---
title: "Stokes' Theorem and the Divergence Theorem"
description: "Both theorems say that integrating a derivative over a region equals integrating the original object over its boundary. With Green's theorem and the fundamental theorem, they are one statement in four"
canonical: https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q
source: https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q.md
retrieved: 2026-09-12
---

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# Stokes' Theorem and the Divergence Theorem

Both theorems say that integrating a derivative over a region equals integrating the original object over its boundary. With Green's theorem and the fundamental theorem, they are one statement in four settings.

Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21
Page: https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q

## Ready when they can

- State each theorem and identify the region and its boundary
- Convert a surface flux integral into a triple integral of divergence
- Explain how all four theorems in the course share one form

## Lesson: Inside sums equal their boundary totals

Both theorems say one thing: integrating a derivative over a region equals integrating the original object over its boundary. The divergence theorem pairs a solid with its closed outer skin: the triple integral of divergence inside equals the outward flux through the skin. Stokes pairs a surface with its rim curve: the flux of curl through the cap equals the circulation around the edge. Together with Green and the fundamental theorem, they are one statement in four settings.

**Example.** Take the field (x, y, z) leaving a cube of side 2. Its divergence is 1 plus 1 plus 1, which is the constant 3. The cube volume is 2 cubed, which is 8. Constant divergence makes the trade trivial: flux is just divergence times volume, so 3 times 8 is 24. Whenever the surface is closed, look inside before parametrising anything.

Stokes lets you swap caps freely: if the surface looks hard, pick an easier one with the same edge, because the answer cannot change. Orientation follows the right hand rule, with fingers curling along the rim curve and thumb pointing along the normal. Positive flux means net outflow, negative means net inflow, and zero means balanced flow.

**Tip.** Match each problem to its pairing before computing. A closed skin asks for the divergence trade, and a rim curve asks for the Stokes trade. And watch orientation hardest of all: one flipped normal changes the sign of the whole answer.

**Recap.** Sum the derivative inside and you get the boundary total for free, so pick the easier side to compute.

## Practice

14 questions on this page, each with its working shown.

## Needs first

- [Surface Integrals and Flux](https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO)
- [The Fundamental Theorem of Calculus](https://lightmysky.com/learn/mathematics/the-fundamental-theorem-of-calculus-mt_EsS_ovdz2A)
- [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE)

## Opens up

- [Displacement Current and Maxwell's Equations](https://lightmysky.com/learn/science/displacement-current-and-maxwells-equations-mt_YR8m8TNfyy)