[proofplan] We differentiate the energy density along the variation and express the result using the pullback Levi-Civita connection on $U^{-1}TN$. Metric compatibility converts the derivative of the local energy density into the contraction of $\nabla V$ with $du$. We then integrate by parts on the compact boundaryless manifold $M$ by applying the divergence identity to the vector field metrically dual to the one-form $X\mapsto h(V,du(X))$. The remaining term is exactly the negative pairing of $V$ with the tension field $\tau(u)=\operatorname{tr}_g(\nabla du)$. [/proofplan]

[step:Differentiate the energy density along the variation]Let $I:=(-\varepsilon,\varepsilon)$, and let $\pi:I\times M\to M$ be the projection defined by $\pi(s,p):=p$. Equip the pullback bundle $U^{-1}TN\to I\times M$ with the connection induced by the Levi-Civita connection of $(N,h)$; denote this connection by $\nabla^U$. For fixed $s\in I$, denote by $\nabla^s$ the induced connection on $U_s^{-1}TN$. Let $p\in M$, and choose a local $g$-orthonormal frame $e_1,\dots,e_m$ on an open neighbourhood $W\subset M$ of $p$, where $m:=\dim M$. We view each $e_i$ as a vector field on $I\times W$ independent of the $s$-variable. The local energy density of $U_s$ is the smooth map $e_U:I\times W\to \mathbb{R}$ defined at each point $(s,q)\in I\times W$ by \begin{align*} e_U(s,q)=\frac{1}{2}\sum_{i=1}^m h\bigl(dU_{(s,q)}(e_i),dU_{(s,q)}(e_i)\bigr). \end{align*} This definition is independent of the chosen local $g$-orthonormal frame, since it is the $g$-trace of the symmetric [bilinear form](/page/Bilinear%20Form) $(X,Y)\mapsto h(dU_s(X),dU_s(Y))$. Thus these local definitions patch to a global smooth energy density $e_U:I\times M\to\mathbb{R}$. Choose $\delta\in(0,\varepsilon)$. Since $M$ is compact and $\partial_s e_U$ is continuous on the compact set $[-\delta,\delta]\times M$, the derivative $\partial_s e_U$ is bounded there. Hence the [differentiation under the integral sign](/page/Differentiation%20Under%20the%20Integral%20Sign) is justified at $s=0$, and \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \partial_s e_U(0,p)\,d\operatorname{vol}_g(p). \end{align*} By metric compatibility of $\nabla^U$ with $h$, \begin{align*} \partial_s e_U = \sum_{i=1}^m h\bigl(\nabla^U_{\partial_s}(dU(e_i)),dU(e_i)\bigr). \end{align*} The torsion-free property of the [Levi-Civita connection](/page/Levi-Civita%20Connection) on $N$ gives the pullback commutation identity \begin{align*} \nabla^U_{\partial_s}(dU(e_i))-\nabla^U_{e_i}(dU(\partial_s)) = dU([\partial_s,e_i]). \end{align*} Because $e_i$ is independent of $s$, $[\partial_s,e_i]=0$, and hence \begin{align*} \nabla^U_{\partial_s}(dU(e_i)) = \nabla^U_{e_i}(dU(\partial_s)). \end{align*} Evaluating at $s=0$ gives $dU(\partial_s)|_{\{0\}\times M}=V$ and $dU(e_i)|_{\{0\}\times M}=du(e_i)$. Therefore \begin{align*} \partial_s e_U(0,p) = \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr), \end{align*} where $\nabla^u$ denotes the pullback connection on $u^{-1}TN$. Thus \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)\,d\operatorname{vol}_g. \end{align*}[/step]

