📂Vector Analysis

Symmetrized Gradient

Definition ¹

Let the Jacobian of $\mathbf{u}$ be simply denoted by $\nabla \mathbf{u}$. The matrix operation $\epsilon (\mathbf{u})$ defined as follows is called the symmetrized gradient. $$ \varepsilon (\mathbf{u}) = {\frac{ 1 }{ 2 }} \left( \nabla \mathbf{u} + \left( \nabla \mathbf{u} \right)^{T} \right) $$

Explanation

The symmetrized gradient is defined as the average of the tensor $\nabla \mathbf{u}$ and its transpose matrix $\left( \nabla \mathbf{u} \right)^{T}$. As the definition itself indicates, $\varepsilon (\mathbf{u})$ is a symmetric matrix.

The symmetrized gradient arises when dealing with partial differential equations, and in particular appears when describing Newton’s law of viscosity. Typically, $\mathbf{u}$ is represented as a three-dimensional vector function. Because the gradient is defined for scalar functions, the gradient of $\mathbf{u}$, when treated as a vector function, is the Jacobian. To express this more explicitly in matrix form, one can write: $$ \nabla \mathbf{u} = \nabla \left( u_{1} , u_{2} , u_{3} \right) = \begin{bmatrix} {\frac{ \partial u_{1} }{ \partial x_{1} }} & {\frac{ \partial u_{1} }{ \partial x_{2} }} & {\frac{ \partial u_{1} }{ \partial x_{3} }} \\ {\frac{ \partial u_{2} }{ \partial x_{1} }} & {\frac{ \partial u_{2} }{ \partial x_{2} }} & {\frac{ \partial u_{2} }{ \partial x_{3} }} \\ {\frac{ \partial u_{3} }{ \partial x_{1} }} & {\frac{ \partial u_{3} }{ \partial x_{2} }} & {\frac{ \partial u_{3} }{ \partial x_{3} }} \end{bmatrix} $$