Taking the partial time derivative of a functional

• I
Gold Member
Let us suppose we have a functional of f such that ##f=f((\vec{r}(t),t)## where ##\vec{r}(t) = a(t)\vec{x}(t)##.

I am trying to derive an equation such that

$$\left.\frac{\partial}{\partial t}\right|_r = \left.\frac{\partial }{\partial t}\right|_x + \left.\frac{\partial \vec{x}}{\partial t}\right|_r \cdot \nabla_x$$

where ##\nabla_r = \frac{1}{a}\nabla_x##

It is actually about coordinate transformation.