I have question about the inner product of the material derivative.
$$\frac{D\mathbf{v}}{Dt}=\frac{d\mathbf{v}}{dt}+\mathbf{v}\cdot\nabla\mathbf{v}.$$
How can you calculate the second term inner product?
$$\mathbf{v}\cdot\nabla\mathbf{v}=\begin{bmatrix}v_1\\v_2\\v_3\end{bmatrix}\begin{bmatrix}a_{11} & \cdots & a_{13} \\\vdots & \ddots & \vdots \\a_{31} & \cdots & a_{33}\end{bmatrix}.$$
I wrote the right dyad as $a$. The length of the vectors and height of the matrix does not match. How can you calculate it?
