--- title: "Change of Variables and the Jacobian" description: "Change coordinates in a multiple integral, and see the Jacobian determinant as the local factor by which area or volume is stretched." canonical: https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft source: https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft.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`. # Change of Variables and the Jacobian Change coordinates in a multiple integral, and see the Jacobian determinant as the local factor by which area or volume is stretched. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/change-of-variables-and-the-jacobian-mt_1_LRE2cJft ## Ready when they can - Compute a Jacobian and use it to transform a double integral to easier coordinates - Recover the polar, cylindrical and spherical area elements as Jacobians rather than as memorised facts - Design a substitution that turns an awkward region into a rectangle ## Lesson: Stretching grids: the Jacobian factor A change of variables swaps old coordinates for new ones, the way u substitution swaps x for u. Each tiny rectangle in the new grid lands stretched in the old grid. The Jacobian, written J, is the determinant of the matrix of first partial derivatives. It measures that local stretching. The area element becomes dx dy equals the absolute value of J times du dv. The absolute value matters because area stays positive even when the determinant comes out negative. Work in three moves. First pick a map that straightens the region, so each awkward boundary becomes u equals a constant or v equals a constant. Then rewrite the integrand and the limits in the new variables. Finally multiply by the absolute value of the Jacobian. Polar coordinates are the most used case: x equals r cos t and y equals r sin t, with Jacobian r. That r is no memorised extra. It drops out of the determinant of the polar partials. **Example.** Take the map x equals 2u and y equals 3v. All partials are constant, and the determinant is 2 times 3 minus 0, which is 6. So one unit of uv area covers 6 units of xy area. The classic error is the polar cousin: writing the area element as dr dt and dropping the r. At radius 5 the Jacobian equals 5, so forgetting it throws away a factor of 5. **Tip.** Choose polar when you see circular regions, radial symmetry, or x squared plus y squared in the integrand. Sketch the region to read off the angle sweep and the inner and outer radii. The same stretching idea gives r in cylindrical coordinates and rho squared sin phi in spherical ones. If the new region is no simpler than the old one, try a different map. **Recap.** You pick a map that straightens the region, rewrite everything, and multiply by the absolute Jacobian that measures the local stretch. ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Double Integrals in Polar Coordinates](https://lightmysky.com/learn/mathematics/double-integrals-in-polar-coordinates-mt_93e-KSzQMT) - [Determinants and What They Measure](https://lightmysky.com/learn/mathematics/determinants-and-what-they-measure-mt_fWr2D_1BdM) - [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT) - [Arc Length and the Area of a Surface of Revolution](https://lightmysky.com/learn/mathematics/arc-length-and-the-area-of-a-surface-of-revolution-mt_W4uujT9Z1t) ## Opens up - [Surface Integrals and Flux](https://lightmysky.com/learn/mathematics/surface-integrals-and-flux-mt_b3FIxOVaTO)