Divergence

Relevant parts to questions…

• In suffix form: $\nabla \cdot \mathbf{u}=\partial u_j/\partial x_j={\nabla }_j{u}_j$.
• Identity: $\nabla \cdot (\nabla \times \mathbf{u})=0$ - ${ϵ}_{ijk}{\partial }_i{\partial }_j{u}_k=0$ by “kill rule”.
• Curvilinear: divergence is the trace of the covariant derivative: $\nabla \cdot \mathbf{A}=A_{\cdot ,i}^i$.

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 (divergence of a vector is a scalar). Positive $\Rightarrow$ source; negative $\Rightarrow$ sink.

Example

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

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

Divergence of a curl ( $\nabla \cdot (\nabla \times \mathbf{u})=0$)

$\nabla \cdot (\nabla \times \mathbf{u})={\partial }_i{ϵ}_{ijk}{\partial }_j{u}_k={ϵ}_{ijk}{\partial }_i{\partial }_j{u}_k$. Mixed partials are symmetric in $i,j$; ${ϵ}_{ijk}$ is antisymmetric in $i,j$. By the kill rule, this is zero.

Vector in, scalar out.