Introduction to the Modeling and Analysis of Complex Systems

230 CHAPTER 13. CONTINUOUS FIELD MODELS I: MODELINGFigure 13.2: Contour f(x, y) = 1/2 (blue solid circle) of a spatial function f(x, y) =e −(x2 +y 2) .Here, n is the number of dimensions of the space. The symbol ∇ is called “del” or “nabla,”which can be considered the following “vector” of differential operators:⎛ ⎞∂∂x 1∂∇ =∂x 2(13.4)⎜ .⎟⎝ ∂ ⎠∂x nA gradient of f at position x is a vector pointing toward the direction of the steepestascending slope of f at x. The length of the vector represents how steep the slope is. Avector field of gradient ∇f defined over a scalar field f is often called a gradient field (Seean example in Fig. 13.3). The gradient is always perpendicular to the contour that goesthrough x (unless the gradient is a zero vector; compare Figs. 13.2 and 13.3).