Important Equations

Good engineers build safe reliable machines that work predictably. The most effective design tools are knowledge and experience.

An accurate understanding of how the physical world works coupled with the ability to mathematically model and predict results, defines good design engineering.

The ability to mathematically model and predict the behavior of mechanical systems is possible in great part because of the combined work of hundreds of scientists over several centuries. Taking the time to understand the chronology of events leading up to the technical sophistication we now enjoy, is a worthwhile endeavor. Here are some important equations and a brief example of their contributions. 

The Pythagorean Theorem

What does it mean: The square of the hypotenuse of a right triangle is equal to the SUM of the squares of its legs.

History: Attributed to Pythagoras, it isn't certain that he first proved it. The first clear proof came from Euclid, and it is possible the concept was known 1000 years before Pythoragas by the Babylonians. 

Importance: The equation is at the core of geometry, links it with algebra, and is the foundation of trigonometry. Without it, accurate surveying, mapmaking, and navigation would be impossible.

Modern use: Triangulation is used to this day to pinpoint relative location for GPS navigation. 

The logarithm and its identities

What does it mean: You can multiply numbers by adding related numbers. 

History: The initial concept was discovered by the Scottish Laird John Napier of Merchiston in an effort to make the multiplication of large numbers, then incredibly tedious and time consuming, easier and faster. It was later refined by Henry Briggs to make reference tables easier to calculate and more useful. 

Importance: Logarithms were revolutionary, making calculation faster and more accurate for engineers and astronomers.

Modern use: Logarithms still inform our understanding of radioactive decay.

The Fundamental Theorem of Calculus

What does it mean?: Allows the calculation of an instantaneous rate of change. 

History: Calculus as we currently know it was described around the same in the late 17th century by Isaac Newton and Gottfried Leibniz. There was a lengthy debate over plagiarism and priority which may never be resolved. We use the leaps of logic and parts of the notation of both men today.

Importance: According to Stewart, "More than any other mathematical technique, it has created the modern world." Calculus is essential in our understanding of how to measure solids, curves, and areas. It is the foundation of many natural laws, and the source of differential equations.

Modern use: Any mathematical problem where an optimal solution is required. Essential to medicine, economics, and computer science.

Newton's universal law of gravitation

What does it mean?: Calculates the force of gravity between two objects. 

History: Isaac Newton derived his laws with help from earlier work by Johannes Kepler. He also used, and possibly plagiarized the work of Robert Hooke. 

Importance: Used techniques of calculus to describe how the world works. Even though it was later supplanted by Einstein's theory of relativity, it is still essential for practical description of how objects interact with each other. We use it to this day to design orbits for satellites and probes.

Value: When we launch space missions, the equation is used to find optimal gravitational "tubes" or pathways so they can be as energy efficient as possible. Also makes satellite TV possible. 

The origin of complex numbers

What does it mean?: The square of an imaginary number is negative.   

History: Imaginary numbers were originally posited by famed gambler/mathematician Girolamo Cardano, then expanded by Rafael Bombelli and John Wallis. They still existed as a peculiar, but essential problem in math until William Hamilton described this definition. 

Importance: According to Stewart ".... most modern technology, from electric lighting to digital cameras could not have been invented without them." Imaginary numbers allow for complex analysis, which allows engineers to solve practical problems working in the plane.

Modern use: Used broadly in electrical engineering and complex mathematic theory.

Euler's formula for polyhedra

What does it mean?: Describes a space's shape or structure regardless of alignment. 

History: The relationship was first described by Descartes, then refined, proved, and published by Leonhard Euler in 1750.

Importance: Fundamental to the development of topography, which extends geometry to any continuous surface. An essential tool for engineers and biologists. 

Modern use: Topography is used to understand the behavior and function of DNA.

The normal distribution

What does it mean?: Defines the standard normal distribution, a bell shaped curve in which the probability of observing a point is greatest near the average, and declines rapidly as one moves away.   

History: The initial work was by Blaise Pascal, but the distribution came into its own with Bernoulli. The bell curve as we currently comes from Belgian mathematician Adolphe Quetelet. 

Importance: The equation is the foundation of modern statistics. Science and social science would not exist in their current form without it. 

Modern use: Used to determine whether drugs are sufficiently effective relative to negative side effects in clinical trials.

The wave equation

What does it mean?: A differential equation that describes the behavior of waves, originally the behavior of a vibrating violin string. 

History: The mathematicians Daniel Bournoulli and Jean D'Alembert were the first to describe this relationship in the 18th century, albeit in slightly different ways. 

Importance: The behavior of waves generalizes to the way sound works, how earthquakes happen, and the behavior of the ocean. 

Modern use: Oil companies set off explosives, then read data from the ensuing sound waves to predict geological formations.

The Fourier transform

What does it mean?: Describes patterns in time as a function of frequency. 

History: Joseph Fourier discovered the equation, which extended from his famous heat flow equation, and the previously described wave equation. 

Importance: The equation allows for complex patterns to be broken up, cleaned up, and analyzed. This is essential in many types of signal analysis. 

Modern use: Used to compress information for the JPEG image format and discover the structure of molecules.

The Navier-Stokes equations

What does it mean?: The left side is the acceleration of a small amount of fluid, the right indicates the forces that act upon it.  

