🎬focus on the tangent vector basis and Christoffel symbols in polar coordinates

Christoffel symbols are not abstract mathematical quantities, but direct geometric measures of how the basis vectors of a coordinate system change from one point to another. In a simple Cartesian (x, y) system, the basis vectors are constant everywhere; they always point along the x and y axes. However, in a curved system like polar coordinates. the demo reveals that Christoffel symbols are the necessary "correction factors" for doing calculus in a coordinate system where the very definition of direction changes at every point.

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