This is my entry to the #SoME3 competition run by ‪@3blue1brown‬ and ‪@LeiosLabs‬. Use the hashtag to check out the many other great entries!

0:00 Fun with bubbles!
0:46 Minimal Surfaces
2:35 Calculus of Variations
6:27 Derivation of EulerLagrange Equation
11:31 The EulerLagrange Equation
13:10 Deriving the Catenoid
15:25 Boundary Conditions

Bubbles naturally try to minimize surface area, and so if we make a wireframe boundary for the bubble to attach to, the big question is what is that minimal surface going to be? And how can we compute it out mathematically? In this video I am going to approach this question from the perspective of the Calculus of Variations. We will see that the surface area for one of the simplest shapes, the catenoid formed between two parallel circles, results in what is called a functional a surface area integral in terms of a function f(x) and its derivative, so our question will be what function minimizes that surface area integral. We will derive the 1 dimensional EulerLagrange equation via the Calculus of Variations and apply it in our case to deduce the f(x).

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