global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -392,19 +392,19 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
 
 \begin{proof}
   Note that the sets \(H^1(W_\gamma) \times T_\gamma H^1(I, M)\) are precisely
-  the images of the canonical charts
+  the images of the charts
   \[
-    \varphi_\gamma :
-    \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+    \varphi_\gamma^{-1} :
+    \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
     \subset T H^1(I, M)
     \to H^1(W_\gamma) \times T_\gamma H^1(I, M)
   \]
-  of \(T H^1(I, M)\) -- i.e. the charts given by\footnote{Once more, we use the
+  of \(T H^1(I, M)\) given by\footnote{Once more, we use the
   canonical identification $T_X H^1(W_\gamma) \cong H^1(\gamma^* TM)$ to apply
   the vector $\phi_\gamma(Y) \in H^1(\gamma^* TM)$ to $(d \exp_\gamma)_X : T_X
   H^1(W_\gamma) \to T_{\exp_\gamma(X)} H^1(I, M)$.}
   \begin{align*}
-    \varphi_\gamma^{-1} : H^1(W_\gamma) \times T_\gamma H^1(I, M)
+    \varphi_\gamma : H^1(W_\gamma) \times T_\gamma H^1(I, M)
     & \to T H^1(I, M) \\
     (X, Y) & \mapsto (d \exp_\gamma)_X \phi_\gamma(Y)
   \end{align*}
@@ -412,16 +412,16 @@ extended to a canonical isomorphism of vector bundles, as seen in\dots
   By composing charts we get a fiber-preserving, fiber-wise linear
   diffeomorphism
   \[
-    \varphi_\gamma^{-1}(H^1(W_\gamma) \times T_\gamma H^1(I, M))
+    \varphi_\gamma(H^1(W_\gamma) \times T_\gamma H^1(I, M))
     \subset T H^1(I, M)
     \isoto
     \psi_{1, \gamma}(H^1(W_\gamma) \times H^1(\gamma^* TM)),
   \]
-  which takes \(\varphi^{-1}(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to
-  \(\psi_{1, \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With
-  enough patience, one can deduce from the fact that \(\varphi_\gamma\) and
+  which takes \(\varphi(X, Y) \in T_{\exp_\gamma(X)} H^1(I, M)\) to \(\psi_{1,
+  \gamma}(X, \phi_\gamma(Y)) \in H^1(\exp_\gamma(X)^* TM)\). With enough
+  patience, one can deduce from the fact that \(\varphi_\gamma^{-1}\) and
   \(\psi_{1, \gamma}^{-1}\) are charts that these maps agree in the
-  intersections of the open subsets \(\varphi_\gamma^{-1}(H^1(W_\gamma) \times
+  intersections of the open subsets \(\varphi_\gamma(H^1(W_\gamma) \times
   T_\gamma H^1(I, M))\), so that they may be glued together into a global
   smooth map \(\Phi : T H^1(I, M) \to \coprod_{\eta \in H^1(I, M)} H^1(\eta^*
   TM)\).