# 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 $$v=0$$.

### :clapper: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.

{% embed url="<https://youtu.be/EceUt0z3AYg>" %}

### :pen\_ballpoint:Mathematical Proof

{% embed url="<https://viadean.notion.site/Vector-Field-Singularities-and-Stokes-Theorem-2591ae7b9a328005a657f888294497d8?source=copy_link>" %}