Biharmonic Equation

 del ^4phi=0.
In Cartesian coordinates, the biharmonic equation is
del ^4phi=del ^2(del ^2)phi
=((partial^2)/(partialx^2)+(partial^2)/(partialy^2)+(partial^2)/(partialz^2))((partial^2)/(partialx^2)+(partial^2)/(partialy^2)+(partial^2)/(partialz^2))phi
=(partial^4phi)/(partialx^4)+(partial^4phi)/(partialy^4)+(partial^4phi)/(partialz^4)+2(partial^4phi)/(partialx^2partialy^2)+2(partial^4phi)/(partialy^2partialz^2)+2(partial^4phi)/(partialx^2partialz^2)
=0

http://mathworld.wolfram.com/BiharmonicEquation.html

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