Consider the 2-rank tensor $Q_{ij}$:
$$Q_{ij} = \begin{bmatrix} Q_{11} & Q_{12} & Q_{13} \\ Q_{21} & Q_{22} & Q_{33} \\ Q_{31} & Q_{32} & Q_{33} \end{bmatrix}.$$
Consider then the integral: $$\int_L{\nabla\cdot (\nabla \cdot Q)*dr}.$$
How would I apply the divergence theorem to this integral? From other answers on stack exchange, I expect:
$$\int_L{\nabla\cdot (\nabla \cdot Q)*dr} = (\nabla \cdot Q)|_a^b$$
However, for the particular problem I am working on, I expect a scalar answer from evaluating this integral (it would be non-sensical for this quantity to be a vector).
In 2 and 3 dimensions, applying the divergence equates the volume integral of the divergence of a quantity to the flux normal to the boundary--keyword being normal. Thus, this problem doesn't seem to arise.
Am I just missing this normal unit vector when applying the 1D divergence theorem, i.e., misapplying the theorem?
Thank you!