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del [tex](\bigtriangleup)[/tex], when applied to a scalar, creates a vector with that scalar as each of the XYZ values. eg

[tex]\bigtriangleup . x = (x,x,x)[/tex]

[tex]\bigtriangleup . 3 = (3,3,3)[/tex]

divergence is applied to a vector, and sums the components of the vector into a scalar. eg

[tex]\bigtriangleup . (x,y,z) = x+y+z [/tex]

[tex]\bigtriangleup . (1,2,3) = 1+2+3 = 6 [/tex]

finally, laplacian. This is the one I'm not as sure about. It's applied to a scalar I think?

[tex]\bigtriangleup ^2 = \bigtriangleup(\bigtriangleup) [/tex]

[tex]\bigtriangleup ^2 . x = \bigtriangleup(\bigtriangleup . x) [/tex]

[tex]= \bigtriangleup((x,x,x)) [/tex]

[tex] = 3x [/tex]

That doesn't seem right (I think I'm meant to end up with a vector). Can laplacian be broken up like that or does it have a special rule?