Euler is the Master of us All | Max Du

As of yesterday, September 18th, Leonhard Euler had left us for exactly 233 years [1]. Euler was a true genius to a degree that some of my friends who major maths in college complained, “I wish he had died earlier.”

2007(1) an image of Euler on a stamp

The title came from a famous French mathematician Pierre-Simon Laplace [2], whose method in calculating matrix determinant should be well acquainted by Linear Algebra students. The original line is “Lisez Euler, lisez Euler, c’est notre maître à tous.” which translate to “Read Euler, read Euler, he is the master of all.” There is even a book titled Euler: The Master of Us All. [3]

How is Euler a true master of us all ? I always wondered.

This week’s learning of his method of solving the first-order ODE (short for ordinary differential equations) helped me clarify this myth.

A first-order ODE looks like this: y’=f(x,y). [4]

To solve any of that geometrically, all you have to do is to plot out the direction field (which tells you the slopes at various points of the solution function) and then sketch out some integral curves accordingly. Any of the integral curves is a solution to the differential equation. (see demonstration below, where short arrows form the direction field and blue integral curves are plotted along the direction of those arrows)

image016

(2) a graph of integral curves plotted over a direction field

Now, the question is how to find the solution function’s expression? Don’t we sketch these curves according to our sense of where the arrows are going? How do I translate my sense into maths?

Euler gave us a simple thought. Just as we have to start somewhere in order to draw each curve, Euler method also has a starting point. When we start drawing, we “follow along” the directions at which the arrows are pointing. Although this is a continuous motion, Euler method points out that we can break the whole motion down into infinitesimal steps, much like the idea in the famous Zeno’s Arrow Paradox.

At the starting point, we have an expression of the slope at that point. Then we take one step infinitesimally to the right (or left), so that the slope doesn’t change too much, and so we can approximate this very next point. The same procedure is then repeated until we reach a desired answer. And if the steps taken are really infinitely small, and the number of steps are taken large enough, then Euler method gives you an approximation infinitely close to the real one.

Without going any further into the exact Euler method, I would like to point out how important its idea is. If you think back to even the most basic calculus, differentiation works exactly the same way. It breaks a continuous curve into infinitesimal pieces and then these pieces magically turn into straight line segments, instead of parts of a curve. Taylor’s expansion has also a similar approach.

It is fair to say that turning something continuous into “million tiny pieces” is really the core thinking, and of course, the foundation of calculus. Euler is a true master of this way of thinking, if not a master of us all.

 

Works Cited:

[1],[2] https://en.wikipedia.org/wiki/Leonhard_Euler

[3] Dunham, William. Euler: The Master of Us All. Washington, D.C.: Mathematical Association of America, 1999. Print.

[4] Prof. Arthur Mattuck. MIT OpenCourseWare. Course number 18.03 Differential Equations (Spring 2010).

 

Images Used:

(1) http://eulerarchive.maa.org/portraits/2007.jpg

(2) http://tutorial.math.lamar.edu/Classes/DE/DirectionFields_files/image016.gif

6 thoughts on “Euler is the Master of us All | Max Du

  1. willmanidis

    As a Euler-Aficionado myself, I can’t help but wonder if you have seen Project Euler? It’s an interesting programming/mathematics challenge site that I did some work with my Sophomore year and provides some really interesting applications of Euler’s work.

    Reply
    1. Max Post author

      Yeah, I know that website. I think, though, Project Euler is more for computer science since it is a huge list of algorithmic problems.

      Reply
  2. dexcoengilbert

    I love this little bit about Euler, even though I hated him when I had to learn his method in class. Your explanations are clear and concise. If you continue to profile some mathematicians, it would be cool if you gave a little background on them, and not just their math.

    Reply
  3. Susan Watehouse

    One amazing thing about Euler is the many different area of math that he worked in, yesterday I was using an example of one of his graph theory formulas. Also, Euler built a foundation of thinking for others to build on, have you looked at other (better?) methods of approximating solutions to differential equations? Are there methods that don’t use the basic thinking that Euler started with here?

    Reply

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