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The Laplacian Operation on Products and Powers of Scalar Functions |
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The Laplacian operator, which is denoted as ∇²( ), is the divergence of the vector field that results from taking the gradient of a scalar field. Thus it is more properly represented as ∇·∇( ).
It is a linear operation in that
However ∇²(fg) is not equal to f∇²(g)+g∇²(f). It is somewhat more complicated. First the gradient of fg is taken and the formula is
The the divergence is taken of the terms on the RHS. The divergence of the first term on the right is
The divergence of the second term is
Thus these two equations combined gives
A power of a function is just a special kind of product. If f=g the previous formula reduces to
For f³
(To be continued.)
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