--- title: "Divergence and Curl" description: "Two derivative operations on a field: divergence measures net outflow at a point, curl measures local rotation. Both are the language of the theorems that follow." canonical: https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE source: https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE.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`. # Divergence and Curl Two derivative operations on a field: divergence measures net outflow at a point, curl measures local rotation. Both are the language of the theorems that follow. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE ## Ready when they can - Compute the divergence and curl of a given field - Describe the physical reading of each at a point - Show that a conservative field has zero curl ## Lesson: Outflow at a point, spin at a point Divergence is the net outflow of a field at a point. You add the three outward rates, one per direction, and the total says how much flow the point creates or swallows. A positive value marks a source, and a negative value marks a sink. No integrals are needed: you read it straight off the partial derivatives. Curl measures the local rotation instead. It assembles the cross derivatives into a vector that points along the spin axis, with a size equal to twice the local rotation rate. Train your eye first: arrows spreading apart signal positive divergence, arrows bunching together signal negative divergence, and arrows looping around signal curl. A field can carry one without the other. **Example.** Try the radial field (x, y, z). Each component differentiates to 1, so the divergence is 1 plus 1 plus 1, which is 3 everywhere: a pure source with no spin. Now try the spin field (minus y, x, 0). Its curl keeps only a k component: dQ/dx minus dP/dy is 1 minus minus 1, which is 2, so the curl is (0, 0, 2). That is pure rotation with no outflow. **Tip.** Use the pair as vocabulary for everything that follows. Zero curl flags a field that could be a gradient, and nonzero divergence flags a source sitting inside. Uniform rotation has curl but no divergence, while radial outflow has divergence but no curl, so always test both before judging a field. **Recap.** Divergence totals the outflow at a point and curl vectors the spin, both read straight from partial derivatives. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK) - [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ) ## Opens up - [Surface Integrals and Flux](https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO) - [Laplace's Equation and the Maximum Principle](https://lightmysky.com/learn/mathematics/laplaces-equation-and-the-maximum-principle-mt_jSEkaTav1V) - [Stokes' Theorem and the Divergence Theorem](https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q)