Appendix A

A.1 Miscellaneous Functions

The Kronecker Delta Function

δij=

{

1 ,j=i

0 ,j=i.

q

∑

j=p

xjδjr=

{

0 ,p≤r≤q,

0 , otherwise.

q

∑

j=p

xjδjr(1−δjr)=xr.

In=[δij]n, the unit matrix,

n

∑

r=1

∣

∣

∣

∣

fr δir

gr δjr

∣

∣

∣

∣

=

∣

∣

∣

∣

fj 1

gi 1

∣

∣

∣

∣

, 1 ≤i, j≤n,

n

∑

r=1

∣ ∣ ∣ ∣ ∣ ∣

aip aiq δir

ajp ajq δjr

akp akq δkr

∣ ∣ ∣ ∣ ∣ ∣

=

∣ ∣ ∣ ∣ ∣ ∣

aip aiq 1

ajp ajq 1

akp akq 1

∣ ∣ ∣ ∣ ∣ ∣

, 1 ≤i, j, k≤n.

δi,even=

{

1 ,ieven,

0 ,iodd.