Divergence

Divergence of a 2D Vector Field

A vector field $\mathbf{F}(x,y) = (F_x(x,y),\; F_y(x,y))$ assigns a vector to every point in the plane.

The divergence measures how much the field "spreads out" at a point:

$\nabla \cdot \mathbf{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y}$

Interpretation

• $\nabla \cdot \mathbf{F} > 0$: the field is a source — fluid flows outward
• $\nabla \cdot \mathbf{F} < 0$: the field is a sink — fluid flows inward
• $\nabla \cdot \mathbf{F} = 0$: incompressible flow — no net gain or loss

The Divergence Theorem

In 2D, the divergence theorem relates the integral of divergence over a region to flow across the boundary:

$\iint_R \nabla \cdot \mathbf{F}\, dA = \oint_{\partial R} \mathbf{F} \cdot \mathbf{n}\, ds$

This is the 2D version of Gauss's law.

Examples

$\mathbf{F} = (x, y)$ (radially outward):

• $\partial F_x / \partial x = 1$, $\partial F_y / \partial y = 1$
• $\nabla \cdot \mathbf{F} = 2$ (constant — uniform spreading)

$\mathbf{F} = (-y, x)$ (rotation):

• $\partial F_x / \partial x = 0$, $\partial F_y / \partial y = 0$
• $\nabla \cdot \mathbf{F} = 0$ (incompressible rotation — no sources or sinks)

$\mathbf{F} = (x^2, y^2)$:

• $\partial F_x / \partial x = 2x$, $\partial F_y / \partial y = 2y$
• $\nabla \cdot \mathbf{F} = 2(x+y)$

Your Task

Implement double divergence_2d(double (*Fx)(double, double), double (*Fy)(double, double), double x, double y, double h) that computes the divergence using central differences.