Update README.md

README.md

@@ -97,15 +97,15 @@ about $\alpha \in \mathcal{M}$. At some nearby point $\alpha + \delta\alpha \in
  $$\ket{\psi(\alpha + \delta\alpha)} = \ket{\psi(\alpha)} + \sum_l \delta\alpha_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha)} + \frac{1}{2} \sum_{kl} \delta\alpha_k \delta\alpha_l \frac{\partial^2}{\partial \alpha_k \partial \alpha_l} \ket{\psi(\alpha)} + \cdots \in \mathcal{H}$$
 where $\delta\alpha_l \in \mathbb{C}$ is an infinitesimal variation in the $l$-th direction, which is approximated to first order as an affine function
  $$\ket{\psi(\alpha + \delta\alpha)} \approx \ket{\psi(\alpha)} + \sum_l \delta\alpha_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha)} \in \mathcal{H}$$
-on $V$. Here, the $\delta\alpha_l$ represent the amount of change in each direction needed to linearly approximate the function $\ket{\psi}$ at $\alpha + \delta\alpha \in V$ from the neighboring point $\alpha \in \mathcal{M}$, so that the affine approximation becomes exact in the limit $\delta\alpha_l \to 0$.
+on $V$. Here, the $\delta\alpha_l$ represent the amount of change to $\alpha$ in each direction needed to linearly approximate the function $\ket{\psi}$ at $\alpha + \delta\alpha \in V$ from the neighboring point $\alpha \in \mathcal{M}$, so that the affine approximation becomes exact in the limit $\delta\alpha_l \to 0$.
 
 Similarly, we define a path $\alpha:[\tau, \tau + \delta \tau] \to \mathcal{M}$ in $\mathcal{M}$ where $\delta\tau > 0$ such that $\alpha(\tau) = \alpha \in \mathcal{M}$ and $\alpha(\tau + \delta \tau) = \alpha + \delta\alpha \in V$, and expand the path about $\tau \in \mathbb{R}$ to see
  $$\alpha(\tau + \delta \tau) = \alpha(\tau) + \delta\tau \frac{d \alpha}{d\tau}(\tau) + \frac{\delta\tau^2}{2} \frac{d^2 \alpha}{d\tau^2}(\tau) + \cdots \approx \alpha(\tau) + \delta\tau \frac{d \alpha}{d\tau}(\tau) \in \mathcal{M}$$
 which is an affine function on the closed interval $[\tau, \tau + \delta \tau] \subset \mathbb{R}$. Evaluating $\ket{\psi}:\mathcal{M} \to \mathcal{H}$ at $\alpha(\tau + \delta \tau) \in V$, we find that
  $$\ket{\psi(\alpha(\tau + \delta \tau))} \approx \ket{\psi(\alpha(\tau))} + \delta\tau \frac{d}{d\tau}\ket{\psi(\alpha(\tau))} \in \mathcal{H}$$
 is an affine function on $V$ which becomes exact in the limit $\delta\tau \to 0$. We may compare the first order terms of $\ket{\psi(\alpha(\tau + \delta \tau))}$ and $\ket{\psi(\alpha + \delta\alpha)}$ to find that
  $$\delta \tau \frac{d}{d\tau} \ket{\psi(\alpha(\tau))} = \delta \tau \sum_l \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha(\tau))} \frac{d \alpha_l}{d\tau}(\tau) = \sum_l \delta\alpha_l(\tau) \frac{\partial}{\partial \alpha_l} \ket{\psi(\alpha(\tau))}$$
-is a linear combination of the tangent vectors $\frac{d \alpha_l}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$, so that the total variation $\delta\alpha(\tau) = \delta \tau \frac{d \alpha}{d\tau}(\tau)$ at time $\tau$ is in the direction of the tangent vector $\frac{d \alpha}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$ such that
+is a linear combination of the tangent vectors $\frac{d \alpha_l}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$, so that the total variation $\delta\alpha(\tau) = \delta \tau \frac{d \alpha}{d\tau}(\tau)$ needed to linearly approximate the function $\ket{\psi}$ at $\alpha(\tau + \delta\tau) \in V$ from the neighboring point $\alpha(\tau) \in \mathcal{M}$ at time $\tau$ is in the direction of the tangent vector $\frac{d \alpha}{d\tau}(\tau)$ at $\alpha(\tau) \in \mathcal{M}$ such that
  $$\alpha(\tau + \delta \tau) \approx \alpha(\tau) + \delta\alpha(\tau) = \alpha(\tau) + \delta \tau \frac{d \alpha}{d\tau}(\tau)$$
 is the infinitesimal change in the parameters $\alpha(\tau) \in \mathcal{M}$ at time $\tau$ over the interval $[\tau, \tau + \delta \tau]$. Letting $T_\alpha \mathcal{M}$ denote the tangent space of $\mathcal{M}$ at $\alpha(\tau) \in \mathcal{M}$ and $T_{\psi(\alpha)} \mathcal{H}$ denote the tangent space of $\mathcal{H}$ at $\ket{\psi(\alpha(\tau))} \in \mathcal{H}$, we push forward the local vector field $\frac{d \alpha}{d\tau}$ on $V \subset \mathcal{M}$ to the local vector field $\frac{d}{d\tau} \ket{\psi(\alpha)}$ on the image $\tilde{V} = \ket{\psi(V)} \subset \mathcal{H}$ of $V$ under the mapping $\ket{\psi}:\mathcal{M} \to \mathcal{H}$, and note that
  $$\frac{d}{d\tau} \ket{\psi(\alpha(\tau))} \in T_{\psi(\alpha)} \tilde{V}$$