Introduction to the Modeling and Analysis of Complex Systems

13.2. FUNDAMENTALS OF VECTOR CALCULUS 229finally facing systems with infinitely many variables!But of course, we are not actually capable of modeling or analyzing systems made ofinfinitely many variables. In order to bring these otherwise infinitely complex mathematicalmodels down to something manageable for us who have finite intelligence and lifespan,we usually assume the smoothness of function f. This is why we can describe the shapeand behavior of the function using well-defined derivatives, which may still allow us tostudy their properties using analytical means 1 .13.2 Fundamentals of Vector CalculusIn order to develop continuous field models, you need to know some basic mathematicalconcepts developed and used in vector calculus. A minimalistic quick review of thoseconcepts is given in the following.ContourA contour is a set of spatial positions x that satisfyf(x) = C (13.2)for a scalar field f, where C is a constant (see an example in Fig. 13.2). Contoursare also called level curves or surfaces.GradientA gradient of a scalar field f is a vector locally defined as⎛ ⎞∂f∂x 1∂f∇f =∂x 2. (13.3)⎜ .⎟⎝ ∂f ⎠∂x n1 But we should also be aware that not all physically important, interesting systems can be represented bysmooth spatial functions. For example, electromagnetic and gravitational fields can have singularities wheresmoothness is broken and state values and/or their derivatives diverge to infinity. While such singularitiesdo play an important role in nature, here we limit ourselves to continuous, smooth spatial functions only.