Divergence theorem on a rectangle in the plane

Statement

Let $a < b$ and $c < d$, and let $R = [a, b] \times [c, d]$. Let $F = (P, Q)$ be a vector field of class $C^1$ on an open set that contains $R$. Then $$\iint_R \left( \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y} \right) dA = \int_c^d \big( P(b, y) - P(a, y) \big) \, dy + \int_a^b \big( Q(x, d) - Q(x, c) \big) \, dx.$$

Remarks

The right side is the outward flux $\oint_{\partial R} F \cdot n \, ds$. Here the outward unit normal $n$ is $(1, 0)$, $(-1, 0)$, $(0, 1)$ and $(0, -1)$ on the right, left, top and bottom sides. Proof: by Fubini's theorem, $\iint_R \partial P / \partial x \, dA = \int_c^d \int_a^b \partial P / \partial x \, dx \, dy$. The fundamental theorem of calculus turns the inner integral into $P(b, y) - P(a, y)$. The $Q$ term is similar. This is the simplest case of the divergence theorem. In the plane, the divergence theorem is equivalent to Green's theorem.

Explanations

L1 human

The total source strength inside a rectangle equals the net flow out through its four sides. In each coordinate direction, this is the fundamental theorem of calculus, applied along each line through the rectangle.