Introduction to the Modeling and Analysis of Complex Systems

13.2. FUNDAMENTALS OF VECTOR CALCULUS 235The gradient field of f shows a diverging pattern where the scalar field f is concavelike a dip or a valley, or a converging pattern where f is convex like a hump or a ridge.Therefore, the Laplacian of f, which is the divergence of the gradient field of f, has apositive value where f is concave, or a negative value where f is convex. This is similarto the second-order derivative of a mathematical function; a concave function has a positivesecond-order derivative while a convex function has a negative one. The Laplacian isa generalization of the same concept, applicable to functions defined over a multidimensionaldomain.Exercise 13.2 Which of the following surface plots correctly illustrates the shapeof a scalar field f(x, y) = xy(x − 1)?Exercise 13.3 Which of the following vector field plots correctly illustrates theflows given by a vector field v(x, y) = (−xy, x − y)?Calculate the gradient field and the Laplacian of each of the fol-Exercise 13.4lowing: