Green's Theorem (Flux-Divergence Form)

Definition

The flux-divergence form of Green’s Theorem (also known as the 2D Divergence Theorem) relates the net outward flux of a vector field across a simple closed curve $C$ to the integral of the field’s divergence over the enclosed region $R$: $\oint_C \mathbf{F} \cdot \mathbf{n} ds = \oint_C M dy - N dx = \iint_R \left( \frac{\partial M}{\partial x} + \frac{\partial N}{\partial y} \right) dA$

• How to read: “The closed line integral of the dot product of F and n with respect to s is equal to the closed line integral of M d y minus N d x, which equals the double integral over the region R of the quantity partial M partial x plus partial N partial y, with respect to area.”
• Meaning: Net outward flux through the boundary equals total divergence (source density) inside — a 2D conservation law.

Why It Matters

This theorem provides a mathematical bridge between local internal changes and global boundary behavior, essential for verifying conservation in any 2D system. It simplifies complex volumetric analysis by allowing researchers to focus solely on the perimeter of a region.

Core Concepts

• Flux: The net flow of the vector field across the boundary curve $C$ in the direction of the outward normal $\mathbf{n}$.
• Divergence: The expression $(\partial M/\partial x + \partial N/\partial y)$ measures the rate at which the field “spreads out” from a point (source) or “contracts” into it (sink).
• Outward Normal: The vector $\mathbf{n}$ points perpendicularly away from the region $R$ at every point on the boundary.

Connected Concepts

• The Divergence Theorem
• Green’s Theorem (Circulation-Curl Form)
• The Binomial Theorem