The total differential of Eq (6.1) can be manipulated to produce relationships between partial derivatives of x, y, and z, or other variables. The technique can be described as the "divide-and-hold-constant" method. It works as follows: Eq (6.1) is divided by a differential (not necessarily dx, dy or dz) and another (dx, dy or dz) is set equal to zero, thus holding constant the variable associated with it. For example, suppose Eq (6.1) is divided by dy with z held constant (i.e., dz = 0). The result is the following relation:
or, because partial derivatives can be inverted, the above equation becomes what is called the cyclic transformation:
In another example of the "divide-and-hold-constant" method, Eq (6.1) can be divided by a fourth variable w with x held constant to yield:
which is the chain rule for partial derivatives.
It is obvious that there are so many combinations of variables to which the "divide-and-hold-constant" method can be applied that attempting to construct an exhaustive catalog of relations such as Eqs (6.6) and (6.7) would be fruitless. The practical approach is to apply the method to suit the needs of particular problems.
Example: If a fluid is heated in a constant-volume container from T to T + AT, what is the pressure rise Ap?
For this problem, the starting function is the equation of state v(T,p), for which the total differential is:
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