Curl theorem

Here a again denotes the outer normal unit vector perpendicular to the surface S. Eq. (A.20) is sometimes also referred to as the curl theorem. 

Across the interface separating two different media the fields may be discontinuous and a boundary condition is associated with each of Maxwell s equations. To derive the boundary conditions, we consider a regular domain D enclosed by a surface S with outward normal unit vector n, and use the curl theorem... 

In the non-Abelian Stokes theorem (482), on the other hand, the boundary conditions are defined because the phase factor is path-dependent, that is, depends on the covariant derivative [50]. On the U(l) level [50], the original Stokes theorem is a mathematical relation between a vector field and its curl. In 0(3) or SU(2) invariant electromagnetism, the non-Abelian Stokes theorem gives the phase change due to a rotation in the internal space. This phase change appears as the integrals... 

Therefore, on the 0(3) level, the magnetic part of the complete free field is defined as a sum of a curl of a vector potential and a vacuum magnetization inherent in the structure of the B cyclic theorem. On the U(l) level, there is no B(3) field by hypothesis. 

Stokes theorem states that the circulation of a vector v around the boundary s of a closed surface A is equal to the flux of the curl of the... 

Helmholtz s theorem states that A general vector field, that vanishes at infinity, can be completely represented as the sum of two independent vector fields, one that is irrotational (zero curl) and another that is solenoidal (zero divergence) [Morse and Feshbach, 1953 Plonsey and CoUin, 1961]. The impressed current density P is a vector field that vanishes at infinity and, according to the theorem, maybe expressed as the sum of two components ... 

Stokas theorem A theorem that is the analogue of the divergence theorem for the curl of a vector. Stokes theorem states that if a surface S, which is smooth and simply connected (i.e. any closed curve on the surface can be contracted continuously into a point without leaving the surface), is bounded by a line L the vector P defined in S satisfies JsCurlf-dS= JiF-df. 

The vorticity equation describes how vorticity is changed by various properties of the flow. Only in very special circumstances would the vorticity be conserved following the flow. Kelvin s circulation theorem describes how an integral measure of vorticity is conserved but is valid only for barotropic flow and furthermore requires a knowledge of the time evolution of material surfaces. There does exist a quantity, referred to as the Ertel potential vorticity, that is conserved under more general conditions than either the vorticity or the circulation. It may be shown by combining the curl of the momentum equation [Eq. (26a)] with the continuity equation [Eq. (26c)] and the thermodynamic equation [Eq. (26b)] expressed in terms of potential temperature 0 that... 

Another formulation of Bernoulli s theorem exists when the flow is irrotational (i.e. the curl of the velocity is zero at every point in the domain). This allows application to unsteady flows such as the propagation of waves on the sea. For a steady, irrotational flow, the head is the same at every point in the domain, and it is no longer necessary to check that the two points between which Bernoulli s theorem is being applied are coimected by a streamline. However, the assumption of an irrotational flow is a substantial restriction it precludes the recirculation depicted in Figure 2.2, for which the problem posed for the application of Bernoulli s theorem was already indicated. 

A different approach to the problem has been given by Pitzer [1955] and Pitzer, Lippmann, Curl, Huggins and Petersen [1955]. These authors also give a theorem of corresponding states for polyatomic molecules, containing three parameters. These parameters however have no simple molecular meaning (such as e, r and but are empirical. They choose the critical temperatute, the critical pressure and the vapour pressure at a specified reduced temperature. Formally... 

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