Stokes’ theorem

These are the two components of the Navier-Stokes equation including fluctuations s., which obey the fluctuation dissipation theorem, valid for incompressible, classical fluids ... 

Papanastasiou et al. (1992) suggested that in order to generate realistic solutions for Navier-Stokes equations the exit conditions should be kept free (i.e. no outflow conditions should be imposed). In this approach application of Green s theorem to the equations corresponding to the exit boundary nodes is avoided. This is eqvrivalent to imposing no exit conditions if elements with... 

Analogous to vector operations the tensorial fonn of Stokes theorem is written as... 

Using the pressure of an ideal fluid, we see that the total force acting on a volume of fluid due to the rest of the fluid is given by — fgP dS. Applying Stokes Theorem, we then have that... 

The Navier-Stokes equations are recovered by substituting this first-order expression for the pressure tensor into the conservation theorem with n — mvi (i.e, into equation 9.55). 

Now we will establish a relationship between the potential U(p) at any point p of the volume V and its values on the spherical surface, surrounding all masses. Fig. 1.11. The reason why we consider this problem is very simple it plays the fundamental role in Stokes s theorem, which allows one to determine the elevation of the geoid with respect to the reference ellipsoid. 

This property is readily established from the definition of Fourier transform and convolution. In scattering theory this theorem is the basis of methods for the separation of (particle) size from distortions (Stokes [27], Warren-Averbach [28,29] lattice distortion, Ruland [30-34] misorientation of anisotropic structural entities) of the scattering pattern. 

Desmearing. In practice, there are two pathways to desmear the measured image. The first is a simple result of the convolution theorem (cf. Sect. 2.7.8) which permits to carry out desmearing by means of Fourier transform, division and back-transformation (Stokes [27])... 

Stokes theorem, geometric phase theory, eigenvector evolution, 14-17 Stueckelberg oscillations, direct molecular dynamics, trajectory surface hopping, 398-399... 

The latter equation can be interpreted to mean that the third Stokes parameter does not vary with time in a circularly polarized beam of light. The particular solution (87) gives the cyclic theorem (9) self-consistently [11-20]. 

In order to understand interferometry at a fundamental level in gauge field theory, the starting point must be the non-Abelian Stokes theorem [4]. The theorem is generated by a round trip or closed loop in Minkowski spacetime using covariant derivatives, and in its most general form is given [17] by... 

To reduce Eq. (153) to the ordinary Stokes theorem, the U(l) covariant derivative is used... 

The space part of this expression is the ordinary, or Abelian, Stokes theorem... 

The integral of /I over a hypersurface in four-dimensions is always zero, a result of the ordinary Stokes theorem in four dimensions ... 

From the foregoing, it becomes clear that fields and potentials are freely intermingled in the symmetry-broken Lagrangians of the Higgs mechanism. To close this section, we address the question of whether potentials are physical (Faraday and Maxwell) or mathematical (Heaviside) using the non-Abelian Stokes theorem for any gauge symmetry ... 

The non-Abelian Stokes theorem is a relation between covariant derivatives for any gauge group symmetry ... 

This is the Stokes theorem as usually found in textbooks. For plane waves, A is always perpendicular to the path, so in free space... 

The aim of this chapter is to present a short review of the non-Abelian Stokes theorem. At first, we will give an account of different formulations of the non-Abelian Stokes theorem and next of various applications of thereof. 

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