I'm very confused with the the following terms.
In fluid mechanics, the gradient of velocity can be written as a $3\times 3$ matrix, which can be split into the sum of two matrices, i.e., the symmetric one and the skew-symmetric one.
People often denote the symmetric one as the rate of strain tensor:
$$ \textbf{S} = \frac{1}{2}(\nabla U + (\nabla U)^T ),$$
or
$$ S_{ij} = \frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i} \right). $$
Therefore, $\textbf{S}$ would be explicitly expressed as the following matrix:
\begin{bmatrix} \frac{\partial u}{\partial x} & \frac{1}{2} (\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}) & \frac{1}{2} (\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}) \\ & \frac{\partial v}{\partial y} & \frac{1}{2} (\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y})\\ symmetry & & \frac{\partial w}{\partial z} \\ \end{bmatrix}
People use a scalar variable, strain rate, defined as the following equation:
$$ S = \sqrt{2 S_{ij}S_{ij}}.$$
This equation seems to be the Euclidian norm of $\textbf{S}$, which I do not understand.
The Euclidian norm of $\textbf{S}$ should be calculated as the following equation:
$$ The \: Euclidian \: norm \: of \: \textbf{S} \equiv \sqrt{\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2 + \left(\frac{\partial w}{\partial z}\right)^2 + \frac{1}{2} \left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)^2 +\frac{1}{2} \left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)^2 +\frac{1}{2} \left(\frac{\partial v}{\partial z}+\frac{\partial w}{\partial y}\right)^2}.$$
If I'm correctly understanding that $S = \sqrt{2 S_{ij}S_{ij}} = \sqrt{\frac{1}{2} \left(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i}\right)^2 }$, how is this expression equal to $ The\: Euclidian \: norm \: of \: \textbf{S} $?