419 Basic Principles of Electrodynamics boundaries of this region are known, then the vector field is uniquely specified (within that domain). This theorem is referred to as the Uniqueness Theorem. Let us therefore consider a vector field K (r ) whose divergence and the curl are given: ∇ ⋅ K (r ) = s (r ), (12.1a) and ∇ × K (r ) = c (r ). (12.1b) We further assume that the normal components K ^ (r b ) of the vector field K (r ) on the boundary, i.e., for all the points on the boundary, are known. The theorem requires us to show that if there is any other vector field K ′ (r ) for which Eqs.12.1a and b hold, and for which K′ ^ (r b ) = K ^ (r b ) at all the points on the boundary, then the difference W (r ) = K (r ) – K ′ (r ) must vanish. In other words, the vector field is uniquely specified by the conditions (i), (ii) and (iii) as stated in this theorem.
- V.D. Kupradze(Author)
- 2012(Publication Date)
- North Holland(Publisher)
CHAPTER III Uniqueness TheoremS In this chapter the Uniqueness Theorems for the basic boundary value and boundary-contact problems of classical elasticity, the couple-stress theory and thermoelasticity are proved. The problems for internal and external (infinite) domains are considered in the case of statics, harmonic oscillations and general dynamics. The proofs are based on the classical principle of energy and its generalization. §1. Static problems in classical elasticity 1. Green's formulas If / = (Ju fi, · · fm) and φ = ( φ ΐ9 φ 2, ..., = ί ^ * ν « + μ { | £ + | £ ). In the conditions of the theorem the integral over D + of the left-hand side of (1.3) exists and by virtue of the Gauss-Ostrogradski formula (see K e l l o g [1]) we have J [vA(dx)u + E(v, « )] dx = j [ Σ ( Σ d yS. D* S ' But (see (I, 2.1) and (I, 13.3)) { Σ ( Σ VpTpq(U J — { Σ ν Ρ ( Σ ΤΡ<ϊ(Μ)^<ϊ^ | = { Σ vp ( T u )p}+ = { v T u V and the theorem is proved. The outward direction with respect to D + is taken here and in what follows for the positive direction of the normal to S. 1.2. Note. The proof as to whether the Gauss-Ostrogradski theorem is applicable in the conditions of Theorem 1.1 is non-elementary (see K e l l o g [1]). The proof is simplified if S E J I (a ) is replaced by S G J J 2( 0).
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