galilean transformation identity

Question

in an excericse im doing the gallilean transformation in 3D of spacetime is defined as $$ \vec{x}=\vec{x}'+\vec{v}t $$ and $$ t=t' $$ and later it says that $ \vec{\nabla}=\vec{\nabla}' $ .That I still understand. But after that he writes $$ \frac{d}{dt}=\frac{dt'}{dt}\frac{d}{dt'}+\frac{d\vec{x}'}{dt}\vec{\nabla}' $$ that I dont get. Im assuming that $\frac{d\vec{x}'}{dt}\vec{\nabla}'=0$ since $\frac{dt'}{dt}\frac{d}{dt'}=\frac{d}{dt}$. But I dont know why that should be true

Answer 1

If $\vec{x}'$ is a function of $t$ (or equivalently $t'$) then, by the chain rule, $$ \frac{d}{dt'}f(t',\vec{x}')=\frac{\partial}{\partial t'}f(t',\vec{x}')+\vec{\nabla}'f(\vec{x}')\cdot\frac{d\vec{x}'}{dt'}. $$ In short, $$ \boxed{\frac{d}{dt'}=\frac{\partial}{\partial t'}+\frac{d\vec{x}'}{dt'}\cdot\vec{\nabla}'\,.} $$ To me it looks like that, the term $$ \frac{dt'}{dt}\frac{d}{dt'} $$ is a bad notation for $\frac{\partial}{\partial t'}\,.$ In this setting one should more carefully distinguish between partial and total time derivatives.