[guided]The energy is an integral of a pointwise quadratic expression in $dU_s$, so the first task is to differentiate that quadratic expression. Let $I:=(-\varepsilon,\varepsilon)$, and let $\pi:I\times M\to M$ be the projection defined by $\pi(s,p):=p$. The map $U:I\times M\to N$ pulls the tangent bundle $TN\to N$ back to a vector bundle $U^{-1}TN\to I\times M$. The Levi-Civita connection of $(N,h)$ induces a connection on this pullback bundle; denote it by $\nabla^U$. Fix $p\in M$. Choose a local $g$-orthonormal frame $e_1,\dots,e_m$ on an open neighbourhood $W\subset M$ of $p$, where $m:=\dim M$. We extend each $e_i$ to $I\times W$ by making it independent of $s$. Define the local energy density as the smooth map $e_U:I\times W\to \mathbb{R}$. For $(s,q)\in I\times W$, set \begin{align*} e_U(s,q)=\frac{1}{2}\sum_{i=1}^m h\bigl(dU_{(s,q)}(e_i),dU_{(s,q)}(e_i)\bigr). \end{align*} This expression is independent of the chosen local orthonormal frame because it is the trace of the bilinear form $(X,Y)\mapsto h(dU_s(X),dU_s(Y))$ with respect to $g$. Choose $\delta\in(0,\varepsilon)$. Since $M$ is compact and $U$ is smooth, the function $e_U$ is smooth on $I\times M$, so $\partial_s e_U$ is continuous on the compact set $[-\delta,\delta]\times M$. Therefore $\partial_s e_U$ is bounded on that compact set. This boundedness is the local domination needed for the [differentiation under the integral sign](/page/Differentiation%20Under%20the%20Integral%20Sign) at $s=0$, and hence \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \partial_s e_U(0,p)\,d\operatorname{vol}_g(p). \end{align*} Now compute $\partial_s e_U$. Metric compatibility of the pullback connection means that for sections $A,B\in\Gamma(U^{-1}TN)$, \begin{align*} \partial_s h(A,B) = h(\nabla^U_{\partial_s}A,B)+h(A,\nabla^U_{\partial_s}B). \end{align*} Apply this with $A=B=dU(e_i)$. The factor $\frac12$ cancels the two equal terms, so \begin{align*} \partial_s e_U = \sum_{i=1}^m h\bigl(\nabla^U_{\partial_s}(dU(e_i)),dU(e_i)\bigr). \end{align*} We next replace $\nabla^U_{\partial_s}(dU(e_i))$ by a derivative of the variation field. This is where the torsion-free property of the [Levi-Civita connection](/page/Levi-Civita%20Connection) on $N$ enters. For vector fields $A,B$ on $I\times M$, the pullback connection satisfies \begin{align*} \nabla^U_A(dU(B))-\nabla^U_B(dU(A)) = dU([A,B]). \end{align*} Taking $A=\partial_s$ and $B=e_i$, and using that the lifted frame $e_i$ is independent of $s$, gives $[\partial_s,e_i]=0$. Hence \begin{align*} \nabla^U_{\partial_s}(dU(e_i)) = \nabla^U_{e_i}(dU(\partial_s)). \end{align*} At $s=0$, the section $dU(\partial_s)$ restricts to the variation field $V\in\Gamma(u^{-1}TN)$, defined by $V(p):=dU_{(0,p)}(\partial_s)$ for $p\in M$, and $dU(e_i)$ restricts to $du(e_i)$. Therefore \begin{align*} \partial_s e_U(0,p) = \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr), \end{align*} where $\nabla^u$ is the pullback connection on $u^{-1}TN$. Substituting this pointwise identity into the differentiated energy gives \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)\,d\operatorname{vol}_g. \end{align*} This formula is the analytic heart of the first variation: the derivative of the energy is the trace pairing between the covariant derivative of the variation field and the differential of $u$.[/guided]

[step:Convert the trace pairing into a divergence plus the tension field]Define the one-form $\alpha\in\Omega^1(M)$ by \begin{align*} \alpha(X):=h(V,du(X)) \end{align*} for every vector field $X\in\mathfrak{X}(M)$. Let $\alpha^\sharp\in\mathfrak{X}(M)$ be the vector field metrically dual to $\alpha$, meaning \begin{align*} g(\alpha^\sharp,X)=\alpha(X) \end{align*} for every $X\in\mathfrak{X}(M)$. At a point $p\in M$, choose the local $g$-orthonormal frame $e_1,\dots,e_m$ so that $\nabla^M_{e_i}e_j=0$ at $p$, where $\nabla^M$ is the Levi-Civita connection of $g$. Then the definition of divergence in the local orthonormal frame gives \begin{align*} \operatorname{div}_g(\alpha^\sharp)(p)=\sum_{i=1}^m e_i\bigl(g(\alpha^\sharp,e_i)\bigr)(p). \end{align*} Since $g(\alpha^\sharp,e_i)=\alpha(e_i)=h(V,du(e_i))$, this becomes \begin{align*} \operatorname{div}_g(\alpha^\sharp)(p)=\sum_{i=1}^m e_i\bigl(h(V,du(e_i))\bigr)(p). \end{align*} Metric compatibility of the pullback connection $\nabla^u$ with $h$ gives \begin{align*} \operatorname{div}_g(\alpha^\sharp)(p)=\sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)(p)+\sum_{i=1}^m h\bigl(V,\nabla^u_{e_i}(du(e_i))\bigr)(p). \end{align*} Because $\nabla^M_{e_i}e_i=0$ at $p$, the second sum is \begin{align*} \sum_{i=1}^m h\bigl(V,(\nabla du)(e_i,e_i)\bigr)(p) = h(V,\tau(u))(p), \end{align*} where \begin{align*} \tau(u):=\operatorname{tr}_g(\nabla du)=\sum_{i=1}^m(\nabla du)(e_i,e_i) \end{align*} is the tension field. Hence, pointwise on $M$, \begin{align*} \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr) = \operatorname{div}_g(\alpha^\sharp)-h(V,\tau(u)). \end{align*}[/step]

