My All 2020 Mathematics A to Z: Delta


I have Dina Yagodich to thank for my inspiration this week. As will happen with these topics about something fundamental, this proved to be a hard topic to think about. I don’t know of any creative or professional projects Yagodich would like me to mention. I’ll pass them on if I learn of any.

Color cartoon illustration of a coati in a beret and neckerchief, holding up a director's megaphone and looking over the Hollywood hills. The megaphone has the symbols + x (division obelus) and = on it. The Hollywood sign is, instead, the letters MATHEMATICS. In the background are spotlights, with several of them crossing so as to make the letters A and Z; one leg of the spotlights has 'TO' in it, so the art reads out, subtly, 'Mathematics A to Z'.
Art by Thomas K Dye, creator of the web comics Projection Edge, Newshounds, Infinity Refugees, and Something Happens. He’s on Twitter as @projectionedge. You can get to read Projection Edge six months early by subscribing to his Patreon.

Delta.

In May 1962 Mercury astronaut Deke Slayton did not orbit the Earth. He had been grounded for (of course) a rare medical condition. Before his grounding he had selected his flight’s callsign and capsule name: Delta 7. His backup, Wally Schirra, who did not fly in Slayton’s place, named his capsule the Sigma 7. Schirra chose sigma for its mathematical and scientific meaning, representing the sum of (in principle) many parts. Slayton said he chose Delta only because he would have been the fourth American into space and Δ is the fourth letter of the Greek alphabet. I believe it, but do notice how D is so prominent a letter in Slayton’s name. And S, Σ, prominent in both Slayton and Schirra’s.

Δ is also a prominent mathematics and engineering symbol. It has several meanings, with several of the most useful ones escaping mathematics and becoming vaguely known things. They blur together, as ideas that are useful and related and not identical will do.

If “Δ” evokes anything mathematical to a person it is “change”. This probably owes to space in the popular imagination. Astronauts talking about the delta-vee needed to return to Earth is some of the most accessible technical talk of Apollo 13, to pick one movie. After that it’s easy to think of pumping the car’s breaks as shedding some delta-vee. It secondarily owes to school, high school algebra classes testing people on their ability to tell how steep a line is. This gets described as the change-in-y over the change-in-x, or the delta-y over delta-x.

Δ prepended to a variable like x or y or v we read as “the change in”. It fits the astronaut and the algebra uses well. The letter Δ by itself means as much as the words “the change in” do. It describes what we’re thinking about, but waits for a noun to complete. We say “the” rather than “a”, I’ve noticed. The change in velocity needed to reach Earth may be one thing. But “the” change in x and y coordinates to find the slope of a line? We can use infinitely many possible changes and get a good result. We must say “the” because we consider one at a time.

Used like this Δ acts like an operator. It means something like “a difference between two values of the variable ” and lets us fill in the blank. How to pick those two values? Sometimes there’s a compelling choice. We often want to study data sampled at some schedule. The Δ then is between one sample’s value and the next. Or between the last sample value and the current one. Which is correct? Ask someone who specializes in difference equations. These are the usually numeric approximations to differential equations. They turn up often in signal processing or in understanding the flows of fluids or the interactions of particles. We like those because computers can solve them.

Δ, as this operator, can even be applied to itself. You read ΔΔ x as “the change in the change in x”. The prose is stilted, but we can understand it. It’s how the change in x has itself changed. We can imagine being interested in this Δ2 x. We can see this as a numerical approximation to the second derivative of x, and this gets us back to differential equations. There are similar results for ΔΔΔ x even if we don’t wish to read it all out.

In principle, Δ x can be any number. In practice, at least for an independent variable, it’s a small number, usually real. Often we’re lured into thinking of it as positive, because a phrase like “x + Δ x” looks like we’re making a number a little bigger than x. When you’re a mathematician or a quality-control tester you remember to consider “what if Δ x is negative”. From testing that learn you wrote your computer code wrong. We’re less likely to assume this positive-ness for the dependent variable. By the time we do enough mathematics to have opinions we’ve seen too many decreasing functions to overlook that Δ y might be negative.

Notice that in that last paragraph I faithfully wrote Δ x and Δ y. Never Δ bare, unless I forgot and cannot find it in copy-editing. I’ve said that Δ means “the change in”; to write it without some variable is like writing √ by itself. We can understand wishing to talk about “the square root of”, as a concept. Still it means something else than √ x does.

