Helmholtz Theorem

9 Key excerpts on "Helmholtz Theorem"

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  Earth's Magnetosphere

  Formed by the Low-Latitude Boundary Layer

  • Walter Heikkila(Author)
  • 2011(Publication Date)
  • Elsevier Science

    (Publisher)

  Chapter 3. Helmholtz’s theorem

  The sources of vector fields are their divergence and curl, by Helmholtz's theorem. Since the electrostatic field has zero curl, it cannot modify the curl of the inductive component, nor the electromotive force. This has many consequences, for example, solar wind plasma transfer across the magnetopause, current thinning during the growth phase, tailward moving plasmoids, bursty bulk flows, omega band auroras, and prompt particle acceleration to high energies discussed throughout the book. We performed a 1D particle simulation [Omura et al., 2003] where a constant electric field was applied along the magnetic field lines. At first all plasma particles are accelerated freely in opposite directions until a Buneman current instability sets in. As the particles gain more energy the instability diminishes; almost free acceleration is again maintained. Cluster 3 spacecraft made an exceptional high-resolution measurement of a beam of electrons with energies up to 400 keV [Taylor et al., 2006]. The distribution evolved from antiparallel, through counterstreaming to parallel over a period of 20 s, agreeing with our simulation. The global conditions of the magnetosphere were analysed and the beam event is clearly associated with a substorm.

  Discovery consists of seeing what everybody has seen and thinking what nobody else has thought.

  Albert Szent-Györgyi, 1957

  In general there seems to be little advantage in the further pursuit of the hydromagnetic–thermodynamic approach when a precise description is required. The particle approach is then necessary for at least some aspects of the problem, and it might as well be adopted for all.

  Colin Hines, 1963

  3.1. Introduction

  The existence and variability of energetic particle populations in the Earth’s magnetosphere has dominated investigations since early satellite observations. These are the key to understanding substorms and plasmoid structures. However, the mechanisms by which these charged particles can obtain such high energies are not well-understood.

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  Mathematical Foundations of Imaging, Tomography and Wavefield Inversion

  • Anthony J. Devaney(Author)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  461 11.2 The Helmholtz Theorem 11.1.2 Maxwell equations in the spatial frequency domain We define the spatial Fourier transform pair E(r) ⇐⇒ ˜ E(K) ˜ E(K) =  d 3 r E(r)e −iK·r , E(r) = 1 (2π ) 3  d 3 K ˜ E(K)e iK·r , with similar definitions for the magnetic field vector and the current and charge densities. As usual we assume that ∂ n ∂ x n j E(r) ⇐⇒ (iK j ) n ˜ E(K), (11.5) at least up to order n = 2, where x j is any Cartesian coordinate of the position vector r and K j is the associated component of the spatial frequency vector K. On taking the spatial Fourier transform of the set of Maxwell equations Eqs. (11.2) we obtain the result i 0 K · ˜ E(K) = ˜ ρ (K), (11.6a) K · ˜ H(K) = 0, (11.6b) K × ˜ E(K) = ωμ 0 ˜ H(K), (11.6c) iK × ˜ H(K) = −iω 0 ˜ E(K) + ˜ J(K). (11.6d) The charge–current conservation equation Eq. (11.1) yields the result K · ˜ J(K) = ω ˜ ρ (K). (11.7) The spatial Fourier transform of the vector Helmholtz equations Eqs. (11.3) are easily obtained directly from Eqs. (11.6). We find that K × K × ˜ E(K) + k 2 0 ˜ E(K) = −iωμ 0 ˜ J(K), (11.8a) K × K × ˜ H(K) + k 2 0 ˜ H(K) = −iK × ˜ J(K). (11.8b) 11.2 The Helmholtz Theorem The Helmholtz Theorem states that any suitably well-behaved vector field V(r) can be uniquely decomposed into the sum of a longitudinal part V L (r) that has zero curl and a transverse part V T (r) that has zero divergence; i.e., V(r) = V L (r) + V T (r), where ∇ × V L (r) = 0, ∇ · V T (r) = 0. (11.9) The proof of the theorem is somewhat tedious in the space domain but there is a simple and elegant proof for vector-valued fields that admit a spatial Fourier decomposition for which the relationship Eq. (11.5) holds and that we will assume in the following treatment. 462 The electromagnetic field To establish the theorem we begin by using the vector identity K × K × ˜ V(K) ≡ K[K · ˜ V(K)] − K 2 ˜ V(K), (11.10) where ˜ V(K) is the spatial Fourier transform of the field V(r).

