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Coordinate transformation of nabla operator

  1. Apr 24, 2009 #1
    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
     
  2. jcsd
  3. Apr 24, 2009 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Use the chain rule. If x'= ax+ d y'= by+ e and z'= cz+ e, or [itex]<x', y', z'>= <ax+ d, by+ d, cz+ e>[/itex] then
    [tex]\frac{\partial f}{\partial x}= \frac{\partial f}{\partial x'}\frac{\partial x'}{\partial x}= a\frac{\partial f}{\partial x}[/tex]
    so that
    [tex]\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>[/tex]
     
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