Emphasis on the coordinate system and the tangent vector basis and the dual basis

Описание: https://viadean.notion.site/Mathemati...

#maths #coordinate #vector #tangents #python

The animation depicts a coordinate system with a tangent vector basis (e.g., yellow dot) and its dual basis (e.g., arrows), illustrating concepts from differential geometry used to describe how vectors behave on curved manifolds, a framework critical in general relativity as supported by studies like Misner, Thorne, and Wheeler's "Gravitation" (1973).
The grid and annotations (e.g., "E1", "1/2") suggest a visual representation of tangent spaces, where each point on a manifold has a unique tangent space, a concept formalized in modern mathematics to generalize tangent lines and planes, with applications in robotics navigation as noted in recent analyses from Borunte.net (2023).
The emphasis on dual basis highlights a mathematical duality where each vector has a corresponding linear functional, a principle rooted in finite-dimensional vector space theory (e.g., Wikipedia's Dual Basis entry, 2025), challenging the intuitive notion that basis vectors alone suffice for all coordinate transformations.