We do write Δ by itself. Even professionals do. Written like this we don’t mean “the change in [ something ]”. We instead mean “a number”. In this role the symbol means the same thing as x or y or t might, a way to refer to a number whose value we might not know. We might not care about. The implication is that it’s small, at least if it’s something to add to the independent variable. We use it when we ponder how things would be different if there were a small change in something.

Small but not tiny. Here we step into mathematics as a language, which can be as quirky and ambiguous as English. Because sometimes we use the lower-case δ. And this also means “a small number”. It connotes a smaller number than Δ. Is 0.01 a suitable value for Δ? Or for δ? Maybe. My inclination would be to think of that as Δ, reserving δ for “a small number of value we don’t care to specify”. This may be my quirk. Others might see it different.

We will use this lowercase δ as an operator too, thinking of things like “x + δ x”. As you’d guess, δ x connotes a small change in x. Smaller than would earn the title Δ x. There is no declaring how much smaller. It’s contextual. As with δ bare, my tendency is to think that Δ x might be a specific number but that δ x is “a perturbation”, the general idea of a small number. We can understand many interesting problems as a small change from something we already understand. That small change often earns such a δ operator.

There are smaller changes than δ x. There are infinitesimal differences. This is our attempt to make sense of “a number as close to zero as you can get without being zero”. We forego the Greek letters for this and revert to Roman letters: dx and dy and dt and the other marks of differential calculus. These are difficult numbers to discuss. It took more than a century of mathematicians’ work to find a way our experience with Δ x could inform us about dx. (We do not use ‘d’ alone to mean an even smaller change than δ. Sometimes we will in analysis write d with a space beside it, waiting for a variable to have its differential taken. I feel unsettled when I see it.)

Much of the completion of work we can credit to Augustin Cauchy, who’s credited with about 800 publications. It’s an intimidating record, even before considering its importance. Cauchy is, per Florian Cajori’s History Mathematical Notations, one of the persons we can credit with the use of Δ as symbol for “the change in”. (Section 610.) He’s not the only one. Leonhardt Euler and Johann Bernoulli (section 640) used Δ to represent a finite difference, the difference between two values.

I’m not aware of an explicit statement why Δ got the pick, as opposed to other letters. It’s hard to imagine a reason besides “difference starts with d”. That an etymology seems obvious does not make it so. It does seem to have a more compelling explanation than the use of “m” for the slope of a line, or \frac{\Delta y}{\Delta x} , though.

Slayton’s Mercury flight, performed by Scott Carpenter, did not involve any appreciable changes in orbit, a Δ v. No crewed spacecraft would until Gemini III. The Mercury flight did involve tests in orienting the spacecraft, in Δ θ and Δ φ on the angles of the spacecraft’s direction. These might have been in Slayton’s mind. He eventually flew into space on the Apollo-Soyuz Test Project, when an accident during landing exposed the crew to toxic gases. The investigation discovered a lesion on Slayton’s lung. A tiny thing, ultimately benign, which discovered earlier could have kicked him off the mission and altered his life so.


Thank you all for reading. I’m gathering all my 2020 A-to-Z essays at this link, and have all my A-to-Z essays of any kind at this link. Here is hoping there’s a good week ahead.

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

3 thoughts on “My All 2020 Mathematics A to Z: Delta”

  1. I got “dump some delta-v” from my ex-husband, who would tell me I needed to dump some delta-v when he thought I wasn’t decelerating fast enough on an exit or something. I like using it, except that part of me always has to start mentally chewing over the fact that decelerating is still taking on rather than getting rid of delta-v.

    Like

    1. It’s true, though; decreasing speed is still taking on some delta-v''. Even in the astronaut context: a rocket will have a certain range of velocity changes it can make, all at once or in smaller pieces, and it runs out of fuel whether it's accelerating or decelerating. There is a bias toward thinking ofdelta v” as increasing v, even though decreasing is just as legitimate.

      There’s a similar bias in physics to thinking of acceleration'' as only an increase in speed, when the use in that context can also mean a decrease in speed or a change in direction. That probably reflects the ordinary-English meaning of the words, where we separate the meanings intoaccelerate” and decelerate'' andturn”. It’s still interesting that delta has the same hint around it.

      Like

Please Write Something Good

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.