Taking the partial time derivative of a functional

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  • Thread starter Arman777
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  • #1
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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.
 

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  • #2
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Your notation doesn't make any sense. Firstly, you wrote an equation of differential operators, not derivatives. Secondly, you evaluate an operator depending on one variable at a location of another variable. What does that mean? If it is only a coordinate transformation, then you need the chain rule, that's all.
 

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