I have the following expression which I want to evaluate:
$$\int_{\Gamma}\nabla\cdot(\beta uv)d\text{l},$$
where $\Gamma \subset \mathbb{R}$ is a straight line segment, arbitrarily oriented in space. Denote its end-points by $E^-$ and $E^+$, the integral going from the latter to the former, and let $\mathbf{t}_{E^-}$ and $\mathbf{t}_{E^+}$ be co-normal vectors in the points $E^-$ and $E^+$ respectively, such that they are parallell with $\Gamma$ and point away from the line segment. $\beta : \Gamma \rightarrow \mathbb{R}^2$ is a two-dimensional velocity field and $u, v:\Gamma \rightarrow \mathbb{R}$ are scalar valued functions.
I want to compute this integral by using the divergence theorem. Is it correct to say that $\partial\Gamma = \{E^-\}\cup \{E^+\}$ and thus $$\int_{\Gamma}\nabla\cdot(\beta uv)d\text{l} = \mathbf{t}_{E^-}\cdot \mathbf{\beta}uv|_{E^-} + \mathbf{t}_{E^+}\cdot \mathbf{\beta}uv|_{E^+}?$$
This feels intuitively correct having the fundamental theorem of calculus in mind, but I feel that it lacks rigor, since the line segment is arbitrary in space.
Edit: For clarification: the $\nabla$ operator on $\Gamma$ is defined using unique smooth extensions $\overline{\mathbf{w}}(\mathbf{x})=\mathbf{w}\circ\mathbf{p}(\mathbf{x})$, where $\mathbf{x}$ belongs to a neighborhood around $\Gamma$ which allows the closest point mapping $\mathbf{p}(\mathbf{x}) = \mathbf{x} - \phi(\mathbf{x})\mathbf{n}(\mathbf{x})$ to be unique. Here, $\phi$ is a signed distance function such that $\Gamma = \{\mathbf{x} \in \mathbb{R}^2: \phi(\mathbf{x}) = 0\}$ and $\mathbf{n}$ is the corresponding unit normal defined by $\mathbf{n} = \frac{\nabla \phi}{\lVert\nabla \phi\rVert}$. The exact definition of $\nabla\cdot$ of a vector field $\mathbf{w}$ on $\Gamma$ is thus $$\nabla\cdot \mathbf{w} = \nabla \cdot \overline{\mathbf{w}}.$$