Gauss-Green Theorem

Theorem B.l (Gauss Theorem). Let S be the surface of G c then we have 

A domain G with a surface 5o, which is parallel to the z-axis is shown in Fig. B.2. The upper surface of G is 5i, and the lower one is S2. The surfaces and S2 are represented by the following equations  

Let a projection of G onto the (x, j)-surface be A. We have a unit outward normal 

Here we have assumed that the function / is sufficiently smooth. We can obtain the same results for the x and y directions, so that we have Gauss Theorem (B.6). Theorem B.2 (Divergence Theorem).  

We note that in the one-dimensional case, (B.8) is equivalent to the following integration-by-parts  

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We integrate Eq. 8.46 by part, with help of the Gauss-Green theorem, to obtain... 

This relationship is called the second Green s formula and it represents Gauss s theorem when the vector X is given by Equation (1.98). In particular, letting ij/ — constant we obtain the first Green s formula ... 

Using the Gauss law and die Green theorem, the electrostatic contribution can be rewritten as... 

FIGURE 1 Definition of surface and volume integral and Gauss-Green divergence theorem. 

Adding Eqs. (63), (64), and (67) produces the Gauss-Green divergence theorem. That is,... 

The Gauss theorem is also known as the divergence theorem. Green s theorem, and Ostrogradsky s theorem [36]. In particular, the vector form of Gauss s theorem is normally referred to as the divergence theorem [18]. 

Divergence theorem (Gauss s theorem or Green s theorem)... 

By taking an inner product between v and (4.19a), applying Gauss-Green s theorem to the first term, and then applying the conditions (4.19c) and (4.22), we have... 

Again, we must apply a Green-Gauss transformation described in Theorem (9.1.2) to the second spatial derivative terms to get... 

The first two terms of the above equation are transformed using the Green-Gauss Theorem (9.1.2) which results i... 

Next, we apply Galerkin s weighted residual method and reduce the order of integration of the various terms in the above equations using the Green-Gauss Theorem (9.1.2) for each element. For a simpler presentation we will deal with each term in the above equations separately. The terms of the x-component (eqn. (9.95)) of the penalty formulation momentum balance become... 

Defining f = Vip into Gauss theorem and using the chain rule for the divergence of the vector, the so-called Green s first identity is obtained (Theorem (10.1.2)). This identity is also valid when we use it for a vector g = when we substract the fist identity for g to the first identity of f we obtain the Green s second identity (Theorem (10.1.3)). 

In order to obtain Green s identities for the flow field (u,p), a vector z is defined as the dot product of the stress tensor a(u, p) and a second solenoidal vector field v (divergence-free). The divergence or Gauss Theorem (10.1.1) is applied to the vector z... 

Making use of a divergence theorem for curved surfaces due to Green, Gauss, and Ostrogradsky (cf. Weatherburn [28]), we obtain ... 

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