We now turn to the diagrammatic representation of the invariant matrix elements which are given in figures 2.5a, or when working both in angular momentum and isospin spaces in figures 2.5b. We have explicitly shown on these diagrams that the upper line corresponds to the bra function. The the understanding that matrix element) at the ~ sign will in general be omitted from the diagram with the bra line must always enter a hatched box (invariant top. An example of this rule, which in fact yields a phase only in the case of half-integer spins, is given below. Finally we also need the insertion of a complete normalized set of states which is represented by the diagram of figure 2.6 which fulfills the completeness relation of figure 2.7. Here a denotes all other quantum numbers (besides) which define the basis set of states. The crossing sign box in Figure 2.7 is not needed for integer angular momentum. Here we collect the invariant forms of vector algebra and vector ana lysis. They are based on the invariant representations (2.42) (2.43) is the unit vector related to the Cartesian unit vector by i.e. the divergence of a vector is obtained by replacing in the vector expression (2.42) the components of the unit vector by the respective components of the gradient. Hence for [10] (2.48) where we have used the Fano-Racah notation for the invariant triple product. The factor ✓2 arises because of the different normalization implied by the Cartesian cross product and the coupling of two vectors to for the above function F we get 1. Generally iv) Special cases We now give the formulae for a few special cases which will be relevant latter on. Let us define the function which diagrammatically is represented by the diagram of figure 2.9 (2.52) (2.53) J' O J J (2.54) From now on we shall read off directly from the diagram the completeness summations and integrations, carry out directly the triangular condition (J'JO) and the integration over the continuous variable q which leads to p = q. = |