Introduction to the Modeling and Analysis of Complex Systems

234 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGLaplacianA Laplacian of a scalar field f is a scalar field defined as⎛∇ 2 f = ∇ · ∇f =⎜⎝∂∂x 1∂∂x 2.∂∂x n(See an example in Fig. 13.5)⎞⎟⎠T ⎛⎜⎝∂∂x 1∂∂x 2.∂∂x n⎞f = ∂2 f∂x 2 1⎟⎠+ ∂2 f∂x 2 2+ . . . + ∂2 f. (13.13)∂x 2 nFigure 13.5: Laplacian (i.e., divergence of a gradient field) of a spatial functionf(x, y) = e −(x2 +y 2) . Compare this figure with Fig. 13.1.Sometimes the Laplacian operator is denoted by a right side up triangle ∆ instead of∇ 2 . This is so confusing, I know, but blame the people who invented this notation. Inthis textbook, we use ∇ 2 instead of ∆ to denote Laplacians, because ∇ 2 is more intuitiveto show it is a second-order differential operator, and also because ∆ is already used torepresent small quantities (e.g., ∆x).