Laplacian | Physics Library

The Laplacian is a vector differential operator. Like all vector operators, it is given in different forms in different coordinate systems. In general it is given by:

2 ∑ -∂fi ∇ f = Δf = ∂x2 i i

where the subscript i refers to the different coordinate components of the vector f.

0.1 Laplacian in Cartesian coordinates

As usual with vector operators, the Cartesian form is the easiest to remember and apply.

∇2 = -∂--+ -∂--+ --∂- ∂x2 ∂y2 ∂z2

0.2 Laplacian in spherical coordinates

( ) ( ) 2 1--∂- 2-∂- ---1----∂- -∂- ---1----∂2- ∇ sph = r2∂r r ∂r + r2sin 𝜃∂𝜃 sin𝜃∂ 𝜃 + r2sin2 𝜃∂ ϕ2

0.3 Laplacian in cylindrical coordinates

1 ∂ ( ∂ ) 1 ∂2 ∂2 ∇2 = ----- r--- + -2---2 + --2- r ∂r ∂r r ∂ 𝜃 ∂z

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