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The Laplace’s equation is often encountered in electrostatics, magnetic, heat and mass transfer theory, fluid mechanics, elasticity, and other areas of engineering and physics. It has in general the form:

           Δu = 0 --------------------- [4286a]

where the Laplacian Δ is defined in Cartesian coordinates by,

           ------------- [4286b]

We may also write Δ = div D. A solution of Laplace's equation is called a harmonic function.

The two-dimensional (2D) Laplace equation can be given by,

           , in the Cartesian coordinate system ------- [4286c]

           , in the polar coordinate system -------- [4286d]

where,

          x = r cosφ -------------------------- [4286e]
          y = r sinφ --------------------------- [4286f]
          ---------------------------[4286g]