Multiplying determinants according to the rule in § 1, we have
a 2 /
+ a 92 /
dv
dw
a* 2
a*ay
dx 2
a* ay
dx
dx
a 2 /
a
+y%
+e e2 {
dv
a w
a*ay
dy 2
dxdy
ay 2
ay
ay
where, by T,
(1) v= a 3f+y3f=Zf^ + ZfZy = ZF
d* ay dx as ay a£ as’
w = /8-^ + 5-
dx dy
af
dv
By the same rule of multiplication of determinants,
\ dx dy,
,af
dv
dv
Applying (1) with/ replaced by dF/dk for the first column
and by dF/dv for the second column, we get
a 2 f
a 2 f
de
dtdv
a 2 f
d 2 F
dvdt
dv 2
Hence the Hessian of the transformed function F equals the
product of the Hessian h of the given function / by the square
of the determinant of the linear transformation. Conse
quently, h is called a covariant of index 2 of /.
For an interpretation of h= 0, see Exs. 4, 5, § 7. In case
/ is the quadratic function / of § 4, h reduces to 4d, where d
is the invariant ac—h 2 .
The Junctional determinant or Jacobian (named after C.
G. J. Jacobi) of two functions f{x, y) and g(x, y) is defined
to be
Let
g b
Hei
I a
l ai
and
tral
if
let
con
case
Let
and
and