--- title: "Conservative Fields and Path Independence" description: "A conservative field has a potential whose gradient equals the field, so vector line integrals depend only on endpoints. Testing mixed partials requires smoothness and domain assumptions." canonical: https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33 source: https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33.md retrieved: 2026-11-12 --- > **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`. # Conservative Fields and Path Independence A conservative field has a potential whose gradient equals the field, so vector line integrals depend only on endpoints. Testing mixed partials requires smoothness and domain assumptions. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33 ## Ready when they can - Apply the mixed-partial test with its smoothness and domain assumptions - Recover and verify a potential function, including the freedom to add a constant - Explain why the integral around any closed loop of a conservative field is zero ## Lesson: Fields that only care about endpoints You have integrated F dot dr along a path. A potential f satisfies grad f = F throughout the domain. Here grad f = (f_x, f_y), the two partial derivatives in x, y order. A field with a potential is conservative. In this lesson fields are continuous on open, connected regions. Paths have finitely many smooth pieces and stay in the domain. Write F = (P, Q), P_y = dP/dy and Q_x = dQ/dx. Suppose P and Q have continuous first partial derivatives on an open region. If every loop there can shrink to a point inside the region, then P_y = Q_x everywhere proves conservativeness. The whole xy-plane meets this domain condition. Unequal mixed partials anywhere rule out a potential on the whole region. *(drawing: These derivatives match everywhere. The polynomial field is defined on the whole plane, which meets the domain condition.)* **Example.** For F = (6xy, 3x² + 4y), integrate P in x: f = 3x²y + g(y). Then f_y = 3x² + g'(y). Matching Q gives g'(y) = 4y, so g(y) = 2y² + C. Thus f = 3x²y + 2y² + C. Check both partials: f_x = 6xy and f_y = 3x² + 4y. They reproduce F everywhere on the plane. For these polynomial fields, integrate P in x while holding y fixed. Include g(y), whose x derivative is zero. Differentiate the candidate in y and match Q. Integrate the resulting g'(y), then verify both partial derivatives against F throughout the domain. This verification already proves conservativeness; a separate preliminary test is optional. Adding a constant leaves the gradient unchanged. *(drawing: The second component determines g'(y). Integrating it supplies the remaining part of the potential.)* For F = grad f, the fundamental theorem of line integrals gives integral from A to B = f(B) - f(A). Evaluate f at both endpoints and subtract start from end. Reversing the path changes the sign. On a closed loop, A = B, so the result is zero. Add integrals for consecutive path pieces. Any intermediate potential values cancel, as do constants added to f. **Example.** Take f = xy + 2y, whose gradient is F = (y, x + 2). From A = (1, 0) to B = (2, 3), f(A) = 0 and f(B) = 2 × 3 + 2 × 3 = 12. The integral is 12 on every allowed path from A to B. Returning along any path contributes -12, giving zero for the full loop. *(drawing: The return route can differ from the outward route. Both integrals still use the same endpoint values.)* Path independence compares every pair of paths with the same endpoints. Equivalently, every closed-loop integral is zero. One zero loop does not prove this. On the plane with the origin removed, a loop surrounding the missing point cannot shrink inside the domain. Equal mixed partials alone then do not prove a global potential. A field there may still be conservative if a potential can be verified throughout its domain. In a fixed Newtonian gravity field, work done by gravity on an object of fixed mass depends only on its starting and ending positions. Friction and other forces can make total work depend on the route. For a conservative force, physical potential energy U satisfies F = -grad U. Its work from A to B is U(A) - U(B). The mathematical potential f uses F = grad f, so f = -U is a valid choice. **Recap.** Check the derivative test's assumptions, or verify a potential directly. For a conservative field, the integral is the ending potential value minus the starting value. *Lesson, questions, worked solutions and flashcards written for this dataset. OpenStax and Khan Academy records informed the mathematical checks; WeBWorK records informed question design.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Vector Fields and Line Integrals](https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO) ## Optional connections - [Directional Derivatives and the Gradient](https://lightmysky.com/learn/mathematics/directional-derivatives-and-the-gradient-mt_LW_KckY5Ad) ## Opens up - [Cauchy's Theorem and Deforming a Contour](https://lightmysky.com/learn/mathematics/cauchys-theorem-and-deforming-a-contour-mt_aacYFVN-HH) - [Parallel transport, Christoffel symbols and the geodesic equation](https://lightmysky.com/learn/science/parallel-transport-christoffel-symbols-and-the-geodesic-equation-mt_bUEL2R91Vy) - [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ)