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Math Help - Simple vector calculus derivation

  1. #1
    Senior Member
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    Simple vector calculus derivation

    I believe this is simple to derive, but I can't seem to do it.

    How does one get from the LHS to RHS

    f\nabla^2u=\nabla \cdot (f\nabla u)-\nabla u\cdot \nabla f

    where f=f(x,y) and u=u(x,y).
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  2. #2
    Senior Member
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    Better not to use word "simple". Everyting is relative.
    At first nabla acts as differential operator of two terms:
    \nabla \cdot (f\nabla u)=(\nabla f , \nabla u)+f(\nabla , \nabla u)
    and
    f(\nabla , \nabla u)=f \nabla^2 u
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  3. #3
    Senior Member
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    It is better to check it directly
    \nabla \cdot (f\nabla u)=div(f \nabla u)=div(f({ \frac{\partial u}{\partial x}},{ \frac{\partial u}{\partial y}}))=

    ={ \frac{\partial }{\partial x}}( f{ \frac{\partial u}{\partial x}})+{ \frac{\partial }{\partial y}}( f{ \frac{\partial u}{\partial y}})=...
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