Determinants and Their Applications in Mathematical Physics

3.6 The Jacobi Identity and Variants 43

But also,

∂

3

A

∂aij∂apq∂auv

=Aipu,jqv. (3.6.11)

Hence,

∣ ∣ ∣ ∣ ∣ ∣

Aij Aiq Aiv

Apj Apq Apv

Auj Auq Auv

∣ ∣ ∣ ∣ ∣ ∣

=A

2

Aipujqv, (3.6.12)

which, when the parameternis restored, is equivalent to (3.6.5). The for-

mula given in the theorem follows by scaling the cofactors. Note that those

Jacobi identities which contain scaled cofactors lack the factorsA,A
2
,

etc., on the right-hand side. This simplification is significant in applications

involving derivatives.

Exercises

1.Prove that

∑

ep{p,q,r}

AptAqr,st=0,

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

rq

A

ri,jq

∣

∣

∣

∣

=

∣

∣

∣

∣

A

rj

A

rp,qj

A

is

A

ip,qs

∣

∣

∣

∣

=

∣

∣

∣

∣

A

iq

A

ir,sq

A

pj

A

pr,sj

∣

∣

∣

∣

3.Prove the Jacobi identity for general values ofrby induction.

3.6.3 Variants.........................

Theorem 3.6.
∣
∣
∣
∣
∣

A

(n)

ip

A

(n+1)

i,n+1

A

(n)

jp A

(n+1)

j,n+1

∣

∣

∣

∣

∣

−AnA

(n+1)

ij;p,n+1

=0, (A)

∣

∣

∣

∣

∣

A

(n)

ip

A

(n)

iq

A

(n+1)

n+1,p

A

(n+1)

n+1,q

∣

∣

∣

∣

∣

−AnA

(n+1)

i,n+1;pq=0, (B)

∣

∣

∣

∣

A

(n)

rr A

(n+1)

rr

A

(n)

nr A

(n+1)

nr

∣

∣

∣

∣

−A

(n+1)

n+1,r

A

(n+1)

rn;r,n+1

=0. (C)

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, namelyn− 1 andn.