Solving Galilean Transform Homework with Gradients in (u,v)

[MarkovMarakov]

Homework Statement

If there is a change of variables:
[tex](\vec x(t),t)\to (\vec u=\vec x+\vec a(t),\,\,\,v=t+b)[/tex] where [itex]b[/itex] is a constant.

Suppose I wish to write the following expression in terms of a gradient in [itex](\vec u, v)[/itex]

[tex]\nabla_\vec x f(\vec x,t)+{d^2\vec a\over dt^2}[/tex] How do I do that?

Homework Equations

Please see above.

The Attempt at a Solution

For the first term, I think
[tex]f(\vec x, t)\to f(\vec u -\vec a, v-b)[/tex]
I am not sure what to do with the second term though.

 

[CFede]

You should use chain rule I think, so

[itex]\frac{\partial }{\partial x}=\frac{\partial u}{\partial x}\frac{\partial}{\partial u}[/itex]

If I understood your question, this is what you are looking for.