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  Mechanics of Deformable Bodies

  Lectures on Theoretical Physics

  • Arnold Sommerfeld(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  C H A P T E R I V V O R T E X T H E O R Y 18. Helmholtz's Vortex Theorems In the paper of 1858 in which Helmholtz gives the kinematic analysis of vortex motion (cf. the footnote on p. 1) he also completes the dynamic theory of vortices in its essentials. Simplifications in method were found in the following decades, but new results were not discovered. The main content of Helmholtz's theory are the conservation laws: 1 It is impossible to produce or destroy vortices, or, expressed in more general terms, the vortex strength is constant in time. This theorem is correct under the following conditions: the fluid is inviscid and incompressible; the external forces possess a single-valued potential within the space filled by the fluid. Apart from the conservation of the vortex strength in time we shall see that there is also a spatial conservation: the vortex strength is constant along each vortex line or vortex tube, which must be either closed or end at the boundary of the fluid. 1. The Differential Form of the Conservation Theorem Following Helmholtz we start from Euler's equations in the form (11.2). Since the external force F has a potential U by hypothesis, Euler's equations can be written The differential quotient on the left side is again the material acceleration, that is, the total rate of change of the velocity of the material particle under consideration. This leads to the form (11.7) of Euler's equations which can now be written in the following way: Corresponding theorems have been found in posthumous papers of Lejeune Dirichlet, +1859, and were published by Dedekind who edited Dirichlet's collected (i) (2) works. 129 130 MECHANICS OF DEFORMABLE BODIES [ I V . 18] W e take the curl of this equation: the right member vanishes because of curl grad = 0. On setting w = 5 curl v we obtain (3) ^ - curl (v X 6>) = 0.

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  Foundations of Classical Mechanics

  • P. C. Deshmukh(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  The above uniqueness theorem ensures that this can indeed be achieved, from the divergence and the curl of each of the two fields, E (r , t) and B (r , t), with suitable boundary conditions and initial values. The Helmholtz Theorem further guarantees that the electromagnetic fields are expressible as a sum of a solenoidal vector field and an irrotational vector field. The vector fields (E (r , t), B (r , t)) are then derivable from appropriate scalar and vector fields, called the scalar and vector potentials, (f(r , t), A (r , t)). 425 Basic Principles of Electrodynamics We shall now introduce the four fundamental equations of electrodynamics, the Maxwell’s equations. These are, as stated above, partial differential coupled equations, in which the dynamics of the electric field intensity E (r , t) and that of the magnetic flux density field B (r , t) is coupled, making them inseparable entities of a larger physical entity, referred to as the electromagnetic field tensor of rank 2, which we shall introduce in the next chapter. Now, the divergence of a vector field is a scalar point function, and the curl of a vector field is a vector point function. The left hand sides of the Maxwell’s equations stand for the divergence and the curl of the electromagnetic fields (E (r , t), B (r , t)), and the right hand sides provide us information about the physical quantities which represent the corresponding scalar point function (in the case of the divergence of the fields), and the vector point function (in the case of the curl of the fields). The equations for the divergence of the electric field intensity E (r , t) and the magnetic flux density field B (r , t) are     ∇ ⋅ = E r t r t ( , ) ( , ) , ρ ε 0 (12.16a) and ∇ ⋅ B (r , t) = 0. (12.16b) The discussion below will associate the first equation of Maxwell, Eq.

