What is Calculus? A Physics Perspective

The geometry of curves obsessed mathematicians long before Newton hid from the plague at Woolsthrope Manor, where the seeds of the calculus took root. Before the late seventeenth century, finding slopes, areas, and volumes under curves was an entirely ad hoc affair. Unlike the lines of squares, triangles, rectangles, and cubes, height and/or depth doesn’t remain constant over any measurable length of curved lines, making calculations difficult. Although Newton and many others have conceptualized the foundations of calculus in more physics-like terms, it became a formalized system so effective in geometric analysis that it’s easy to mistake the study of mathematical change (calculus) and the geometry of curves as one in the same. This is not the case. 

 

Calculus is the study of continuously changing mathematical functions. A function, as you should know, is an expression of a variable(s) in terms of one or more others—e.g., Y=X ^2, Z=CV^2dx (^ means raised to some power). Continuous change is uninterrupted change over a specified region, usually of time or position along the X axis. Continuous is the opposite of discrete, where change occurs at distinctly separate points--like driving a car for twenty minutes at one speed, followed by a quick acceleration to another speed for the next twenty minutes before stopping. Unlike the behavior of the majority of continuous functions, like many found in engineering applications, most graduate and undergraduate calculus/analysis courses focus on functions where the variables maintain a constant value/expression of their counterparts following processes known as differentiation and integration—e.g., the derivative of dy/dx is always X ^2; the integral of Ydx is always X ^3. While the geometric view is essential in understanding the technical and historical foundations of the calculus, this post will present the alternative conception.

 

Understanding things from different points of view is an essential part of learning, but it’s best that one start with whichever is most comfortable for him/her before moving on to another. The geometric presentation might work best for most people—but not everyone. As a writer with poor nonverbal abilities, I’m what you might call a conceptual learner or “narrative learner.” I only feel like I understand a mathematical problem or concept if I can narrate all the steps in the most literal way possible, especially since I prefer applied rather than theoretical mathematics. I didn’t feel like I understood what calculus was until I was able to write the following.     

 

Physics is the study of the Universe's foundational physical phenomena, what they are and how/why they behave as they do. Over the centuries, with help from chemists and engineers, physicists have investigated the nature of heat, light, sound, forces, motion, space, time, electricity, matter, and celestial bodies.  

All measurable physical phenomena, by dint of being physical and measurable, must have the capacity to undergo and/or produce quantifiable change. One must understand how these things change in order to understand the properties themselves and how they interact with the rest of the Universe. A living thing must be able to acquire, regulate, store, and use energy--an accumulated quantity driven by a force, a rate of change in the momentum of an object with respect to time, over a distance. Society’s function likewise depends upon the acquisition, regulation, storage, and use of energy. The rates and accumulations of energy and its correlates’ changes are, naturally, of great interest to scientists and engineers. 

 

Thus, one could also define the calculus as the study of the rates and accumulations of continuous change.

 

As mathematical functions, they are defined by one or more changing variables expressed either by the rate of change of a variable(s) with respect to another or the products/quotients of one or more changing variables over a specified range. Examples of phenomena include acceleration, change in position with respect to time; force, change in momentum with respect to change in time; work, force multiplied by change in position; and entropy, change in heat content divided by temperature. 

 

Without a means of evaluating continuous change, physics could only systematically evaluate functions undergoing linear change or change at a constant rate. Since the area under a perfect diagonal represents the only continuous function that can be evaluated with an average, physics would be a rather ineffectual field without calculus.

 

Rudimentary calculus has two primary branches: derivative calculus, based on rates of change; and integral calculus, premised on the analyses of areas and volumes under curves, sums of products/quotients, or (as I sometimes call it) accumulations of multiplicative quantities (all three are equivalent). Favoring the second and third definitions, this intro will focus on the more interesting and important method—integration, from the point of view of a summing process in kinematics. 

 

Distance is the integral of velocity multiplied by change in time; work is the integral of force multiplied by change in position; and entropy is the integral of the reciprocal of temperature multiplied by the increase in heat content. Conceptually, as you can see, the integral involves multiplicative quantities and is the equivalent to an area if the “fixed variable” (e.g., velocity in change in distance) called the integrand is plotted next to what I’LL call the “differential variable” (the changing variable, such as time). When a single integral, or area, is calculated, the fixed variable is always expressed as a function of the differential variable: e.g., Y[x]=X is written as Xdx; Y[X]=X^2 is written X ^2(dx). Sometimes, integration involves another variable(s), and a volume in three or more dimensions must be calculated.  

