Time-dependent vector field
= Time-dependent vector field
{title2=$\partial_t\rho_t=Y_t\circ\rho_t$}
A smooth family $Y_t$ assigns a tangent vector at each space-time point. Its <flow maps> solve $\partial_t\rho_t=Y_t\circ\rho_t$, with $\rho_0$ the identity. For a varying <differential form>, differentiation gives $\partial_t(\rho_t^*\alpha_t)=\rho_t^*(\dot\alpha_t+\mathcal L_{Y_t}\alpha_t)$.