I’m fond of calculus. It opens up a very broad array of potential applications. Pretty much all of twentieth-century physics depends directly on calculus concepts. Neither Einstein’s highly-regarded theory of gravity or quantum mechanics would be possible without calculus. Fluid mechanics, thermodynamics, and other fields in physics also depend on calculus operations. Statistics, chemistry, economics, and other fields also adapt calculus operations. Moreover, calculus has this huge impact with the introduction of only one new concept, yielding two operations which are inverses of each other.
The first operation is the derivative. My high school calculus teacher had us memorize the following description:
The derivative of a curve is defined as the slope of the curve at any point. The slope of a curve at a single point is defined as the slope of the line tangent to the curve at that point. The slope of the tangent line may be found by taking the limit of the slope of the secant line, as the secant line’s points of intersection with the curve are moved closer to the point where the tangent line touches the curve.
(Apologies to Joe Millet if I’ve altered the wording. He was a great teacher, but it has been well over twenty years since I sat in his class. I thought math was just a “useful tool” until he awakened in me an interest in math for its own beauty.)
The reason derivatives are powerful is that functions (“curves”) are everywhere once you start making quantitative descriptions of the world. Intuitively, we often think of quantities and their derivatives as separate-but-related concepts: The derivative of a function is its rate of change : Your car’s velocity (direction and speed) is the derivative of its position. The acceleration is the rate of change of its velocity. Since force is mass times acceleration, if we have a history of an object’s position over time, we can use derivatives to calculate the forces on the object at every time. For waves (on the ocean, electromagnetic waves, sound waves, quantum mechanical waves… any waves), the forces on the wave are set by the wave’s amplitude, shape, and other environmental factors. This means that once you know the shape of an initial wave, you can calculate how it will initially change - using derivatives. Once you know the mass, position, and velocity of the Earth & Moon, you can calculate gravitational force and from there see how their positions and velocities will change. Once you have a sufficiently detailed description of any system, the derivative tells you how that system is about to change. Given the equation for a curve, also derivatives let you find the highest and lowest points.
As you learn Calculus I, you will go from problems that are based directly on that initial description and the definition of a derivative to working with a “table of derivatives”: a table the shows the derivatives of common functions. Every item in that table is derived from the initial definition, and can be worked out fairly easily again should you ever forget the table. The table is a mnemonic, a shorthand tool. It’s generally easier to memorize a short table than to work everything out from first principles, but understanding the principles helps you see how to apply calculus to a wide variety of situations.
That brings me to the second operation in calculus, integration or “taking an integral”, often called the “antiderivative” when it is first introduced. An Integral is just the inverse operation of a derivative: Derivatives & integrals are related the same way as addition & subtraction, as multiplication & derivative. Just as a derivative is described as the “slope of a curve”, an integral is described as the “area under a curve”. Just as multiplication is usually easier than division, derivatives are usually easier than integrals. Just as a derivative tells you how fast a function is changing, an integral lets you add up tiny changes over an arbitrary history, to get the accumulated changes (and full state) at any time, given only history of the changes. Suppose you’re given the history of water flowing into a tub, and the volume of the tub. The integral tells you when the tub is full, when it is overflowing, the amount of water in the tub at any time, and how much water has spilled onto the floor at any time.
Because integrals are harder than derivatives, they are introduced later, and once you learn the concept, you can find and use huge tables of integrals to match many different functions. Again, these tables are tools. They may appear intimidating, but they don’t contain anything fundamentally new or important. Go back to the fundamentals whenever the tools are overwhelming or you can’t find what you need.
The examples given for derivatives and integrals so far may seem fairly academic, but there are a few additional insights that are introduced one at a time as you learn calculus. Much of the power of calculus comes from combining these insights with the seemingly-academic derivative and integral. Here are some of them:
Spatial derivatives and integrals: The examples above used “time” as the independent coordinate, but any dimension will work. The slope of a hill is the slope of a function. The area inside a two-dimensional shape is the same as the area under some curve. The derivatives and integrals can also reference other independent variables such as the price of a given product, household income, or the percentage of people voting in favor of a particular candidate. The math stays the same.
Multivariate calculus: Instead of taxing a derivative along a single dimension, take a derivative along several simultaneously to get the gradient - the direction and slope of maximum steepness. Instead of integrating to find the area within a curve, find the volume of a solid object, or of hypothetical “hypervolumes” involving multiple non-spatial, non-temporal dimensions.
Differential equations: Wave and diffusion equations typically relate a quantity to its derivatives. Working out the original function based on that relationship and based on spatial and temporal boundaries can be challenging, but it uses the same fundamental concepts - derivatives and integrals - to test plausible functions, work out their derivatives, and extrapolate from known conditions to other places and times. Differential equations are used throughout physics to describe technical concepts, and to use those concepts to make firm, detailed predictions.
The universality of these concepts gives calculus its power and depth. The power combined with the conceptual simplicity of the core idea gives calculus mathematical beauty. (That core idea is either the indented paragraph near the top of this post, or perhaps “The slope of a function at a point is the limit of the slope of the secant line.”)
The mathematical concept of beauty, of broad & deep power based on a single intuitive definition is why I love calculus.