 

Formally, even the areas of discrete functions are treated as integrals, as in the case of a simple distance-rate problem. The integral of Xdx, represented geometrically as a triangle, is the one continuous integral that can be calculated using an average and, therefore, without calculus. I call it Newton’s integral, for it is thought to have prompted him to explore general methods of integration, which can’t be evaluated with an average.    

 

In integrals where V (velocity) as a function of T (time) is greater than T to the first, there’s a complication. Since dt equals D/V, changing T necessarily changes V; and if V is changing at an increasing rate, the line on the V-T graph curves, eliminating the possibility of calculating an average. If you multiplied the INITIAL V by the change in time of any interval of nonzero size, it would amount to less distance than actually traveled in that interval, given V would clearly be below the average velocity over that interval. If you were to use the FINAL one, it would just as clearly amount to too much change in distance. 

 

But what if you cut the intervals up to ever greater numbers, making their sizes ever smaller? Clearly, even in the simplest distance-rate problems, you could calculate total change in distance using as many intervals as desired. For example, you could cut twenty seconds traveling at one hundred feet per second into five intervals of four seconds and multiple 100 X 4 five times and get the same answer--and you could do the same thing for ten, twenty, or any number of intervals less than infinity.  

 

Back to a continuous function. As the number of total intervals approaches infinity, the size of each interval approaches zero. When the number of intervals approaches infinity in the case where you use initial velocity, that velocity will draw ever closer to the interval’s average, and the error or shortage of calculated distance at each interval, and in total, approaches zero. If you used final velocity, the total and interval errors approach zero. 

 

One can find an answer doing the following: Use a “summing process” that reaches a certain total value of distance traveled as a function of time plus or minus a value of total errors and show that total distance couldn’t get SMALLER than a certain amount (using initial velocity) or GREATER than a certain amount (using final). In other words, each summation would converge on the same expression of T, and you could prove that it would be impossible for it to be any other value of T. 

 

At this juncture, mathematics pedagogy often moves to calculating Newton’s integral and the integral of Y=x ^2 using something called the Reimann sums, which also illustrates the importance of a concept called a “limit.” This is a wonderful exercise for both the geometric viewpoint, and the one here, but there are many resources for this; the author has nothing to add; and, again, this is just an introduction to what calculus is (not to mention he doesn’t have the knowledge or resources to type out all the necessary notations). 

 

While I’m not exactly an authority, to me, “proving calculus,” or (hypothetically) discovering it, means deriving what’s called the power rule for derivatives and integrals: nX ^ n-1 for derivatives and 1/n+1 multiplied by X ^n+1 for integrals. Direct and indirect application of these formulas are prevalent in physics and mathematical analysis.        

 

You can prove the power rules using differentiation (taking derivatives). It should be intuitive that the change of the sum/integral relative to dx should be the final integrand, making it the derivate of the integral. This means that the integral is whatever’s derivative is equal to the integrand. For example, if the derivative of 1/3 X ^3 relative to dx is X ^2, then the former is the integral of X ^2 dx. From here, you prove that the derivatives of X ^ n, X ^ 1/n, and X ^ m/n (where m is a whole number) are nX^n-1, showing it’s the power rule for derivatives. Since the derivative of 1/n(X^n + 1) is the power rule of derivatives, than that’s the power rule for integrals.

 

Thus, the derivative and integral are inverse processes.  

 

 

Some say mathematics doesn’t begin until calculus. But this isn’t just because it’s the first challenging course for talented students. The hidden beauty of calculus is that it’s at once theoretical, practical, abstract, and conceptual, allowing students to see how they all fit together regardless of the inclinations they brought in. It can grab the interest of a conceptual, nonvisual learner, then help him see how mathematics can be conceived more abstractly, making the transition into more theoretical branches of mathematics easier (if only I were smart enough to get past the content herein).

 

But there must be a hook that grabs these learners, which standard calculus pedagogy seems to lack. Hence this post.