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  Theory and Computation of Electromagnetic Fields

  • Jian-Ming Jin(Author)
  • 2015(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Equation (1.1.37) is referred to as the generalized Gauss’ theorem , from which we can easily derive many integral theorems. For example, if we let T ( ̃ ∇) = ̃ ∇ ⋅ 𝐟 = ∇ ⋅ 𝐟 , we obtain the standard Gauss’ theorem in Equation (1.1.5). If we let T ( ̃ ∇) = ̃ ∇ × 𝐟 = ∇ × 𝐟 , we obtain the so-called curl theorem ∫∫∫ V ∇ × 𝐟 d V = ∫ ∫ S d 𝐒 × 𝐟 (1.1.38) from which we can also derive Stokes’ theorem given in Equation (1.1.11) by applying it to a surface with a vanishing thickness. ◾ EXAMPLE 1.1 Using the generalized Gauss’ theorem, derive a new integral theorem ∫∫∫ V ( 𝐛 ∇ ⋅ 𝐚 + 𝐚 ⋅ ∇ 𝐛 ) d V = ∫ ∫ S ( ̂ n ⋅ 𝐚 ) 𝐛 d S . Solution Based on the expression of the right-hand side, we let T ( ̂ n ) = ( ̂ n ⋅ 𝐚 ) 𝐛 . The corresponding symbolic expression is T ( ̃ ∇) = ( ̃ ∇ ⋅ 𝐚 ) 𝐛 , which can further be written as T ( ̃ ∇) = ( ̃ ∇ a ⋅ 𝐚 ) 𝐛 + ( ̃ ∇ b ⋅ 𝐚 ) 𝐛 = ( ̃ ∇ a ⋅ 𝐚 ) 𝐛 + ( 𝐚 ⋅ ̃ ∇ b ) 𝐛 = 𝐛 ∇ ⋅ 𝐚 + 𝐚 ⋅ ∇ 𝐛 REVIEW OF VECTOR ANALYSIS 9 where we have applied the chain rule in Equation (1.1.28) and the relationship between ̃ ∇ and the divergence and gradient operations. The new integral theorem is then obtained by substituting the expressions of T ( ̃ ∇) and T ( ̂ n ) into the generalized Gauss’ theorem in Equation (1.1.37). 1.1.3 Helmholtz Decomposition Theorem In vector analysis, there are two special vectors. One is called the irrotational vector, whose curl vanishes. Denoting this vector as 𝐅 i , we have ∇ × 𝐅 i = 0 , ∇ ⋅ 𝐅 i ≠ 0 . (1.1.39) Another special vector is called the solenoidal vector, whose divergence is zero. Denoting this vector as 𝐅 s , we have ∇ ⋅ 𝐅 s = 0 , ∇ × 𝐅 s ≠ 0 . (1.1.40) Using the symbolic vector method, we can easily prove the following two very important vector identities: ∇ × (∇ 𝜑 ) = 0 (1.1.41) ∇ ⋅ (∇ × 𝐀 ) = 0 . (1.1.42) These identities are valid for any continuous and differentiable scalar function 𝜑 and vector function 𝐀 . Clearly, ∇ 𝜑 is an irrotational vector and ∇ × 𝐀 is a solenoidal vector.

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  The Theory of Electromagnetism

