| ∂ | ( | ∂ƒ | ) = | ∂ | ( | ∂ƒ | ). |
| ∂x | ∂y | ∂y | ∂x |
(i.)
This theorem is not true without limitation. The conditions for its validity have been investigated very completely by H. A. Schwarz (see his Ges. math. Abhandlungen, Bd. 2, Berlin, 1890, p. 275). It is a sufficient, though not a necessary, condition that all the differential coefficients concerned should be continuous functions of x, y. In consequence of the relation (i.) the differential coefficients expressed in the two members of this relation are written
| ∂2ƒ | or | ∂2ƒ | . |
| ∂x∂y | ∂y∂x |
The differential coefficient
| ∂nƒ | , |
| ∂xp ∂yq ∂zr |
in which p + g + r = n, is formed by differentiating p times with respect to x, q times with respect to y, r times with respect to z, the differentiations being performed in any order. Abbreviated notations are sometimes used in such forms as
| ƒ xp yq zr or ƒ | (p, q, r) | . |
| x, y, z |
Differentials of higher orders are introduced by the defining equation