---
title: Curvature and Circulation
format: 
    pdf:
        include-in-header: 
            file: ../latex-header.tex
---

# Definitions

- Let $L$ be a small rectangle in the $(u,v)$ plane, $\hat{L}$ its image on $H^2$.
- Let $V = (P,Q)$ be a vector field on the $(u,v)$ plane induced by the metric on $H^2$:
    - $P =  \frac{\partial_v(A)}{B}$
    - $Q = -\frac{\partial_u(B)}{A}$ 
- The **circulation** of $V$ around the boundary of $L$ is defined as:
  $$ C_V(L) = \oint_{L} V \cdot d\vec{s} = \oint_L P du + Q dv $$

# Calculation

$$ C_V(L) = \oint_L V \cdot dr = \oint_L \frac{\partial_v(A)}{B} du  -\frac{\partial_u(B)}{A} dv $$

In the hyperbolic plane, $A = B = \frac{1}{v}$, so

$$ C_V(L) = \oint_L v \partial_v\left(\frac{1}{v}\right) du - v \partial_u\left(\frac{1}{v}\right) dv $$

and this equals

$$ C_V(L) = \oint_L -\frac{1}{v} du + 0 dv = \oint_L -\frac{1}{v} du $$

This reduces the circulation calculation to a line integral of a single-variable function around the rectangle $L$.

Let $L$ have corners at $(u,v)$, $(u+\delta u, v)$, $(u+\delta u, v+\delta v)$, and $(u, v+\delta v)$. Then these integrals become

\begin{align*}
C_V(L) &= \int_{u}^{u+\delta u} -\frac{1}{v} du + \int_{v}^{v+\delta v} 0 dv + \int_{u+\delta u}^{u} -\frac{1}{v+\delta v} du + \int_{v+\delta v}^{v} 0 dv \\
&= -\frac{\delta u}{v} + \frac{\delta u}{v+\delta v} \\
&= \delta u \left( \frac{1}{v+\delta v} - \frac{1}{v} \right) \\
&= \delta u \left( \frac{v - (v+\delta v)}{v(v+\delta v)} \right) \\
&= -\frac{\delta u \delta v}{v(v+\delta v)}
\end{align*}

Since the circulation per unit area is ultimately the curvature at a point, we divide by the area of $\hat{L}$, which is given by  
$$ \text{Area}(\hat{L}) = \iint_L \frac{1}{v^2} du dv = \frac{\delta u \delta v}{v^2} $$

So the curvature at the point $(u,v)$ is given by
\begin{align*}
K(u,v) &= \lim_{\text{Area}(\hat{L}) \to 0} \frac{C_V(L)}{\text{Area}(\hat{L})} \\
&= \lim_{\delta u, \delta v \to 0} \frac{-\frac{\delta u \delta v}{v(v+\delta v)}}{\frac{\delta u \delta v}{v^2}} \\
&= \lim_{\delta v \to 0} -\frac{v^2}{v(v+\delta v)} \\      
&= \lim_{\delta v \to 0} -\frac{v}{v+\delta v} \\
&= -1
\end{align*}





