🧄Vector Field Singularities and Stokes' Theorem

The hyperbolic coordinate system is non-orthogonal because its coordinate lines (rays and hyperbolas) do not generally intersect at right angles. This is mathematically confirmed by the non-zero inner product of its tangent basis vectors. The system is only orthogonal under two specific conditions: along the ray where $u=0$ and at the origin where $v=0$.

🎬how the non-orthogonal grid is formed by the hyperbolic and radial lines

The animation visually confirms that the hyperbolic coordinate system is non-orthogonal in general, as the lines of constant u (rays) and constant v (hyperbolas) intersect at angles other than 90 degrees. However, it also highlights the two specific conditions where orthogonality is achieved: along the ray where u=0, and at the origin itself where v=0.

🖊️Mathematical Proof

Vector Field Singularities and Stokes' Theorem | Proof and Derivationviadean on Notion