Vector Field Singularities and Stokes' Theorem

Hyperbolic coordinates represent a non-orthogonal transformation of the first quadrant where the grid is composed of hyperbolas (constant $v$ ) and radial lines (constant $u$ ). Unlike standard polar or Cartesian systems, the tangent vectors $E_u$ and $E_v$ are not perpendicular, as evidenced by their non-zero dot product, which depends on both the scale $v$ and the hyperbolic angle $u$. This lack of orthogonality is a defining characteristic of the system; it means the metric tensor contains off-diagonal components, and the dual (contravariant) basis vectors are not simply normalized versions of the tangent (covariant) vectors. Ultimately, this system provides a specialized way to map the $x^1 x^2>0$ region that highlights Lorentz-like symmetries rather than rotational ones.

🎬Narrated Video

https://youtu.be/Uky__iXGMjA

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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