Conversely, when this equation is satisfied there exists a function ƒ which is such that
dƒ= Xdx + Ydy.
The expression Xdx + Ydy in which X and Y are connected by the relation (ii.) is often described as a “perfect differential.” The theory of the perfect differential can be extended to functions of n variables, and in this case there are ½n(n − 1) such relations as (ii.).
In the case of a function of two variables x, y an abbreviated notation is often adopted for differential coefficients. The function being denoted by z, we write
| p, q, r, s, t for | ∂z | , | ∂z | , | ∂2z | , | ∂2z | , | ∂2z | . |
| ∂x | ∂y | ∂x2 | ∂x∂y | ∂y2 |
Partial differential coefficients of the second order are important in geometry as expressing the curvature of surfaces. When a surface is given by an equation of the form z = ƒ(x, y), the lines of curvature are determined by the equation
{ (l + q2)s − pqt} (dy)2 + { (1 + q2)r − (1 + p2)t } dxdy − { (1 + p2)s − pqr} (dx)2 = 0,
and the principal radii of curvature are the values of R which satisfy the equation
R2(rt − s2) − R { (1 + q2)r − 2pqs + (1 + p2)t } √(1 + p2 + q2) + (1 + p2 + q2)2 = 0.
44. Change of variables.The problem of change of variables was first considered by Brook Taylor in his Methodus incrementorum. In the case considered by Taylor y is expressed as a function of z, and z as a function of x, and it is desired to express the differential coefficients of y with respect to x without eliminating z. The result can be obtained at once by the rules for differentiating a product and a function of a function. We have