  • D. S. Jones, I. N. Sneddon, S. Ulam, M. Stark(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  In most applications we need the rate at which energy is crossing a closed surface. H A R M O N I C W A V E S 1.26 Helmholtz's theorem Let us now consider the fields which are produced by currents and charges whose variation with time is simple harmonic. Suppose that the period of oscillation is 2π/ω a n d t h a t ρ (R, t) = f (R) cos (cot + oc). Then, from (105), ' / (R') cos { ωί -(ω/ν) |R - R'| + oc) άτ' V(K,t)=--L -f 4 π ε J We may also write | R -R ' T i(a>t + a) /· j /-D/x F ( R , i ) = * ^ r / -i ^ ^ T T e-f «-!«-»> d r ' 4πε J |R — R'| T where i = ]/— 1 and 0t means take the real part. The vector potential may be written similarly. It is advantageous, as in the treatment of forced oscillations in the theory of mechanics, to omit the symbol 0t during analysis and to supply it at the end. Thus we write the field as E A (R)e ÎÛ,i , H^(R) ?coi with the understanding that the real part is to be taken. Here Ε Λ and H h may be complex functions of R because of the phase oc and the time derivative of the vector potential. This represen-tation is permissible provided that only linear operations are involved, i.e. operations in which it is immaterial whether taking the real part is done before or after the operation. Typical examples of such linear operations are addition, subtraction, integration and taking a derivative. Clearly the operations involved in forming Maxwell's equations are linear. Accordingly, in a harmonic field, we put ρ = ρ Α β < * ί > J = J Ä e i e and similarly for other quantities, the suffix h indicating the part independent of time. The equation of continuity ot 1 C. 0. HINES, loc. cit. and J. Geophys. Res., 56, 535 (1951) has also argued that the Poynting vector does not supply the correct propagation of energy by a wave packet, but his criticism is scarcely valid since he considers only the energy of the individual waves and not how they combine to form a packet. See also §10.10. T.O.E.J. 5

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  Symmetry and Separation of Variables

  • Willard Miller(Author)
  • 1984(Publication Date)
  • Cambridge University Press

    (Publisher)

  CHAPTER 1 The Helmholtz Equation 1.0 Introduction The main ideas relating the symmetry group of a linear partial differen- tial equation and the coordinate systems in which the equation admits separable solutions are most easily understood through examples. Perhaps the simplest nontrivial example that exhibits the features we wish to illustrate is the Helmholtz, or reduced wave, equation (A 2 + co 2 )*(;c,.y)=0 (0.1) where co is a positive real constant and (Here d xx ^ is the second partial derivative of SP" with respect to x.) In this chapter we will study the symmetry group and separated solu- tions of (0.1) and related equations in great detail, thereby laying the groundwork for similar treatments of much more complicated problems in the chapters to follow. For the present we consider only those solutions SP of (0.1) which are defined and analytic in the real variables x 9 y for some common open connected set ^D in the plane R 2 . (For example, ^D can be chosen as the plane itself.) The set of all such solutions >P forms a (complex) vector space %; that is, if *£% and a E ^ , then {a^){x,y) = a^{x,y)^% 9 and (ty l +y 2 )(x,y) = < }' l (x,y) + y i' 2 (x,y)G% whenever ty l9 ty 2 E%. Considering ^D as fixed throughout our discussion, we call % the solution space of (0.1). Let § be the vector space of all complex-valued functions defined and 8 ENCYCLOPEDIA OF MATHEMATICS and Its Applications, Gian-Carlo Rota (ed.). 2 Vol. 4: Willard Miller, Jr., Symmetry and Separation of Variables o Copyright © 1977 by Addison-Wesley Publishing Company, Inc., Advanced Book Program. © All rights reserved. No part of this publication may be reproduced, stored in a retrieval Z system, or transmitted, in any form or by any means, electronic, mechanical photocopying, 22 recording, or otherwise, without the prior permission of the publisher. 1 2 The Helmholtz Equation 1.1. real analytic on ^D and let Q be the partial differential operator £ = A 2 +

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  Inverse Problems and Inverse Scattering of Plane Waves

  • D.N. Roy, L. S. Couchman(Authors)
  • 2001(Publication Date)
  • Academic Press

    (Publisher)

