Curvilinear Coordinates in Differential Geometry

The equations of General Relativity are written in the language of curvilinear coordinates, where mathematical objects like Basis Vectors, Metric Tensors, and Christoffel Symbols dominate the landscape. But where do these concepts come from and how can we connect them to our physical intuition? Join us to find out. Plus -- hotwheels! (Note: some familiarity with vector calculus is highly recommended.)

Please consider helping support us on Patreon:
https://www.patreon.com/dialect_philosophy

Join our Discord!:
https://discord.gg/mtFa2SHWuc

Conceptualizing the Christoffel Symbols Video:
https://youtu.be/TvFvL_sMg4g

Metric Tensor Trilogy:
https://www.youtube.com/playlist?list=PL__fY7tXwodmfntSAAyBDxZ4_eE3ZwbFE

Contents:
00:00 - Intro
00:50 - What are Curvilinear Coordinates?
03:28 - Basis Vectors & Parametric Basis
08:00 - Coordinate Acceleration & Levi-Civita Condition
11:47 - The Christoffel Symbols
12:53 - Characterization of Arbitrary Coordinates
14:42 - Characterization of Polar Coordinates
19:12 - Geodesics
20:31 - Curved Surfaces