I have some problems in acting with this operator algebrically. If I define: $$\boldsymbol p(\boldsymbol x)\in \mathbb R^3 \ \ ,\ \ P_{\alpha\beta}:=\delta_{\alpha\beta}-p_\alpha p_\beta$$ how does it affect for instance this kind of terms?
$$\boldsymbol{p}\cdot\nabla\boldsymbol{p}\ \\ \nabla^2\boldsymbol{p}\\ \boldsymbol{p}(\nabla\cdot\boldsymbol{p})$$