17 Equations that Changed the World

In Pursuit of the Unknown:
17 Equations that Changed the World,
by Ian Stewart.

Ian Stewart’s book In Pursuit of the Unknown: 17 Equations that Changed the World “is the story of the ascent of humanity, told through 17 equations.” Of course, my first thought was “I wonder how many of those equations are in the 4th edition of Intermediate Physics for Medicine and Biology?” Let’s see.

1. Pythagorean theorem: a²+b²=c². In Appendix B of IPMB, Russ Hobbie and I discuss vectors, and quote Pythagoras’ theorem when relating a vector’s x and y components to its magnitude.

2. Logarithms: log(xy)=log(x)+log(y). In Appendix C, we present many of the properties of logarithms, including this sum/product rule as Eq. C6. Log-log plots are discussed extensively in Chapter 2 (Exponential Growth and Decay).

3. Definition of the derivative: df/dt = limit h → 0 (f(t+h)-f(t))/h. We assume the reader has taken introductory calculus (the preface states “Calculus is used without apology”), so we don’t define the derivative or consider what it means to take a limit. However, in Appendix D we present the Taylor series through its first two terms, which is essentially the same equation as the definition of the derivative, just rearranged.

4. Newton’s law of gravity: F = Gm₁m₂/d². Russ and I are ruthless about focusing exclusively on physics that has implications for biology and medicine. Almost all organisms live at the surface of the earth. Therefore, we discuss the acceleration of gravity, g, starting in Chapter 1 (Mechanics), but not Newton’s law of Gravity.

5. The square root of minus one: i² = -1. Russ and I generally avoid complex numbers, but they are mentioned in Chapter 11 (The Method of Least Squares and Signal Analysis) as an alternative way to formulate the Fourier series. We write the equation as i = √-1, which is the same thing as i² = -1.

6. Euler’s formula for polyhedra: F – E + V = 2. We never come close to mentioning it.

7. Normal distribution: P(x) = 1/√(2πσ) exp[-(x-μ)²/2σ²]. Appendix I is about the Gaussian (or normal) probability distribution, which is introduced in Eq. I.4.

8. Wave equation: ∂²u/∂t² = c² ∂²u/∂x². Russ and I introduce the wave equation (Eq. 13.5) in Chapter 13 (Sound and Ultrasound).

9. Fourier transform: f(k) = ∫ f(x) e^-2πixk dx. In Chapter 11 (The Method of Least Squares and Signal Analysis) we develop the Fourier transform in detail (Eq. 11.57), and then use it in Chapter 12 (Images) to do tomography.

10. Navier-Stokes equation: ρ (∂v/∂t + v ⋅∇ v) = -∇ p + ∇ ⋅ T + f. Russ and I analyze biological fluid mechanics in Chapter 1 (Mechanics), and write down a simplified version of the Navier-Stokes equation in Problem 28.

11. Maxwell’s equations: ∇ ⋅ E = 0, ∇ × E = -1/c ∂H/∂t, ∇ ⋅ H = 0, and ∇ × H = 1/c ∂E/∂t. Chapter 6 (Impulses in Nerve and Muscle Cells), Chapter 7 (The Exterior Potential and the Electrocardiogram), and Chapter 8 (Biomagnetism) discuss each of Maxwell’s equations. In Problem 22 of Chapter 8, Russ and I ask the reader to collect all these equations together. Yes, I own a tee shirt with Maxwell’s equations on it.

12. Second law of thermodynamics: dS ≥ 0. In Chapter 3 (Systems of Many Particles), Russ and I discuss the second law of thermodynamics. We derive entropy from statistical considerations (I would have chosen S = k_B lnΩ rather than dS ≥ 0 to sum up the second law). We state in words “the total entropy remains the same or increases,” although we don’t actually write dS ≥ 0.

13. Relativity: E = mc². We don’t discuss special relativity in much detail, but we do need E = mc² occasionally, most notably when discussing pair production in Chapter 15 (Interaction of Photons and Charged Particles with Matter).

14. Schrödinger’s equation: i ħ ∂Ψ/∂t = Ĥ Ψ. Russ and I don’t write down or analyze Schrödinger’s equation, but we do mention it by name, particularly at the start of Chapter 3 (Systems of Many Particles).

15. Information theory: H = - Σ p(x) log p(x). Not mentioned whatsoever.

16. Chaos theory: x_i+1 = k x_i (1-x_i). Russ and I analyze chaotic behavior in Chapter 10 (Feedback and Control), including the logistic map x_i+1=kx_i(1-x_i) (Eq. 10.36).

17. Black-Scholes equation: ½ σ²S² ∂²V/∂S² + rS ∂V/∂S + ∂V/∂t – rV = 0. Never heard of it. Something about economics and the 2008 financial crash. Nothing about it in IPMB.

Seventeen is a strange number of equations to select (a medium sized prime number). If I were to round it out to twenty, then I would have three to select on my own. My first thought is Newton’s second law, F=ma, but Stewart mentions that this relationship underlies both the Navier-Stokes equation and the wave equation, so I guess it is already present implicitly. Here are my three:

18. Exponential equation with constant input: dy/dt = a – by. Chapter 2 of IPMB (Exponential Growth and Decay) is dedicated to the exponential function. This equation appears over and over throughout the book. Stewart discusses the exponential function briefly in his chapter on logarithms, but I am inclined to add the differential equation leading to the exponential function to the list. Among its many uses, this function is crucial for understanding the decay of radioactive isotopes in Chapter 17 (Nuclear Physics and Nuclear Medicine).

19. Diffusion equation: ∂C/∂t = D ∂²C/∂x². To his credit, Stewart introduces the diffusion equation in his chapter on the Fourier transform, and indeed it was Fourier’s study of the heat equation (the same as the diffusion equation, with T for temperature replacing C for concentration) that motivated the development of the Fourier series. Nevertheless, the diffusion equation is so central to biology, and discussed in such detail in Chapter 4 (Transport in an Infinite Medium) of IPMB, that I had to include it. Some may argue that if we include both the wave equation and the diffusion equation, we also should add Laplace’s equation, but I consider that a special case of Maxwell’s equations, so it is already in the list.

20. Light quanta: E = hν: Although Stewart included Schrodinger’s equation of quantum mechanics, I would include this second equation containing Planck’s constant h. It summarizes the wave-particle duality of light, and is crucially important in Chapters 14 (Atoms and Light), 15 (Interaction of Photons and Charged Particles with Matter), and 16 (Medical Uses of X Rays).

Runners up include the Bloch equations since I need something from Chapter 18 (Magnetic Resonance Imaging), the Boltzmann factor (except that it is a factor, not an equation), Stokes law, the ideal gas law and its analog the van’t Hoff’s law from Chapter 5 (Transport through Neutral Membranes), the Hodgkin and Huxley equations, the Poisson-Boltzmann equation in Chapter 9 (Electricity and Magnetism at the Cellular Level), the Poisson probability distribution, and Planck’s blackbody radiation law (perhaps in place of E=hν).

Overall, I think studying the 4th edition of Intermediate Physics for Medicine and Biology introduces the reader to most of the critical equations that have indeed changed the world.