Coordinate transformation of nabla operator

[Marin]

Hi all!

I am studying the Galilean group of transformations and I'm not sure how to transform the Nabla operator.

Consider the 2 transformations:

(x,t)->(x+s,t)
(x,t)->(Dx,t)

and the expression "nabla (x)"

where D is a matrix and x, s are vectors

I am pretty sure that I have to substitute x+s or Dx for x, but what about the nabla operator. How am I supposed to transform it?

And another question: If I have a transformation which somehow changes time (t->t+T), and a second derivative dx(t)/dt of a function, then does the derivative change its variable from t to t+T [dx(t+T)/d(t+T)] under the transformation or not?




thanks a lot in advance

best regards, marin

 

[HallsofIvy]

Use the chain rule. If x'= ax+ d y'= by+ e and z'= cz+ e, or $<x', y', z'>= <ax+ d, by+ d, cz+ e>$ then
$$\frac{\partial f}{\partial x}= \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial x}= a\frac{\partial f}{\partial x}$$
so that
$$\nabla \left<f, g, \right>= \eft< a\frac{\partial f}{\partial x'}, b\frac{\partial f}{\partial y'}, c\frac{\partial f}{\partial z'}\right>$$