The Math of Bubbles // Minimal Surfaces & the Calculus of Variations #SoME3

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 Euler-Lagrange Equation 11:31 The Euler-Lagrange Equation 13:10 Deriving the Catenoid 15:25 Boundary Conditions Bubbles naturally try to minimize surface area, and so if we make a wire-frame 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 Euler-Lagrange equation via the Calculus of Variations and apply it in our case to deduce the f(x). Check out my MATH MERCH line in collaboration with Beautiful Equations ► COURSE PLAYLISTS: ►DISCRETE MATH:    • Discrete Math (Full Course: Sets, Logic, P...   ►LINEAR ALGEBRA:    • Linear Algebra (Full Course)   ►CALCULUS I:    • Calculus I (Limits, Derivative, Integrals)...   ► CALCULUS II:    • Calculus II (Integration Methods, Series, ...   ►MULTIVARIABLE CALCULUS (Calc III):    • Calculus III: Multivariable Calculus (Vect...   ►VECTOR CALCULUS (Calc IV)    • Calculus IV: Vector Calculus (Line Integra...   ►DIFFERENTIAL EQUATIONS:    • Ordinary Differential Equations (ODEs)   ►LAPLACE TRANSFORM:    • Laplace Transforms and Solving ODEs   ►GAME THEORY:    • Game Theory   OTHER PLAYLISTS: ► Learning Math Series    • 5 Tips To Make Math Practice Problems Actu...   ►Cool Math Series:    • Cool Math Series   BECOME A MEMBER: ►Join:    / @drtrefor   MATH BOOKS I LOVE (affilliate link): ► https://www.amazon.com/shop/treforbazett SOCIALS: ►Twitter (math based):   / treforbazett   ►Instagram (photography based):   / treforphotography