42. Jacobians.Many problems in which partial differential coefficients occur are simplified by the introduction of certain determinants called “Jacobians” or “functional determinants.” They were introduced into Analysis by C. G. J. Jacobi (J. f. Math., Crelle, Bd. 22, 1841, p. 319). The Jacobian of u1, u2, ... un with respect to x1, x2, ... xn is the determinant

∂u1   ∂u1... ∂u1
∂x1 ∂x2∂xn
∂u2   ∂u2... ∂u2
∂x1 ∂x2∂xn
 :
∂un   ∂un... ∂un
∂x1 ∂x2∂xn

in which the constituents of the rth row are the n partial differential coefficients of ur, with respect to the n variables x. This determinant is expressed shortly by

∂(u1, u2, ..., un).
∂(x1, x2, ..., xn)

Jacobians possess many properties analogous to those of ordinary differential coefficients, for example, the following:—

∂(u1, u2, ..., un)× ∂(x1, x2, ..., xn)= 1,
∂(x1, x2, ..., xn) ∂(u1, u2, ..., un)
∂(u1, u2, ..., un)× ∂(y1, y2, ..., yn)= ∂(u1, u2, ..., un).
∂(y1, y2, ..., yn) ∂(x1, x2, ..., xn)∂(x1, x2, ..., xn)

If n functions (u1, u2, ... un) of n variables (x1, x2, ..., xn) are not independent, but are connected by a relation ƒ(u1, u2, ... un) = 0, then

∂(u1, u2, ..., un)= 0;
∂(x1, x2, ..., xn)

and, conversely, when this condition is satisfied identically the functions u1, u2 ..., un are not independent.

43. Partial differential coefficients of the second and higher Interchange of order of differentiations. orders can be formed in the same way as those of the first order. For example, when there are two variables x, y, the first partial derivatives ∂ƒ/∂x and ∂ƒ/∂y are functions of x and y, which we may seek to differentiate partially with respect to x or y. The most important theorem in relation to partial differential coefficients of orders higher than the first is the theorem that the values of such coefficients do not depend upon the order in which the differentiations are performed. For example, we have the equation