Mathematical interlude. More properties of exact differentials The cyclic rule

4 MATHEMATICAL INTERLUDE. MORE PROPERTIES OF EXACT DIFFERENTIALS. THE CYCLIC RULE 

The total differential of a function of two variables /(x, y) is written in the form 

If we form second derivatives of the function/(x, y), there are several possibilities (df /dx) can be differentiated with respect to either x or y, and the same is true of (df /dy). We get 

Of these four, only three are distinct. It can be shown that for a function of several variables, the order of differentiation with respect to two variables such as x and y does not matter and the mixed derivatives are equal that is. 

Differentiating the first of Eqs. (9.11) with respect to y, and the second with respect to x, we obtain 

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