Calculation of Lie derivative

Question

Picture below is from 4th page of A mean curvature type flow in space forms, I try to calculate the red line, but I got a different result. $$ \mathcal L_V d\rho^2 = d(\mathcal L _V \rho^2)=d(\phi(\rho)\partial_\rho(\rho^2)) =d(2\rho \phi(\rho))=2\rho \phi' d\rho+2\phi d\rho $$ why the $2\phi d\rho$ vanish ?

Answer 1

This is how you can prove the red line. $$\mathcal L_V(d\rho \otimes d\rho) = \mathcal L_V(d\rho) \otimes d\rho + d\rho\otimes \mathcal L_V(d\rho) $$ Since $V=\phi \frac \partial {\partial \rho}$ exploiting the commutativity of $\mathcal{L}$ and $d$ we have that $\mathcal{L}_V(d\rho)=d \mathcal{L}_V(\rho)=d(V(\rho))=d\phi=\frac \partial {\partial \rho}\phi \ d\rho.$ Thus $$\mathcal L_V(d\rho \otimes d\rho) = \mathcal L_V(d\rho) \otimes d\rho + d\rho\otimes \mathcal L_V(d\rho) = 2 \frac \partial {\partial \rho}\phi \ d\rho\otimes d\rho $$