My father was intrigued by the mathematics he knew, but never understood what “higher” math was good for. Calculus was a mystery to him and he couldn’t conceive of more abstract mathematics.
He was a high school industrial arts teacher, very practical, very hands-on. He could do simple geometry and make measurements for straight forward calculations. Formulas for determining the area and volume of a few shapes were not foreign to him.
Occasionally, he may have used the Pythagorean theorem, trigonometry tables and some basic algebra. He was proud he understood the power of compound interest in finance.
But what, he would ask, was all this other mathematical stuff I was studying good for?
Most of his math was for acquiring and manipulating information about things that were static. The area or volume of some object he was making wasn’t going to change. He drew up plans. The dimensions of each piece conformed to the values in his plan. The final product had the shape and volume he wanted.
These endeavors didn’t confront him with things that were dynamic or changing. He did, however, understand compound interest rates. This was a little different in that investments grow (change) each month at a fixed rate.
An initial investment of $100 is worth $102 a month later and $104.04 the following month if the interest rate is 2 percent per month. Things are no longer static. Since the rate of growth is applied to both the principle and interest earned in the preceding month, your earnings compound. The concept of a rate has a dynamic quality.
Casual observation demonstrates that most things have dynamic qualities. Calculus addresses the behavior of things undergoing change. It was invented, essentially simultaneously, by the famous physicists — they were natural philosophers in their day — Isaac Newton and Gottfried Leibniz.
Though calculus isn’t limited to changes that are a function of time, they are the easiest to envision. So, my explanation focuses on rates of change with respect to time. I’ll also discuss how the methods used in calculus are generalized to examining changes that are themselves undergoing changes.
Velocity is the most intuitive. Velocity is a quantitative change in location per unit of time. That is, it is the distance traversed per unit time as in miles per hour, or feet per second. If velocity remains constant, the distance traveled is given by multiplying velocity by time.
When velocity does change, the rate at which it changes is called acceleration. It is the change in velocity per unit time. The velocity of something falling in a gravitational field continually increases (ignoring air resistance) at a rate of 32 feet per second per second.
Though one talks in terms of seconds, minutes or hours, of course this acceleration is happening in each and every instant of its fall. It starts with zero velocity and a simple equation shows its velocity anytime afterward.
That equation is V = G x T. Here V represents velocity, G is gravitational acceleration (itself a constant) and T is the time since the object’s release.
In these examples, velocity is said to be the “first derivative” of distance with respect to time. Acceleration is the “second derivative” of distance, again, with respect to time or the first derivative of velocity with respect to time.
It is generally the case that the value of something is variable with respect to things other than time. This is to say, it is a function of those other things. Perhaps the strength of a magnetic field varies from one location to another. In this case we could talk about the strength of the field with respect to changes in location.
Differential calculus deals with the dynamic behavior of all kinds of systems. That behavior is determined by the rates of change of the variables we use to characterize them. In some instances variables don’t change independently, they are linked.
For instance, physicists describe the thermodynamic properties of systems (heat engines, atmospheres, chemical reactions) by variables such as heat, temperature, entropy, pressure and the like. Through experimentation, scientists have established how changes in any one of these properties is linked to changes in the others.
Those dynamic relationships can be described with mathematical equations made up of the appropriate derivatives. If one knows all the pertinent differential equations, and one of the variables changes value, there are methods for calculating the net result.
This is the other side of the coin dubbed integral calculus. Essentially, its methods enable one to add up the incrementally small changes occurring in the thing you’re interested in as something it depends on changes.
I often wonder if powerful mathematical tools are little more than sophisticated bookkeeping tools or do they reflect more profound truisms about how nature works. I’m fairly certain that, in some instances, there is a deeper significance — a kind of mathematical universe.
It takes a lot of time and effort to get to where one can appreciate the elegant nature of “higher math.” The practical value of these insights isn’t immediately apparent. For my dad, it wasn’t likely to hold his interest if it didn’t help him build something. But, for me, math has provided a lifetime of wonderment.
Steve Luckstead is a medical physicist living in Walla Walla. He can be reached at firstname.lastname@example.org.