Stokes theorem properties

Barrett [50] has interestingly reviewed and compared the properties of the Abelian and non-Abelian Stokes theorems, a review and comparison that makes it clear that the Abelian and non-Abelian Stokes theorems must not be confused [83,95]. The Abelian, or original, Stokes theorem states that if A(x) is a vector field, S is an open, orientable surface, C is the closed curve bounding S, dl is a line element of C, n is the normal to S, and C is traversed in a right-handed (positive direction) relative to n, then the line integral of A is equal to the surface integral over 5 of V x A-n ... 

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. 

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 ... 

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. 

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