History: Leonhard Euler made the first attempt at modeling fluid movement, French engineer Claude-Louis Navier and Irish mathematician George Stokes made the leap to the model still used today

Importance: Once computers became powerful enough to solve this equation, it opened up a complex and very useful field of physics. It is particularly useful in making vehicles more aerodynamic.

Modern use: Among other things, allowed for the development of modern passenger jets.

Maxwell's equations

What does it mean?: Maps out the relationship between electric and magnetic fields.

History: Michael Faraday did pioneering work on the connection between electricity and magnetism, James Clerk Maxwell translated it into equations, fundamentally altering physics. 

Importance: Helped predict and aid the understanding of electromagnetic waves, helping to create many technologies we use today.

Modern use: Radar, television, and modern communications.

Second law of thermodynamics

What does it mean?: Energy and heat dissipate over time.   

History: Sadi Carnot first posited that nature does not have reversible processes. Mathematician Ludwig Boltzmann extended the law, and William Thomson formally stated it. 

Importance: Essential to our understanding of energy and the universe via the concept of entropy. It helps us realize the limits on extracting work from heat, and helped lead to a better steam engine. 

Modern use: Helped prove that matter is made of atoms, which has been somewhat useful.

Einstein's theory of relativity

What does it mean?: Energy equals mass times the speed of light squared.

History: The less known (among non-physicists) genesis of Einstein's equation was an experiment by Albert Michelson and Edward Morley that proved light did not move in a Newtonian manner in comparison to changing frames of reference. Einstein followed up on this insight with his famous papers on special relativity (1905) and general relativity (1915). 

Importance: Probably the most famous equation in history. Completely changed our view of matter and reality.

Modern use: Helped lead to nuclear weapons, and if GPS didn't account for it, your directions would be off thousands of yards.

The Schrödinger equation

What does it mean?: Models matter as a wave, rather than a particle. 

History: Louis-Victor de Broglie pinpointed the dual nature of matter in 1924. The equation you see was derived by Erwin Schrodinger in 1927, building off of the work of physicists like Werner Heisenberg.

Importance: Revolutionized the view of physics at small scales. The insight that particles at that level exist at a range of probable states was revolutionary. 

Modern use: Essential to the use of the semiconductor and transistor, and thus, most modern computer technology.

Shannon's information entropy

What does it mean?: Estimates the amount of data in a piece of code by the probabilities of its component symbols. 

History: Developed by Bell Labs engineer Claude Shannon in the years after World War 2. 

Importance: According to Stewart, "It is the equation that ushered in the information age." By stopping engineers from seeking codes that were too efficient, it established the boundaries that made everything from CDs to digital communication possible. 

Modern use: Pretty much anything that involves error detection in coding. Quality testing software, etc.

The logistic model for population growth

What does it mean?: Estimates the change in a population of creatures across generations with limited resources. 

History: Robert May was the first to point out that this model of population growth could produce chaos in 1975. Important work by mathematicians Vladimir Arnold and Stephen Smale helped with the realization that chaos is a consequence of differential equations. 

Importance: Helped in the development of chaos theory, which has completely changed our understanding of the way that natural systems work. 

Modern use: Used to model earthquakes and forecast the weather.

The Black–Scholes model

What does it mean?: Prices a derivative based on the assumption that it is riskless and that there is no arbitrage opportunity when it is priced correctly. 

History: Developed by Fischer Black and Myron Scholes, then expanded by Robert Merton. The latter two won the 1997 Nobel Prize in Economics for the discovery.  

Importance: Helped create the now multi trillion-dollar derivatives market. It is argued that improper use of the formula (and its descendants) contributed to the financial crisis. In particular, the equation maintains several assumptions that do not hold true in real financial markets. 

Modern use: Variants are still used to price most derivatives, even after the financial crisis.

The Hodgkin-Huxley equations

The Hodgkin-Huxley equation boosted mathematical biology by giving equations to model the way nerve cells send signals to each other. This equation formed the basis of theoretical neuroscience.

Source: In Pursuit of the Unknown: 17 Equations That Changed the World by Ian Stewart

Ideal Gas Law

What does it mean?: The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. The law that the product of the pressure and the volume of one gram molecule of an ideal gas is equal to the product of the absolute temperature of the gas and the universal gas constant.

History: Emil Claperyon is credited with having formulated the ideal gas law in 1834. Emil observed the algebra that describes the relationship between Pressure, Volume, Temperature, and the quantity of gas in a system. The simple statement of Ideal Gas Law represents the culmination of the work of dozens of scientists over hundreds of years and it includes the discoveries of the scientists such as Amadeo Avagadro, Jacques Charles, Robert Boyle, Blaise Pascal and others.

Importance: The ideal gas law is a valuable tool in understanding state relationships in gaseous systems. For example, in a system of constant temperature and pressure, the addition of more gas molecules results in increased volume.

Modern use: Leak testing for quality control in manufacturing of medical devices, aeronautical, automotive, electronics and more to separate good parts from bad part. During leak testing, changes in pressure in the test part are of primary interest. When a test part is filled with air, it at first expands inside the part to occupy its volume. To detect the leakage, the changes in pressure must be taken into consideration. Since parameters like pressure, volume, and amount of air are involved in this process, it is governed by the ideal gas law.