Formulae second Green

Making here the mutual replacements of u x) and v[x) and subtracting the resulting equation from (50), we establish the second Green formula in a more general form... 

For any y and u subject to the homogeneous boundary conditions yg = y = 0 and Uq = =0, the second Green formula takes a very elegant... 

The second Green formula. In the integral calculus the following formula... 

Substracting (9) from (8) we derive the difference analog of the second Green formula... 

To prove this fact, we make use of the second Green formula for the homogeneous boundary conditions (10 ) ... 

Due to the second Green formula it is self-adjoint. In turn, the first Green formula assures us of the validity of the relation... 

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 ... 

Here Lqp is the distance between points q and p. Note that G q, p) is called a Green s function. There are an infinite number of such functions and all of them have a singularity at the observation point p. Inasmuch as the second Green s formula has been derived assuming that singularities of the functions U and G are absent within volume V, we cannot directly use this function G in Equation (1.99). To avoid this obstacle, let us surround the point by a small spherical surface S and apply Equation (1.99) to the volume enclosed by surfaces S and S, as is shown in Fig. 1.9. Further we will be mainly interested by only cases, when masses are absent inside the volume V, that is. 

Taking into account Equations (1.101 and 1.103), the volume integral in the second Green s formula vanishes and we obtain... 

Some difference formulae. In the sequel, when dealing with various difference expressions, we shall need the formulae for difference differentiating of a product, for summation by parts and difference Green s formulae. In this section we derive these formulae within the framework similar to the appropriate apparatus of the differential calculus. Similar expressions were obtained in Section 2 of Chapter 1 in studying second-order difference operators, but there other notations have been used. It performs no difficulty to establish a relationship between formulae from Section 2 of Chapter 1 and those of the present section. 

The second reduction step of Fe(II)(TPP) yields an extremely air-sensitive green product which can be assigned the formula [Fe(I)(TPP) ] because of the red shift of the Soret band in the UV-VIS spectrum. The pure material is diamagnetic (S = 0) but this does not allow one to distinguish between the three possible descriptions as an iron(O) / -porpyhrin, a spin-coupled S = 1/2 iron(I)-porphyrin... 

The table shows the structural formulae of some sulfonamides. Some sulfonamides are antibacterial agents (shown in green in the first column of the table) and some (in the second column, in red) have no antibacterial activity. By examining both sets of structures you can see that the structural fragment that makes sulfonamide active is ... 

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