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

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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:
OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:
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MATH BOOKS I LOVE (affilliate link):
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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:
OTHER PLAYLISTS:
► Learning Math Series
►Cool Math Series:
BECOME A MEMBER:
MATH BOOKS I LOVE (affilliate link):
SOCIALS:
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