Integration theorems over surfaces

The second term is a divergence, so that its integral becomes, by the divergence theorem, a surface integral over the boundary of "K of an integrand containing as a factor the normal component of the material flux vector pwa and this vanishes, according to our boundary conditions. 

Now integrate over all space. By the vector theorem (j), Table 1.4.2, and with v ExH, J d3rV v — J d2r v n for any vector v, being the outer surface normal, the first term is transformed to a surface integral extending over infinitely remote boundaries and thus vanishes. This leaves... 

This will then be converted from the integral over the whole surface area of the region to the volume integral of the divergence j A according to Gauss integral theorem. 

Green s theorem (Helmholtz-Kirchhoff integral theorem) Representation of the field by means of an integral extending over a closed surface containing the field point. 

The three most us ul equations for determining the density at the gas-liquid surface are the tirst YBG equation (4.34), the integral equation over the direct correlation function (4.S2), and the potential distribution theorem (4.69). At a planar surface with K- 0 these can be re-written in simpler forms. If the direction normal to the surface (the heig ) is that of the z-axis, then the only non-zero gradient is d/dz and we have cylindrical symmetry about this axis. The volume element can then be more usefully expressed in cylindrical or spherical polar coordinates. 

The surface integral in Eq. (3.32b) can be transformed into a volume integral over the enclosed domain by using Gauss s integral theorem, Eq. (3.7). Equating with Eq. (3.32a) leads to the electrostatic equilibrium condition, which is known as one of Maxwell s equations in integral form ... 

Equation (3.9) is the Reynolds transport theorem. It displays how the operation of a time derivative over an integral whose limits of integration depend on time can be distributed over the integral and the limits of integration, i.e. the surface, S. The result may appear to be an abstract mathematical operation, but we shall use it to obtain our control volume relations. 

The divergence of the flux vector is therefore the net rate of accumulation of the quantity which is transported in and out of the volume element dK This can be integrated over an arbitrary volume Cl limited by the surface I to give the divergence theorem of Gauss... 

The boundary surface of a region in space is an important physical quantity. The integral of a field in the region is related by the fundamental theorem of calculus to an integral over its boundary surface. A surface integral can be approximated by summing quantities associated with a subdivision of the surface into patches. In the present work, the surface patches are taken to be the (approximate) exposed surface area of atom in a molecule. 

Using Green s theorem, it can be converted into a volume integral over fir, the tip side from the separation surface. Noticing that the sample wavefunction ip satisfies Schrodinger s equation, Eq. (3.2), in fir, and that the Green s function satisfies Eq. (3.8), we obtain immediately... 

The Gauss-Bonnet theorem, which relates integrals of Gaussian curvature (1/(/ii) in three dimensions) over a surface to integrals of mean curvature (1/iii + I/R2 in three dimensions) over boundaries of the surface, is particularly simple in two dimensions. In two dimensions, the (N — 6)-rule is equivalent to the Gauss-Bonnet theorem. 

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 ... 

The notation Wtl defines a Wronskian integral over ct/2. By the surface matching theorem, xm = X> the interior component of 4> Since 4r is a solution of the Lippmann-Schwinger equation, this implies xout = Xm when evaluated in the interior of Tjj. This is a particular statement of the tail-cancellation condition. To show this in detail, after integration by parts... 

This problem can be avoided by expressing the MST equations in terms of the square matrix S C = CVS, Hermitian in consequence of the surface-matching theorem. This matrix has full rank because it is contracted over the larger index lg. From the definitions of the C and S matrices, the matrix product S C is a specific integral involving the Helmholtz Green function [281],... 

The first boundary condition at the surface is provided by integrating the Poisson Eqs. (la) and (lb) over the volume of a flat box, which includes the surface, with the large sides parallel to the surface and a vanishingly thin width. After using the Gauss theorem, one obtains ... 

Now integrate over all space. By Gauss theorem, / d r V (f x Ti) may be transformed into a surface integral enveloping the fields the surface integration can be extended to infinitely remote boundaries where the fields ultimately vanish. This term therefore drops out, leaving... 

With this theorem we can introduce the measure of the difference between the observed and predicted fields as the energy flow of the residual field through the surface of observations, integrated over time t ... 

Note that, due to radiation conditions, the surface integral over the sphere On goes to zero if the radius R tends to infinity, and the volume integral is equal to G (r r" w). As a result we arrive at the following important theorem. 

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