---
tag: 0GZ3D8F921
kind: statement
role: proposition
name: Divergence theorem on a rectangle in the plane
aliases: [planar-divergence-theorem-rectangle]
msc: [26B20]
uses: [02NSMQVQWK, 0R0T1VWH5X, 0NKBMFC18R, 0JX86PSSA9, 0HR3N02317]
sources:
  - url: https://en.wikipedia.org/wiki/Divergence_theorem
    locator: Mathematical statement
  - url: https://en.wikipedia.org/wiki/Green%27s_theorem
    locator: Relationship to the divergence theorem
  - url: https://en.wikipedia.org/wiki/Green%27s_theorem
    locator: Proof when D is a simple region
---

## 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|0R0T1VWH5X]], $\iint_R \partial P / \partial x \, dA = \int_c^d \int_a^b \partial P / \partial x \, dx \, dy$. 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|0JX86PSSA9]]. In the plane, the divergence theorem is equivalent to [[Green's theorem|0HR3N02317]].

## 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.