diff --git a/5-tensors.tex b/5-tensors.tex index 27b01a5..fe9567e 100644 --- a/5-tensors.tex +++ b/5-tensors.tex @@ -454,9 +454,9 @@ \section{Tensor bundles} \end{example} \begin{exercise}[\textit{[homework 3]}] - Let $F:N\to P$ and $G:M\to N$ two diffeomorphisms of smooth manifolds. + Let $F:M\to N$ and $G:N\to P$ two diffeomorphisms of smooth manifolds. \begin{enumerate} - \item Show that the chain rule $(F\circ G)_* = F_* \circ G_*$ holds. + \item Show that the chain rule $(G\circ F)_* = G_* \circ F_*$ holds. \item Show that our previous definition\footnote{That is, Definition~\ref{def:pullback0s} -- which includes the pullback from Definition~\ref{def:pullback1f}.} of pullback is a particular case of the following general definition of a \emph{pullback of $(r,s)$-tensor fields by $F$}: \begin{equation} F^* := (F^{-1})_* : \cT_s^r(N) \to \cT_s^r(M).