I see some notation like \begin{align*} \int \nabla \mathbf{u} : \nabla \mathbf{v} \; dx \end{align*}
Here I think the two vectors $\mathbf{u}$ and $\mathbf{v}$ should be column vectors, i.e. $\mathbf{u} = [u_1,u_2,...,u_n]^T$ and $\mathbf{v} = [v_1,v_2,...,v_n]^T$. Is that right?
So, here $\nabla \mathbf{u}$ will be a Jacobian matrix, and $\nabla \mathbf{v}$ as well. Right?
Then, does the operation $\nabla \mathbf{u} : \nabla \mathbf{v}$ mean the element-wise multiplication such that \begin{align*} \nabla \mathbf{u} : \nabla \mathbf{v} &= (\nabla \otimes \mathbf{u}) : (\nabla \otimes \mathbf{v}) \\ &= \left( \begin{bmatrix} \nabla_1 \\ \nabla_2 \\ \vdots \\ \nabla_n \end{bmatrix} \begin{bmatrix} u_1 & u_2 & \cdots & u_n \end{bmatrix} \right) : \left( \begin{bmatrix} \nabla_1 \\ \nabla_2 \\ \vdots \\ \nabla_n \end{bmatrix} \begin{bmatrix} v_1 & v_2 & \cdots & v_n \end{bmatrix} \right) \\ &= \begin{bmatrix} \nabla_1 u_1 \nabla_1 v_1 & \nabla_1 u_2 \nabla_1 v_2 & \cdots & \nabla_1 u_n \nabla_1 v_n \\ \nabla_2 u_1 \nabla_2 v_1 & \nabla_2 u_2 \nabla_2 v_2 & \cdots & \nabla_2 u_n \nabla_2 v_n \\ \vdots & \vdots & \vdots & \vdots \\ \nabla_n u_1 \nabla_n v_1 & \nabla_n u_2 \nabla_n v_2 & \cdots & \nabla_n u_n \nabla_n v_n \end{bmatrix}. \end{align*} Am I right?
I didn't ever see "$:$" before, but I think it is so-called the Frobenius inner product, though its Wiki page doesn't mention this notation.
$\endgroup$ 2 Reset to default
