Let $d\in\mathbb N$, $M$ be a bounded $d$-dimensional properly embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, $\sigma_{\partial\Omega}$ denote the surface measure on $\partial\Omega$, $\operatorname P_{\partial\Omega}(x)$ denote the orthogonal projection of $\mathbb R^d$ onto $T_x\:\partial\Omega$ for $x\in\partial\Omega$, $$\operatorname{div}_{\partial\Omega}f(x):=\operatorname{tr}\left({\rm D}f(x)\circ\operatorname P_{\partial\Omega}(x)\right)\;\;\;\text{for }x\in\partial\Omega$$ for $f\in C^1(\mathbb R^d,\mathbb R^d)$, $E$ be a $\mathbb R$-Banach space, $f\in C^1(\mathbb R^d,E)$, $\Theta\subseteq C^{0,\:1}(\mathbb R^d,\mathbb R^d)$ be a $\mathbb R$-Banach space continuously embedded into $C^1(\mathbb R^d,\mathbb R^d)$.
I would like to show that $$g(\theta):=\int {\rm D}f(x)\theta(x)+f(x)(\operatorname{div}_{\partial\Omega}\theta)(x)\:\sigma_{\partial\Omega}({\rm d}x)$$ for $\theta\in\Theta$ is continuous.
The trace functional and any orthogonal projection have operator norm $1$. So, we should have $$\left|(\operatorname{div}_{\partial\Omega}\theta)(x)\right|\le\left\|{\rm D}\theta(x)\operatorname P_{\partial\Omega}(x)\right\|_{\mathfrak L(\mathbb R^d)}\le\left\|{\rm D}\theta(x)\right\|_{\mathfrak L(\mathbb R^d)}\le\left\|\left.\theta\right|_{\partial\Omega}\right\|_{C^1(\partial\Omega,\:\mathbb R^d)}\tag1$$ for all $x\in\partial\Omega$.
Now, $\partial\Omega$ is compact, but I'm not sure how we need to use the assumption that $\Theta$ is continuously embedded into $C^1(\mathbb R^d,\mathbb R^d)$.
Does this assumption imply that for all compact $K\subseteq\mathbb R^d$, there is a $c\ge0$ with $$\left\|\left.\theta\right|_K\right\|_{C^1(K,\:\mathbb R^d)}\le c\left\|\theta\right\|_{\Theta}?\tag2$$