Euler's Method...

From Notes on Diffy Qs: Differential Equations for Engineers, by Jirí Lebl

"What if we want to find the value of the solution at some particular x? Or perhaps we want to produce a graph of the solution to inspect the behavior. In this section we will learn about the basics of numerical approximation of solutions.

The simplest method for approximating a solution is Euler’s method. It works as follows: We take x0 and compute the slope k = f (x₀; y₀). The slope is the change in y per unit change in x. We follow the line for an interval of length h on the x axis. Hence if y = y₀ at x₀, then we will say that y₁ (the approximate value of y at x₁ = x₀ + h) will be y₁ = y₀ + hk. Rinse, repeat! That is, compute x₂ and y₂ using x₁ and y₁." See Notes on Diffy Qs above (under graphic)

Topics: Differential Equations, Diversity in Science, Mathematics, Women in Science

Okay, this is the LAST time I'll talk about Hidden Figures (although I did order the book).

Not to spoil it for you, but Dr. Katherine Johnson (played by Taraji P. Henson) mentioned an "old method" of mathematics. What both the actress and NASA scientist referred to is something you're taught usually sophomore year in a STEM major. Euler's Method is named after Leonhard Euler, and it's used to numerically approximate differential equations, something in the movie and the embed below alludes to is now done by what we now know as computers (the laptop kind, not female mathematicians).

It is important to understand the steps, derivation and mathematics behind computer calculation. How do you KNOW it's right? I'm often challenged as to "when I ever use Calculus" at work. Most often they're right, I don't. There's a software package designed with the equations embed within them to literally SPIT out an answer. The program doesn't have imagination nor does it visualize an expected end result. "The answer" is the end of a calculation without any notion of its consequences if incorrect.

Part of its practicality is essentially how the study of mathematics and physics organizes one's thinking. I use systematic approaches to solving just about any problem in life. However in Hidden Figures, it was initially the NASA scientists and eventually Dr. Johnson knowing the mathematics and relying on human insight and intuition that averted catastrophe, not that it doesn't happen when launching humans on the top of essentially systematic staged bombs to achieve Earth orbit.

The old riddle "which came first: the chicken or the egg?" can easily be answered with regards to computers and humans. The Singularity will have a ways yet.

François Arago said of him (Euler) "He calculated just as men breathe, as eagles sustain themselves in the air" (Beckmann 1971, p. 143; Boyer 1968, p. 482). [1]

In a testament to Euler's proficiency in all branches of mathematics, the great French mathematician and celestial mechanic Laplace told his students, "Liesez Euler, Liesez Euler, c'est notre maître à tous" ("Read Euler, read Euler, he is our master in everything" (Beckmann 1971, p. 153). [2]

1, 2: Scienceworld.Wolfram.com: Euler
LA Times:
Meet the ‘Hidden Figures’ mathematician who helped send Americans into space, Amina Khan