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Helmholtz Theorem (vector Calculus)
There are several theorems known as the Helmholtz theorem: * Helmholtz decomposition, also known as the fundamental theorem of vector calculus * Helmholtz reciprocity in optics * Helmholtz theorem (classical mechanics) * Helmholtz's theorems in fluid mechanics * Helmholtz minimum dissipation theorem In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868) states that ''the steady Stokes flow, Stokes flow motion of an Incompressible flow, incompressible fluid has the smallest rate of ... See also * Helmholtz–Thévenin theorem {{mathematical disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Decomposition
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational ( curl-free) vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz. Definition For a vector field \mathbf \in C^1(V, \mathbb^n) defined on a domain V \subseteq \mathbb^n, a Helmholtz decomposition is a pair of vector fields \mathbf \in C^1(V, \mathbb^n) and \mathbf \in C^1(V, \mathbb^n) such that: \begin \mathbf(\mathbf) &= \mathbf(\mathbf) + \mathbf(\mathbf), \\ \mathbf(\mathbf) &= - \nabla \Phi(\mathbf), \\ \nabla \cdot \mathbf(\mathbf) &= 0. \end Here, \Phi \in C^2(V, \mathbb) is a scalar potential, \nabla \Phi is its gradient, and \nabla \cdot \mathbf is the divergence of the vector fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Reciprocity
The Helmholtz reciprocity principle describes how a ray of light and its reverse ray encounter matched optical adventures, such as reflections, refractions, and absorptions in a passive medium, or at an interface. It does not apply to moving, non-linear, or magnetic media. For example, incoming and outgoing light can be considered as reversals of each other,Hapke, B. (1993). ''Theory of Reflectance and Emittance Spectroscopy'', Cambridge University Press, Cambridge UK, , Section 10C, pages 263-264. without affecting the bidirectional reflectance distribution function (BRDF) outcome. If light was measured with a sensor and that light reflected on a material with a BRDF that obeys the Helmholtz reciprocity principle one would be able to swap the sensor and light source and the measurement of flux would remain equal. In the computer graphics scheme of global illumination, the Helmholtz reciprocity principle is important if the global illumination algorithm reverses light paths ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Theorem (classical Mechanics)
The Helmholtz theorem of classical mechanics reads as follows: Let H(x,p;V) = K(p) + \varphi(x;V) be the Hamiltonian of a one-dimensional system, where K = \frac is the kinetic energy and \varphi(x;V) is a "U-shaped" potential energy profile which depends on a parameter V. Let \left\langle \cdot \right\rangle _ denote the time average. Let *E = K + \varphi, *T = 2\left\langle K\right\rangle _, *P = \left\langle -\frac\right\rangle _, *S(E,V)=\log \oint \sqrt\,dx. Then dS = \frac. Remarks The thesis of this theorem of classical mechanics reads exactly as the heat theorem of thermodynamics. This fact shows that thermodynamic-like relations exist between certain mechanical quantities. This in turn allows to define the "thermodynamic state" of a one-dimensional mechanical system. In particular the temperature T is given by time average of the kinetic energy, and the entropy S by the logarithm of the action (i.e., \oint dx \sqrt). The importance of this theorem has been reco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz's Theorems
In fluid mechanics, Helmholtz's theorems, named after Hermann von Helmholtz, describe the three-dimensional motion of fluid in the vicinity of vortex lines. These theorems apply to inviscid flows and flows where the influence of viscous forces are small and can be ignored. Helmholtz's three theorems are as follows: ;Helmholtz's first theorem: :The strength of a vortex line is constant along its length. ;Helmholtz's second theorem: :A vortex line cannot end in a fluid; it must extend to the boundaries of the fluid or form a closed path. ;Helmholtz's third theorem: :A fluid element that is initially irrotational remains irrotational. Helmholtz's theorems apply to inviscid flows. In observations of vortices in real fluids the strength of the vortices always decays gradually due to the dissipative effect of viscous forces. Alternative expressions of the three theorems are as follows: # The strength of a vortex tube does not vary with time. # Fluid elements lying on a vortex line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Minimum Dissipation Theorem
In fluid mechanics, Helmholtz minimum dissipation theorem (named after Hermann von Helmholtz who published it in 1868) states that ''the steady Stokes flow, Stokes flow motion of an Incompressible flow, incompressible fluid has the smallest rate of dissipation than any other incompressible motion with the same velocity on the boundary''. The theorem also has been studied by Diederik Korteweg in 1883 and by Lord Rayleigh in 1913. This theorem is, in fact, true for any fluid motion where the nonlinear term of the incompressible Navier-Stokes equations can be neglected or equivalently when \nabla\times\nabla\times\boldsymbol=0, where \boldsymbol is the vorticity vector. For example, the theorem also applies to unidirectional flows such as Couette flow and Hagen–Poiseuille equation, Hagen–Poiseuille flow, where nonlinear terms disappear automatically. Mathematical proof Let \mathbf,\ p and E=\frac(\nabla\mathbf+(\nabla\mathbf)^T) be the velocity, pressure and strain rate tensor of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |