§5] 
EXAMPLES OF COVARIANTS 
13 
jmn 
the 
uare 
nse- 
case 
re d 
■ C. 
ined 
Let the above transformation T replace / by F(£, f), and 
g by G(Ç, rj). By means of (1), we get 
d{F, G) 
9(f, v) 
df af 
dx dy ' a P _ a 9(/, g) 
dg dg y 8 d(x, y)' 
dx dy 
Hence the Jacobian of / and g is a covariant of index unity of 
/ and g. For example, the Jacobian of the linear functions 
l and L in § 4 is their resultant r; they are proportional if 
and only if the invariant r is zero. The last fact is an illus 
tration of the 
Theorem. Two Junctions f and g of x and y are dependent 
if and only if their Jacobian is identically zero. 
First, if g = 4>(f), the Jacobian of / and g is 
Next, to prove the second or converse part of the theorem, 
let the Jacobian of / and g be identically zero. If g is a 
constant, it is a (constant) function of /. In the contrary 
case, the partial derivatives of g are not both identically zero. 
Let, for example, dg/dx be not zero identically. Consider g 
and y as new variables in place of x and y. Thus /=F(g, y) 
and the Jacobian is 
jmn 
the 
aare 
nse- 
case 
re d 
■ C. 
ined 
dFdg dFdg^dF 
dg dx dg dy dy 
dg dg 
dx dy 
dg dg 
dx dy