Stoke’s theorem

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. 

Integrating this equation over a face S of the prism, bounded by a rectangular contour L, and using Stokes s theorem, we find... 

Stokes s theorem Let 5 be a surface bounded by a closed curve C and dr be the tangent vector. Then... 

The path L can have an arbitrary shape, and it can intersect media characterized by various physical properties. In particular, it can be completely contained within a conducting medium. Because of the fact that the electromotive force caused by electric charges is zero, a Coulomb force field can cause an electric current by itself. This is the reason why non-Coulomb forces must be considered in order to understand the creation of current flow. Equation 1.46 is the first Maxwell equation for electric fields which do not vary with time, given in its integral form, and relates the values of the field various points in the medium. To obtain eq. 1.46 in differential form, we will make use of Stoke s theorem, according to which for any vector A having first spatial derivatives, the following relationship holds ... 

As has been previously mentioned, Stoke s theorem (eq. 1.46) is valid only when the first spatial derivative exist. Thus, this equation cannot be used at points where one of the components is a discontinuous function of position. In order to obtain a differential form of eq. 1.46 valid at such points, we will apply this equation along an elementary path... 

Figure 1.19. Application of Stoke s theorem along an elementary path.

Numerous experiments have shown the appropriateness of selecting the vector X in this form. This quantity was called a displacement current. As follows from the second Maxwell equation, there are two sources for the magnetic field conduction currents and displacement currents. Applying Stoke s theorem, we obtain the integral form of the second Maxwell equation ... 

Next, making use of Stoke s theorem, we obtain the differential form of the first Maxwell equation ... 

In previous sections, by making use of Gauss s and Stoke s theorems, we have developed the basic laws for the electromagnetic fields in the form of equations. In accord with these laws the electromagnetic field must satisfy the following set of equations ... 

The fact that (8-49) vanishes via Stokes s theorem implies the following three scalar equations ... 

Physical interpretation of vorticity is facilitated by application of Stokes s theorem. This theorem states that for an area A with normal vector n enclosed by a curve or path P with contour element dr,... 

Therefore, the circulation is the line integral about the contour P of the component of the velocity tangent to the contour and is another measure of the rotation of the fluid. Stokes s theorem provides the relationship to vorticity the circulation divided by the area equals the average normal component of vorticity in the area. 

By using Stokes s theorem, we can express the line integral of A as a surface integral of B which shows that the geometric phase (/ g) is equal to the flux of the magnetic field through the surface S enclosed by C... 

If S is an orientable surface bounded by a closed curve C, the orientation of the closed curve C is chosen to be consistent with the orientation of the surface S. Then we have Stokes s theorem ... 

Integrating the seeond equation over an area S which is bounded by a contour C and using Stokes s theorem leads to... 

Maxwell s equations can also be put in integral form. Specifically, integrating both sides of each equation, over a volume enclosed by a surface S for the first two equations and over a surface enclosed by a curve C for the last two equations, and using Gauss s theorem or Stokes s theorem (see Appendix G) as appropriate, we find that the equations in polarizable matter take the form... 

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... 

In two dimensions, Stokes theorem reduces to Green s theorem (10.75) ... 

Note that Faxen s theorem is valid for any flow field uj(r) satisfying the Stokes equations. Using Eq. (9.23) in Eq. (9.24) and comparing with Eq. (9.21), one obtains... 

Taking into account certain restrictions originating from the symmetry properties of isotropic liquids at equilibrium (Curie s theorem), after some tensor algebra we obtain the Navier-Stokes equations for single-component atomic fluids. 

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

Before we engage in the non-Abelian Stokes theorem it seems reasonable to recall its Abelian version. The (Abelian) Stokes theorem says (see, e.g., Ref. 1 for an excellent introduction to the subject) that we can convert an integral around a closed curve C bounding some surface S into an integral defined on this surface. Specifically, in three dimensions... 

A physicist would view the expression (10) as typical in quantum mechanics and as corresponding to the evolution operator. Equations (8) and (9) are, incidentally, very typical in gauge theory, such as in QCD. Thus, guided by our intuition, we can reformulate our chief problem as a quantum-mechanical one. In other words, the approaches to the l.h.s. of the non-Abelian Stokes theorem are analogous to the approaches to the evolution operator in quantum mechanics. There are the two main approaches to quantum mechanics, especially to the construction of the evolution operator opearator approach and path-integral approach. Both can be applied to the non-Abelian Stokes theorem successfully, and both provide two different formulations of the non-Abelian Stokes theorem. 

Unfortunately, it is not possible to automatically generalize the Abelian Stokes theorem [e.g., Eq. (4)] to the non-Abelian one. In the non-Abelian case one faces a qualitatively different situation because the integrand on the l.h.s. assumes values in a Lie algebra g rather than in the field of real or complex numbers. The picture simplifies significantly if one switches from the local language to a global one [see Eq. (5)]. Therefore we should consider the holonomy (7) around a closed curve C ... 

We will apply the Abelian Stokes theorem to the exponent of the integrand of the path integral yielding the r.h.s. of the non-Abelian Stokes theorem [a counterpart of the r.h.s. in Eq (8)]. 

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