Partial Differentiation

In science we often have to deal with two or more variables that have no necessary dependence on one another. For example, although Boyle’s law defines the following relationship between the pressure p, volume V and temperature T of a mole of a perfect gas: $$ pV=RT $$ (where R = 8.31 J mol−1 K−1 is the gas constant), it does not prevent us from varying any two of the three variables independently. We can have whatever pressure we like at whatever temperature we like, provided we accept whatever volume results, etc. In this sort of circumstance we cannot define an ordinary derivative, such as dV/dT, because this has no definite value unless we specify how p is to change with T. For example, if we decided that p was to be a constant independent of T we would have $$ \frac{dV}{dT}=\frac{R}{p} $$ but if we set up the system so that p increased in proportion to T, we would instead find $$ \frac{dV}{dT}=0 $$ and other expressions would be given by other relationships between p and T.