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 $v=0$.

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✍️Mathematical Proof

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Non-Orthogonal Nature

Unlike common systems like polar or cylindrical coordinates, hyperbolic coordinates are not orthogonal. The inner product of the tangent basis vectors, $E_u \cdot E_v$, is not zero, which means the coordinate lines for constant $u$ and constant $v$ do not intersect at right angles. This is a fundamental characteristic that distinguishes them from more familiar orthogonal systems.

Geometric Interpretation

The coordinate lines have distinct geometric shapes in the Cartesian plane. The lines of constant $u$ are rays from the origin, while the lines of constant $v$ are hyperbolas given by the equation $x^1 x^2=v^2$. This unique geometry is a direct result of the exponential transformation relations.

Transformation Relations

The system is defined by the transformations $x^1=v e^u$ and $x^2=v e^{-u}$. The inverse transformations, which are crucial for expressing $u$ and $v$ in terms of Cartesian coordinates, are $u=\frac{1}{2} \ln \left(\frac{x^1}{x^2}\right)$ and $v=\sqrt{x^1 x^2}$.

Basis Vectors

The tangent basis vectors, $E_u$ and $E_v$, and the dual basis vectors, $E^u$ and $E^v$, are explicitly calculated. Their non-orthogonality is evident from the non-zero inner products, such as $E_u$. $E_v=2 v \sinh (2 u)$. These basis vectors are essential for performing vector operations and understanding the local geometry of the coordinate system.

🎬Demonstration

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 $v=0$.

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