[proofplan] We express $\mathcal{H}^m(\Gamma_f(U))$ as the integral of the $m$-dimensional Jacobian of the graph parametrization $\Phi(x) = (x, f(x))$. The Jacobian matrix $J\Phi_x \in \mathbb{R}^{(m+1) \times m}$ has a specific block structure that allows explicit computation of $J_m(D\Phi_x) = \sqrt{\det(J\Phi_x^\top J\Phi_x)}$. We verify that this determinant equals $1 + |\nabla f(x)|^2$, then apply the area formula for injective $C^1$ maps to conclude. [/proofplan] [step:Define the graph parametrization and compute its Jacobian matrix] Define the graph parametrization \begin{align*} \Phi: U &\to \mathbb{R}^{m+1} \\ x &\mapsto (x, f(x)). \end{align*} Since $f \in C^1(U; \mathbb{R})$, the map $\Phi$ is $C^1$ on $U$. The Jacobian matrix of $\Phi$ at $x \in U$ is the $(m+1) \times m$ matrix \begin{align*} J\Phi_x = \begin{pmatrix} I_m \\ \nabla f(x)^\top \end{pmatrix} \in \mathbb{R}^{(m+1) \times m}, \end{align*} where $I_m$ is the $m \times m$ identity matrix and $\nabla f(x)^\top = (\partial_{x_1} f(x), \ldots, \partial_{x_m} f(x))$ is the $1 \times m$ row vector of partial derivatives. [/step] [step:Compute the $m$-dimensional Jacobian $J_m(D\Phi_x) = \sqrt{1 + |\nabla f(x)|^2}$] The $m$-dimensional Jacobian of $D\Phi_x$ is defined as \begin{align*} J_m(D\Phi_x) = \sqrt{\det(J\Phi_x^\top J\Phi_x)}. \end{align*} We compute the $m \times m$ matrix $J\Phi_x^\top J\Phi_x$. Writing $v = \nabla f(x) \in \mathbb{R}^m$ (as a column vector), we have \begin{align*} J\Phi_x^\top J\Phi_x = \begin{pmatrix} I_m & v \end{pmatrix} \begin{pmatrix} I_m \\ v^\top \end{pmatrix} = I_m \cdot I_m + v \cdot v^\top = I_m + v v^\top. \end{align*} The matrix $vv^\top$ is the rank-one matrix with entries $(vv^\top)_{ij} = v_i v_j$, where $v_i = \partial_{x_i} f(x)$. To compute $\det(I_m + vv^\top)$, we use the matrix determinant lemma: for any column vector $v \in \mathbb{R}^m$, \begin{align*} \det(I_m + vv^\top) = 1 + v^\top v = 1 + |v|^2. \end{align*} Therefore \begin{align*} J_m(D\Phi_x) = \sqrt{1 + |v|^2} = \sqrt{1 + |\nabla f(x)|^2}. \end{align*} [/step] [step:Apply the area formula to obtain the integral representation] The map $\Phi: U \to \mathbb{R}^{m+1}$ is $C^1$ and injective (if $\Phi(x) = \Phi(y)$, then the first $m$ components give $x = y$). The area formula for injective $C^1$ maps from an open subset of $\mathbb{R}^m$ to $\mathbb{R}^{m+1}$ states \begin{align*} \mathcal{H}^m(\Phi(U)) = \int_U J_m(D\Phi_x) \, d\mathcal{L}^m(x). \end{align*} Since $\Phi(U) = \Gamma_f(U)$ and $J_m(D\Phi_x) = \sqrt{1 + |\nabla f(x)|^2}$ by the computation above: \begin{align*} \mathcal{H}^m(\Gamma_f(U)) = \int_U \sqrt{1 + |\nabla f(x)|^2} \, d\mathcal{L}^m(x). \end{align*} [/step]