Practical Electron Microscopy and Database

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]