Divergence

The Divergence of a vector field $\mathbf{u}$ is $\nabla \cdot \mathbf{u}=\frac{\partial u_1}{\partial x_1}+\frac{\partial u_2}{\partial x_2}+\frac{\partial u_3}{\partial x_3}=\frac{\partial u_j}{\partial x_j}={\nabla }_i{u}_i$. Note the dummy index $j$ for the summation $j=1,2,3$, and the complete lack of free indices (as the divergence of a vector is a scalar quantity). If positive, it indicates a source, if negative, it indicates a sink.

Example

Find $\nabla \cdot \mathbf{A}$ at the point $Q=(2,1,2)$ for $\mathbf{A}=\frac{1}{2}{x}_1^3{x}_2\mathbf{i}-(4x_1{x}_2^5+1)\mathbf{j}+x_2{x}_3^3\mathbf{k}$…

$\nabla \cdot \mathbf{A}=\frac{3}{2}{x}_1^2{x}_2+20x_1{x}_2^4+3x_2{x}_3^2:\nabla \cdot \mathbf{A}{|}_{(2,1,2)}=6+40+12=\boxed{58}$.

Vector in, scalar out.