Introduction to the Modeling and Analysis of Complex Systems

13.2. FUNDAMENTALS OF VECTOR CALCULUS 23121y0-1-2-2 -1 0 1 2Figure 13.3: Gradient field of a spatial function f(x, y) = e −(x2 +y 2) .xDivergenceA divergence of a vector field v is a scalar field defined as⎛∇ · v =⎜⎝∂∂x 1∂∂x 2.∂∂x n⎞⎟⎠T⎛⎜⎝⎞v 1v 2⎟.v n⎠ = ∂v 1+ ∂v 2+ . . . + ∂v n. (13.5)∂x 1 ∂x 2 ∂x nNote the tiny dot between ∇ and v, which indicates that this is an “inner product” of them.Don’t confuse this divergence with the gradient discussed above!The physical meaning of divergence is not so straightforward to understand, but anyway,it literally quantifies how much the vector field v is “diverging” from the location x.Let’s go through an example for a 2-D space. Assume that v is representing flows ofsome “stuff” moving around in a 2-D space. The stuff is made of a large number of particles(like a gas made of molecules), and the flow v = (v x , v y ) means how many particles