What is the meaning of notation $ \nabla u + \nabla u^T$?

Question

In my exercises it appears ($u:\mathbb{R}^3\rightarrow\mathbb{R}^3$):

$$ \nabla u + \nabla u^T$$

What is the meaning of this notation when writing it in terms of partials of $u_i$?

Answer 1

Typically, when $u:\Bbb R^3 \to \Bbb R^3$, we define $$ \newcommand{\pwrt}[2]{\frac{\partial #1}{\partial #2}}\\ \nabla u = \pmatrix{\pwrt {u_1}{x_1} & \pwrt {u_1}{x_2} & \pwrt {u_1}{x_3}\\ \pwrt {u_2}{x_1} & \pwrt {u_2}{x_2} & \pwrt {u_2}{x_3}\\ \pwrt {u_3}{x_1} & \pwrt {u_3}{x_2} & \pwrt {u_3}{x_3}\\} $$ In some contexts, this is referred to instead as the Jacobian of $u$.

$M^T$ refers to the transpose of the matrix $M$.

Answer 2

$M=\nabla u \; \;$ is a $(3\times 3)$ matrix with $$M_{i,j}=\frac{\partial u_i}{\partial x_j}$$

and

$$\nabla u^T=M^T=N$$ with $$N_{i,j}=M_{j,i}=\frac{\partial u_j}{\partial x_i}$$ is the transpose matrix of $M$.