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Rotation of an equation

  1. Feb 29, 2012 #1
    This is sort of a homework question but I'm not looking for an answer. I'm just trying to understand exactly what's going on. It says "Among all the equations of the form [the general second order linear homogeneous partial differential equation], show that the only ones that are unchanged under all rotations (rotationally invariant) havce the form a(uxx + uyy) + bu =0.

    What exactly does it mean for an equation to be rotated? I don't understand what's going on here very well.
     
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  3. Feb 29, 2012 #2

    AlephZero

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    I would guess it means if you rotate the coordinate system through an arbitrary angle, the form of the equation stays the same, i.e. you don't get a uxy term.
     
  4. Mar 1, 2012 #3

    HallsofIvy

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    Let [itex]x= x'cos(\theta)+ y' sin(\theta)[/itex], [itex]y= -x'sin(\theta)+ y'cos(\theta)[/itex], so that [itex]x'= xcos(\theta)- ysin(\theta)[/itex] and [itex]y'= xsin(\theta)+ ycos(\theta)[/itex], and use the chain rule to replace [itex]u_{xx}[/itex] and [itex]u_{yy}[/itex] with derivatives in terms of x' and y' rather than x and y.

    For example, [itex]u_x= u_x'(x'_x)+ u_y'(y'_x)= cos(\theta)u_x'+ sin(\theta)u_y'[/itex].
     
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