Page 58 - Determinants and Their Applications in Mathematical Physics

3.6 The Jacobi Identity and Variants 43

But also,
3
∂ A
= A ipu,jqv . (3.6.11)
∂a ij ∂a pq ∂a uv
Hence,

A ij A iq
A iv 2
= A A ipujqv , (3.6.12)
A pj A pq
A pv
A uj A uq A uv
which, when the parameter n is restored, is equivalent to (3.6.5). The for-
mula given in the theorem follows by scaling the cofactors. Note that those
2
Jacobi identities which contain scaled cofactors lack the factors A, A ,
etc., on the right-hand side. This simplification is significant in applications
involving derivatives.

Exercises
1. Prove that

A pt A qr,st =0,
ep{p,q,r}
where the symbol ep{p, q, r} denotes that the sum is carried out over
all even permutations of {p, q, r}, including the identity permutation
(Appendix A.2).
2. Prove that

A ps A pi,js A rj A rp,qj A iq A ir,sq

= = .
A rq A ri,jq A is A ip,qs A pj A pr,sj

3. Prove the Jacobi identity for general values of r by induction.
3.6.3 Variants

Theorem 3.6.
(n)
A A (n+1) (n+1)

ip i,n+1 − A n A =0, (A)
(n) (n+1) ij;p,n+1
A A
j,n+1
jp

(n)
A A (n) (n+1)

ip iq − A n A =0, (B)
(n+1) (n+1) i,n+1;pq
A A
n+1,p n+1,q

(n)
(n+1) (n+1) (n+1)
A rr A rr − A A =0. (C)
(n) rn;r,n+1
n+1,r
(n+1)
A nr A nr
These three identities are consequences of the Jacobi identity but are dis-
tinct from it since the elements in each of the second-order determinants
are cofactors of two different orders, namely n − 1 and n.