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Vector calculus identity proof.

  1. Sep 19, 2009 #1


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    1. The problem statement, all variables and given/known data

    Let f(x,y,z) be a function of three variables and G(x,y,z) be a vector field defined in 3D space. Prove the identity:

    div(fG)= f*div(G)+G*grad(f)

    2. Relevant equations

    For F=Pi +Qj+Rk

    div(F)=dF/dx + dQ/dy + dR/dz

    grad(F)=dF/dx i + dQ/dy j + dR/dz k

    3. The attempt at a solution

    My problem starts with how do I find fG? Because I am thinking that f is a vector with i,j,k components and so is G. So fG should be the dot product of f and G, which gives a scalar, and one can't get the divergence of a scalar :confused: (Since this is an identity, I know somewhere I am missing some elementary fact)
  2. jcsd
  3. Sep 19, 2009 #2


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    No, [itex]f(x,y,z)[/itex] is a scalar function, [itex]\textbf{G}(x,y,z)[/itex] is a vector field.

    So, [tex]f\textbf{G}=f(x,y,z)G_x(x,y,z)\textbf{i}+f(x,y,z)G_y(x,y,z)\textbf{j}+f(x,y,z)G_z(x,y,z)\textbf{k}[/tex]
  4. Sep 19, 2009 #3


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    :biggrin: I got it now!
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