Summary: We use Stokes' theorem to calculate the area of the Mandelbrot set from the series expansion found for its boundary.
Content:
0:00 Elementary area calcuations
0:35 Area from Calculus
0:55 The Area of the Mandelbrot set
2:25 Branding
2:44 Stokes' Theorem
4:20 Circle Mappings
6:48 Arrow Representation
8:31 Stokes' Theorem for a Circle
10:44 Stokes' Theorem for Circle Mappings
11:45 Exercise
13:06 Stokes' Theorem for the 10th Iteration
14:28 Stokes' Theorem on the Riemann Sphere
References:
Daniel Bittner, Long Cheong, Dante Gates and Hieu D. Nguyen, New approximations for the area of the Mandelbrot set, 2017
https://projecteuclid.org/journals/involve-a-journal-of-mathematics/volume-10/issue-4/New-approximations-for-the-area-of-the-Mandelbrot-set/10.2140/involve.2017.10.555.pdf