Operations with Derivatives and Integrals

The following derivatives and integrals are commonly encountered, where x, r, w, and v are variables, a is a constant, and n an integer. 

2 Total Differentials and Relationships Between Partial Derivatives 

Let us consider a function Z that depends on two variables, X and Y, and signify this with the notation Z = f(X, Y). In addition to designating Z as a function, we may also refer to Z as the dependent variable, and X and Y as the independent variables. We can write a differential expression dZ that tells us the change in the dependent variable Z arising from small changes in the independent variables, dX and d Y. The result is 

The quantities dX and d Y are called differentials, the coefficients in front of dX and dT are called partial derivatives,11 and dZ is referred to as a total differential because it gives the total change in Z arising from changes in both X and Y. If Z were to depend upon additional variables, additional terms would be included in equation (A 1.1) to represent the changes in Z arising from changes in those variables. For much of our discussion, two variables describe the processes of interest, and therefore, we will limit our discussion to two independent variables, with the exception of the description of Pfaffian differentials in 

Section A 1.5 where it will be useful to consider a more general case involving three independent variables. 

---