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An Introduction to Curvilinear Coordinates in Differential Geometry 5 месяцев назад


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An Introduction to 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:   / dialect_philosophy   Join our Discord!:   / discord   Conceptualizing the Christoffel Symbols Video:    • Conceptualizing the Christoffel Symbo...   Metric Tensor Trilogy:    • The Metric Tensor   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

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