[guided]We want to rewrite the trace pairing $\sum_{i=1}^m h(\nabla^u_{e_i}V,du(e_i))$ as a divergence plus a geometric error term. Define the one-form $\alpha\in\Omega^1(M)$ by \begin{align*} \alpha(X):=h(V,du(X)) \end{align*} for every vector field $X\in\mathfrak{X}(M)$. Let $\alpha^\sharp\in\mathfrak{X}(M)$ be the vector field metrically dual to $\alpha$, so that \begin{align*} g(\alpha^\sharp,X)=\alpha(X) \end{align*} for every $X\in\mathfrak{X}(M)$. Fix $p\in M$. Choose a local $g$-orthonormal frame $e_1,\dots,e_m$ near $p$ such that $\nabla^M_{e_i}e_j=0$ at $p$, where $\nabla^M$ is the Levi-Civita connection of $g$. This choice is available by taking a normal orthonormal frame at the point. The divergence of $\alpha^\sharp$ at $p$ is then \begin{align*} \operatorname{div}_g(\alpha^\sharp)(p)=\sum_{i=1}^m e_i\bigl(g(\alpha^\sharp,e_i)\bigr)(p). \end{align*} Since $g(\alpha^\sharp,e_i)=\alpha(e_i)=h(V,du(e_i))$, metric compatibility of the pullback connection $\nabla^u$ gives \begin{align*} \operatorname{div}_g(\alpha^\sharp)(p) = \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)(p) + \sum_{i=1}^m h\bigl(V,\nabla^u_{e_i}(du(e_i))\bigr)(p). \end{align*} The second sum is where the tension field appears. The covariant derivative of $du$ is defined by \begin{align*} (\nabla du)(X,Y):=\nabla^u_X(du(Y))-du(\nabla^M_XY) \end{align*} for vector fields $X,Y\in\mathfrak{X}(M)$. Because $\nabla^M_{e_i}e_i=0$ at $p$, we have \begin{align*} \nabla^u_{e_i}(du(e_i))(p)=(\nabla du)(e_i,e_i)(p). \end{align*} Therefore \begin{align*} \sum_{i=1}^m h\bigl(V,\nabla^u_{e_i}(du(e_i))\bigr)(p) = h(V,\tau(u))(p), \end{align*} where $\tau(u):=\operatorname{tr}_g(\nabla du)=\sum_{i=1}^m(\nabla du)(e_i,e_i)$ is the tension field. Rearranging the divergence identity gives, pointwise on $M$, \begin{align*} \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr) = \operatorname{div}_g(\alpha^\sharp)-h(V,\tau(u)). \end{align*}[/guided]

[step:Integrate the divergence term away on the compact boundaryless domain]Since $M$ is compact and has no boundary, the [divergence theorem](/page/Divergence%20Theorem) implies that the integral of the divergence of any smooth vector field over $M$ vanishes. Applying this to $\alpha^\sharp\in\mathfrak{X}(M)$ gives \begin{align*} \int_M \operatorname{div}_g(\alpha^\sharp)\,d\operatorname{vol}_g=0. \end{align*} Combining this with the identity from the previous step, first we have \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)\,d\operatorname{vol}_g. \end{align*} Substituting the pointwise identity for the trace pairing gives \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \operatorname{div}_g(\alpha^\sharp)\,d\operatorname{vol}_g-\int_M h(V,\tau(u))\,d\operatorname{vol}_g. \end{align*} The divergence integral vanishes, so \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=-\int_M h(V,\tau(u))\,d\operatorname{vol}_g. \end{align*} By symmetry of the Riemannian metric $h$, $h(V,\tau(u))=h(\tau(u),V)$. Therefore \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s] = -\int_M h(\tau(u),V)\,d\operatorname{vol}_g, \end{align*} which is the desired [first variation formula](/theorems/2728).[/step]

[guided]We now integrate the pointwise identity from the previous step. The vector field $\alpha^\sharp$ is smooth because $u$, $V$, $g$, and $h$ are smooth. Since $M$ is compact and has no boundary, the [divergence theorem](/page/Divergence%20Theorem) applies to $\alpha^\sharp$ and gives \begin{align*} \int_M \operatorname{div}_g(\alpha^\sharp)\,d\operatorname{vol}_g=0. \end{align*} From the first step we already proved \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s]=\int_M \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr)\,d\operatorname{vol}_g. \end{align*} Substitute the identity \begin{align*} \sum_{i=1}^m h\bigl(\nabla^u_{e_i}V,du(e_i)\bigr) = \operatorname{div}_g(\alpha^\sharp)-h(V,\tau(u)) \end{align*} into this integral. We obtain \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s] = \int_M \operatorname{div}_g(\alpha^\sharp)\,d\operatorname{vol}_g - \int_M h(V,\tau(u))\,d\operatorname{vol}_g. \end{align*} The divergence integral is zero, so \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s] = -\int_M h(V,\tau(u))\,d\operatorname{vol}_g. \end{align*} Finally, $h$ is a Riemannian metric and hence symmetric, so $h(V,\tau(u))=h(\tau(u),V)$. Therefore \begin{align*} \frac{d}{ds}\Big|_{s=0}E[U_s] = -\int_M h(\tau(u),V)\,d\operatorname{vol}_g, \end{align*} which is exactly the first variation formula in the theorem statement.[/guided]