A mathematical hierarchy

There is a certain hierarchy in the structure of equations. Their degree of mathematical complexity depends on the type of things one wants to describe. In high school most of us have been confronted with equations of some sort or another.

The simplest equations are algebraic equations which only involve algebraic manipulations such as addition, multiplication and so on.

A simple algebraic equation that plays an important role in physics is the equation of state of an ideal gas. It is usually written as PV=RT and expresses the phenomenological relation between the pressure P, the volume of a body of gas V, its temperature T and the fundamental gas constant R, which is just a known number.

Such an equation can be used in many ways. It asserts that the three variables P, V, and T that characterize the state of a fixed amount of gas are not independent, because they have to satisfy the given relation. One obvious use of the equation is that if we have the actual values of two of the three variables, we can calculate the third variable. But the relation gives a lot more information of a qualitative kind (see figure). It follows, for example, that if one increases the pressure P while keeping the volume constant, then the temperature has to go up as well. Similarly, one may conclude that decreasing the volume while keeping the temperature constant, we have to go along one of the curves increasing the pressure.

You may be familiar with the way an equation like this one can be rewritten in different but equivalent forms, using the recipe: ‘What you do to the left hand side, you’ve got to do to the right hand side.’ So by subtracting RT from both sides we may write PV–RT=0, or by dividing both sides by T we get PV/T=R. Depending on the question one wants to answer, one or the other form may be more convenient. Even though the equation may look different, the message remains the same. Most of the equations we will discuss are not of this simple algebraic type, but use more sophisticated notions like derivatives. Therefore, before embarking on the story of the fundamental equations, we have to introduce some of the frequently recurring mathematical symbols and explain their meaning.