--- title: "Green's Theorem in the Plane" description: "Trade a line integral around a closed plane curve for a double integral of a derivative expression over the region inside it. The first of three theorems with the same shape." canonical: https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ source: https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Green's Theorem in the Plane Trade a line integral around a closed plane curve for a double integral of a derivative expression over the region inside it. The first of three theorems with the same shape. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ ## Ready when they can - State the theorem with the orientation convention for the boundary - Convert a circulation integral into a double integral and evaluate it - Use the theorem to compute an area from a boundary integral ## Lesson: Trading a rim walk for the inside Green's theorem trades a circulation integral around a closed curve for a double integral over the inside. The line integral of P dx plus Q dy equals the double integral of dQ/dx minus dP/dy over the enclosed region. Picture yourself walking the edge and adding up the tangential push of the field at each step. The theorem says that edge total equals the sum of the local swirl density over the whole inside. **Example.** Take the field (0, x) around the unit square from 0 to 1 in each direction. Write P = 0 and Q = x, then form dQ/dx minus dP/dy, which is 1 minus 0. The circulation integral becomes the double integral of 1 over the square, and that is just the area, which is 1. A hard walk around four edges collapses into one easy area integral. Orientation decides the sign, so traverse the boundary counterclockwise, keeping the region on your left. That is the positive direction. Flip to clockwise and the whole answer flips sign too. The theorem also demands a closed curve and a field that is smooth on the whole region, so check for holes and spikes before you use it. **Tip.** The theorem pays off twice. When dQ/dx minus dP/dy simplifies nicely, a nasty circulation integral turns into an easy area integral. And areas themselves come from boundary walks: pick P and Q with dQ/dx minus dP/dy equal to 1, such as minus y over 2 and x over 2, and the rim integral reports the area. **Recap.** A counterclockwise rim integral of P dx plus Q dy equals the inside sum of dQ/dx minus dP/dy. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33) - [Double Integrals over General Regions](https://lightmysky.com/learn/mathematics/double-integrals-over-general-regions-mt_UKS9_AhAWw) ## Opens up - [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH) - [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE)