Change Is the Only Constant

An interesting book consisting of fascinating stories about calculus. This is definitely not a calculus textbook but if you ever want to know what calculus was, what it is used for and some interesting facts and stories involving calculus, then this would be a book to read.

There are too many chapters to give a chapter by chapter summary. But the book is divided into two sections based on the two main mathematical parts that make up calculus.

The first section covers "Differentiation" and the derivative, or the idea that a derivate is an 'instantaneous change' in an object, be it time, position, and so on. It builds on that by using the example of Newton considering the moon constantly falling towards the earth sideways. Based on how much it 'falls' as it moves to remain in orbit around the earth, its speed can be calculated. The derivative is also the rate of change of a quantity. For example, given your position over time, the 'First derivative' of it would be speed (change in position over time), the Second derivative would be acceleration (change in speed over time). Both Newton and Gottfried Leibnitz provided a notation for derivatives, but Leibnitz's notation proved to be more flexible than Newton's, showing that a proper notation can increase the flexibility for what derivatives can be used for.

Examples of the powers of derivatives are also given, staring with a danger in extrapolating trends via derivatives from too little data. A fascinating example is then given, via a story of using it to determine the direction a bicycle is moving.

A mystery or puzzle at the heart of derivatives is the question of how to 'approach the limits' in derivatives. Initially based on ideas from geometry, the book then shows lines and movements that cannot be differentiated (like Brownian motion), showing that there are limits to finding derivatives by geometric methods

Next, the derivative is used to show how it can find the maximum or the minimum point of a curve and how it can fail when a curve has no maximum. The story of the infamous Laffer Curve is also provided, showing its influence on supply side economics and tax policies. An interesting story is then told of a dog that can apparently apply calculus to the problem of find the best position to jump into the water to fetch a ball. Finally, the books shows that derivatives are used as a standard calculating tool.

The next section covers the "Integral" and how it can be used to calculate the area of a circle by 'slicing' it up into tiny sections and summing them up. Tolstoy's "War and Peace" is then used as a metaphor for integration: the sum of each tiny human experience making up the whole tapestry of history.

The integral is then introduction for calculating the area under a function. It is then shown that integration and derivation are 'fundamental theorems of calculus' and opposites of each other. But despite being opposites, differentiation has well defined methods but integration does not: integration depends on a bag of tools of various ways to perform integration. Some integration problems will only yield to certain tools.

One thing that integrals feature is the "Constant of Integration", an arbitrary number that usually appears and is unknown without knowing an initial condition. The book then looks at Einstein's regret in introducing such a constant into his equation for General Relativity to recreate a static universe, only to discover it was unnecessary in it original form but is now needed to explain the accelerating expansion of the universe.

Archimedes and his geometrical approach to integration is then shown via calculations of the volume of a pyramid and a cone. It is contrasted with using the algebraic formulation of calculus as a way to 'mechanize' calculus.

The paradox of 'Gabriel's horn' is then shown, a geometric shape with finite volume but infinite surface area. Finally, unlike differentiation, some Integrals have not formal solution and is still a constant struggle for current day mathematicians.