  (6.24) Equation (6.23) or (6.24) is referred to as the Helmholtz equation which plays a key role in acoustics, optics, heat flow, diffusion, wave processes, and so on. 3 In a general way, the Helmholtz equation is a special case of the so-called telegrapher’s equation (Koshlyakov et al. , 1964) which is Δ φ = α 0 ¨ φ + 2 α 1 ˙ φ + α 2 φ. (6.25) Separating the variables, i.e., writing the solution as φ ( x, t ) = ψ ( x ) ζ ( t ) and then substituting back in Eq. (6.25) leads to the Helmholtz equation (Δ + k 2 ) ψ ( x ) = 0 , and α 0 ¨ ζ + 2 α 1 ˙ ζ + ( α 2 − k 2 ) ζ = 0 . The constant k is the usual constant that arises when the equations are separated in their individual variables. Various equations are obtained by manipulating the coefficients α 0 , α 1 and α 2 . For example, the wave equation is recovered 3 In many texts, especially, the ones that are more mathematically inclined, the Helmholtz operator is written as − Δ − k 2 instead of Δ + k 2 . There are certain theoretical reasons for writing − Δ instead of Δ. It automatically ensures the positivity of many quantities. For example, for a positive source function f in Poisson’s equation written as − Δ u = f , u = 0 on Γ, the solution is also positive. The corresponding Green’s function is positive as also the eigenvalues and the Fourier symbol . Also an elliptic equation often describes the steady-state of an otherwise transient system in the asymptotic limit of large t . The principle of limiting amplitude formulation of the radiation condition (see Section 6.6) serves as an example. The time-dependent equation of which the elliptic equation is the steady-state limit, is often of the form ˙ u − Δ u = f , resulting in Poisson’s equation with a negative Laplacian. One further point is worth mentioning. If the Helmholtz operator is written as Δ − k 2 instead of Δ+ k 2 , then the corresponding Helmholtz equation (Δ − k 2 ) u = 0 is called the mod-ified Helmholtz equation.

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  Mechanics of Deformable Bodies

  Lectures on Theoretical Physics, Vol. 2

  • Arnold Sommerfeld(Author)
  • 2016(Publication Date)
  • Academic Press

    (Publisher)

  [IV. 20] VORTEX THEORY 147 mann Eqs. (5), which is not the case because of the factor p in (24). Also, Φ and Ψ would have to satisfy the equation d 2 u/dp 2 + d 2 u/dz 2 = 0, but actually Φ satisfies Eq. (23) and Ψ the equation obtained from curl v = Oby (25): (26) ' o 7 ^ ; ^ + l? = = 0 · The powerful tool of the theory of complex functions cannot be used in three-dimensional potential theory. 7 20. A Fundamental Theorem of Vector Analysis The theorem which we wish to prove here is this: A continuous vector field V, defined everywhere in space and vanishing at infinity together with its first derivatives, can be represented as the sum of an irrotational field V 2 and a solenoidal field V 2 : (1) V = V t + V 2 , where (la) curl V! = 0, div V 2 = 0. Our decomposition charges all sources and sinks of the given field V to the component field V! and all vortices to the component field V 2 so that: (lb) div Vi = div V, curl V 2 = curl V. The representation (1) is unique except for a vectonal constant. This fundamental theorem was proved in its essentials by Stokes 8 in The great mathematician David Hilbert (1862-1943) sharply characterized the futility of all attempts in this direction by the following remark: time is one-dimensional, space is three-dimensional, however the number, that is, the perfect complex number, has two dimensions. 8 In his big paper: On the dynamical theory of diffraction, Trans. Cambridge Phil. Soc. Vol. 9, p. 1, reprinted in Math, and Phys. Papers, Vol. II, Cambridge, 1883, p. 243 and particularly p. 254 ff. Stokes does not yet give an explicit definition of the vector potential A. Instead, V 2 is calculated by the following formula obtained through a combination of our equations (5) and (9) . » r C curl curl V , 47rVa = i r dT · For a rigorous proof see: O. Blumen thai, Ueber die Zerlegung unendlicher Vektorfelder, (Mathem. Annalen 61, 235, 1905).

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