An Introduction to Curvilinear Coordinates in Differential Geometry

3-Second Blueprint (English)

1. Curvilinear coordinate curves define local tangent basis vectors that form an infinitesimal linear frame at each point.
2. Taking dot products of local basis vectors via the law of cosines constructs the metric tensor for infinitesimal distances.
3. Projecting normalized second partial derivatives of basis vectors onto the parametric frame derives Christoffel symbols.