Let $(M,g)$ be a compact smooth Riemannian manifold without boundary, let $(N,h)$ be a smooth Riemannian manifold, and let $u:M\to N$ be a smooth map. Define the Dirichlet energy by
\begin{align*}
U:(-\varepsilon,\varepsilon)\times M &\to N
\end{align*}
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be a smooth variation of $u$, meaning $U(0,p)=u(p)$ for every $p\in M$. For each $s\in(-\varepsilon,\varepsilon)$, define $U_s:M\to N$ by $U_s(p):=U(s,p)$, and define the variation field