diff --git a/sections/structure.tex b/sections/structure.tex
@@ -33,16 +33,16 @@ Finally, we may define\dots
From now on we fix \(I = [0, 1]\).
\end{note}
-We should note that every peace-wise smooth curve \(\gamma : I \to M\) is a
+We should note that every piece-wise smooth curve \(\gamma : I \to M\) is a
class \(H^1\) curve. This answer raises and additional question though: why
class \(H^1\) curves? The classical theory of the calculus of variations -- as
described in \cite[ch.~5]{gorodski} for instance -- is usually exclusively
-concerned with the study of peace-wise smooth curves, so the fact that we are
+concerned with the study of piece-wise smooth curves, so the fact that we are
now interested a larger class of curves, highly non-smooth curves in fact,
\emph{should} come as a surprise to the reader.
To answer this second question we return to the case of \(M = \RR^n\). Denote
-by \({C'}^\infty(I, \RR^n)\) the space of peace-wise curves in \(\RR^n\). As
+by \({C'}^\infty(I, \RR^n)\) the space of piece-wise curves in \(\RR^n\). As
described in section~\ref{sec:introduction}, we would like \({C'}^\infty(I,
\RR^n)\) to be a Banach manifold under which both the energy functional and the
length functional are smooth maps. As most function spaces, \({C'}^\infty(I,
@@ -141,7 +141,7 @@ find\dots
\begin{proposition}
Given an Euclidean bundle \(E \to I\) -- i.e. a vector bundle endowed with a
Riemannian metric -- the space \(H^1(E)\) of all class class \(H^1\) sections
- of \(E\) is the completion of the space \({C'}^\infty(E)\) of peace-wise
+ of \(E\) is the completion of the space \({C'}^\infty(E)\) of piece-wise
smooth sections of \(E\) under the inner product given by
\[
\langle \xi, \eta \rangle_1
@@ -199,7 +199,7 @@ find\dots
\end{note}
We are particularly interested in the case of the pullback bundle \(E =
-\gamma^* TM \to I\), where \(\gamma : I \to M\) is a peace-wise smooth curve.
+\gamma^* TM \to I\), where \(\gamma : I \to M\) is a piece-wise smooth curve.
\begin{center}
\begin{tikzcd}
\gamma^* TM \arrow{r} \arrow[swap]{d}{\pi} & TM \arrow{d}{\pi} \\