San José State University
Thayer Watkins
Silicon Valley
& Tornado Alley

The Laplacian Operation on Products
and Powers of Scalar Functions

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

∇²(f + g) = ∇²(f) + ∇²(g)

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

∇(fg) = (∇(f))g + f∇(g)

The the divergence is taken of the terms on the RHS. The divergence of the first term on the right is

∇·((∇(f))g) = (∇²(f))g + (∇(f))·(∇(g))

The divergence of the second term is

∇·(f(∇(g))) = (∇(f))·(∇(g)) + f∇²(g)

Thus these two equations combined gives

∇²(fg) = (∇²(f))g + 2(∇(f))·(∇(g)) + f∇²(g)

The Laplacian of Powers of a Function

A power of a function is just a special kind of product. If f=g the previous formula reduces to

∇²(f²) = 2∇²(f) + 2∇(f))·(∇(f)

For f³

∇²(f³) = ∇²(f²f)
= ∇²(f²)) + 2∇[(∇f)·(∇f)]·(∇f) + ∇²(f)

(To be continued.)

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