--- title: "Surface Integrals and Flux" description: "Parameterise a surface, build its area element from a cross product, and integrate a field through it to measure flux." canonical: https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO source: https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO.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`. # Surface Integrals and Flux Parameterise a surface, build its area element from a cross product, and integrate a field through it to measure flux. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO ## Ready when they can - Parameterise a surface and compute its area element - Choose an orientation and say what the sign of the flux means - Evaluate the flux of a field through a plane or a sphere ## Lesson: Adding things up over curved skins A surface integral adds something over a curved surface in space. The scalar version multiplies a function by the area element dS, which gives things like the mass of a shell from its density. The vector version is flux: you dot the field with the unit normal and integrate, which measures the net flow through the surface. Think of flux as the volume of fluid crossing the skin per unit time. **Example.** Everything rests on a parametrisation r(u, v). Its two tangent vectors cross to give r sub u cross r sub v: the length of that cross product is the patch area, and the vector itself is the oriented patch. So the drill is: parametrise, substitute into the field, dot with the normal for flux (or multiply by its length for the scalar case), and evaluate the ordinary double integral over the flat parameter domain. For a graph z = g(x, y), the area element shortens to the square root of 1 plus gx squared plus gy squared, times dx dy. Orientation is chosen, never given. The normal has two possible directions, and picking one fixes the sign of every flux number you compute. Field lined up with the normal pours through and counts positive, field against it counts negative, and field sliding along the surface contributes nothing. For closed surfaces the convention is outward, and you must keep the normal consistent or patches will cancel by mistake. **Tip.** Test your setup with integrand 1 before trusting anything fancier: with 1 your machinery must reproduce the plain geometric area. Planes and spheres make ideal checks because their areas are known in advance. If the area comes out wrong, the normal or the parameter domain is at fault. **Recap.** Parametrise the skin, dot the field with your chosen normal, and integrate over the flat parameter domain. ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Change of Variables and the Jacobian](https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft) - [Divergence and Curl](https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE) - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK) - [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT) ## Opens up - [Stokes' Theorem and the Divergence Theorem](https://lightmysky.com/learn/mathematics/stokes-theorem-and-the-divergence-theorem-mt_sRojZcrw-q)