In these web notes on solve PDEs bu using the finite element method (FEM), when going from the 5th to 6th equation from the top, the Authors write the following identity
$$
\int_\Omega \nabla \cdot (\nabla \mathbf{T}) \cdot \mathbf{s} d\Omega = -\int_\Omega \nabla \mathbf{T} \cdot \nabla \mathbf{s} d\Omega + \int_\Omega \nabla \cdot (\nabla \mathbf{T}\cdot \mathbf{s})d\Omega
$$
where $\mathbf{T}, \mathbf{s} \in \mathbb{R}^3$ are vectors.
I am confused by the term containing $\nabla \mathbf{T} \cdot \nabla \mathbf{s}$: precisely both $\nabla \mathbf{T}$ and $\nabla \mathbf{s}$ are both the gradient of a vector, which is a matrix.
What is their dot product then?
I think the output is supposed to be a scalar: then shouldn't it be $\nabla \mathbf{T} : \nabla \mathbf{s}$?