Conceptualizing the Christoffel Symbols: An Adventure in Curvilinear Coordinates

What are the Christoffel Symbols, and why do we need them? Our exploration into the world of differential geometry continues, as we strive to lay bare the complex mathematical machinery behind General Relativity. Here we calculate the Christoffel symbols for the most common of curvilinear coordinates system — polar coordinates — in order to build a preliminary intuition for what these symbols represent and how they relate to the metric tensor. As a bonus, we unearth the meaning of concepts like the Levi-Civita connection and parallel transport along the way. Plus: bears! Thank you for watching, and please consider supporting us on Patreon -- it makes a difference. Contents: 00:00 - Intro 00:54 - Cartesian vs. Polar Land 03:31 - The Metric Tensor 10:16 - The Levi-Civita Connection 14:15 - The Christoffel Symbols 21:14 - A Polar Geodesic 22:27 - Future Agenda