Let $ \overrightarrow v=v_i \overrightarrow e_i \thinspace,\nabla \cdot v=0$ , show that $\nabla \cdot(\nabla v)=0 $
Gradient is defined as : $$\nabla v=\frac{\partial v}{\partial x_j} \otimes e_j=\frac{\partial (v_ie_i)}{\partial x_j}\otimes e_j=\frac{\partial v_i}{\partial x_j}e_i\otimes e_j$$
Divergence is defined as:
$$\nabla \cdot v=\frac{\partial v}{\partial x_j}\cdot e_j=\frac{\partial (v_ie_i)}{\partial x_j}\cdot e_j=\frac{\partial v_i}{\partial x_j}(e_i\cdot e_j)= \frac{\partial v_i}{\partial x_j}(\delta_{ij})=\frac{\partial v_i}{\partial x_i}$$ I got confused abut proving this in Einstein notation so It would be great if this made clear.