☁️3/3-Unlocking the Secrets of Shape and Space: A Glimpse into Differential Geometry & Manifolds

Tangent vectors can be understood in multiple ways, including as equivalence classes of curves, derivations on smooth functions, or velocities of curves, all of which are fundamental to understanding vector fields and other complex concepts like the Lie bracket and torsion tensor in differential geometry.

This cloud computing explores various concepts within differential geometry, including equivalence classes of curves, derivations on smooth functions, vector fields on 2D and 3D spheres, tangent vectors, and torsion tensors, through animated results, plotting, and computations.

🎬Animated result

Equivalence Classes of Curves

Derivations on Smooth Functions

the flow of points along the vector field lines on an equirectangular projection

Vector Field on the 3D Sphere

Tangent Vectors as Velocities of Curves

Directional Derivative via Curves

Directional Derivative in Coordinates

Lie Bracket of Vector Fields