--- title: "Vector Fields and Line Integrals" description: "Attach a vector to every point of a region, then integrate along a curve to accumulate work done by the field or mass along a wire." canonical: https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO source: https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO.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`. # Vector Fields and Line Integrals Attach a vector to every point of a region, then integrate along a curve to accumulate work done by the field or mass along a wire. Subject: Mathematics · Area: Calculus & Analysis · Ages 20 to 21 Page: https://lightmysky.com/learn/mathematics/vector-fields-and-line-integrals-mt_VZxKEmsVgO ## Ready when they can - Sketch a simple vector field and describe its flow - Parameterise a curve and evaluate the work integral along it - Distinguish integrating a scalar along a curve from integrating a field along it ## Lesson: Arrows everywhere, added along a path A vector field assigns a vector to each point. For F(x, y) = (P(x, y), Q(x, y)), evaluate both components at the point. The first is horizontal; the second is vertical. On standard axes, positive x points right and positive y points up. Draw sampled vectors with their tails at their points, using one arrow scale. Magnitude gives speed for a velocity field and force strength for a force field. **Example.** A parametrisation r(t) = (x(t), y(t)), a <= t <= b, lists points in travel order. Check its endpoints and curve equation. For r(t) = (2t, t²), t = 1/4 gives (1/2, 1/16). Differentiate each component: constants give 0, ct gives c, and ct² gives 2ct. Here r'(t) = (2, 2t). A segment from A to B can use r(t) = A + t(B - A), 0 <= t <= 1. For continuous F on a piecewise smooth oriented curve r(t), a <= t <= b, use ∫C F dot dr = ∫a^b F(r(t)) dot r'(t) dt. Here dr = r'(t) dt includes displacement direction and length. In two dimensions, the dot product is P x' + Q y'. For a force field, this integral is work. Aligned force contributes positively, opposing force negatively, and perpendicular force zero. Integrating |F| ds loses this signed tangential contribution. *(drawing: This calculation uses constant force and a straight displacement. The cells show arithmetic, not a drawing of the vectors.)* For continuous f along a piecewise smooth curve, ∫C f ds = ∫a^b f(r(t)) |r'(t)| dt. Speed |r'| is the square root of the sum of squared derivative components. For nonnegative mass per unit length f, this gives wire mass. With r(t) = (0, 3t), 0 <= t <= 1, and f = 1 + y, speed is 3 and the integrand is 3 + 9t. Its antiderivative is 3t + 9t²/2, giving mass 15/2. Reverse the same traversal while keeping the field or scalar function fixed. Each displacement reverses, so ∫C F dot dr is multiplied by -1. In ∫C f ds, the function values and positive arc lengths stay the same, so the result is unchanged. These statements assume the integrals exist and no part of the traversal is added or removed. **Example.** Let F(x, y) = (2y, x) and r(t) = (t, t²), 0 <= t <= 2. Substitution gives F(r(t)) = (2t², t). Differentiating gives r'(t) = (1, 2t). Their dot product is 2t² × 1 + t × 2t = 4t². An antiderivative is 4t³/3. Evaluate at the parameter endpoints: 4 × 2³/3 - 0 = 32/3. This is the work if F represents force. In a velocity-field sketch, arrows pointing outward around a point suggest a source pattern; inward arrows suggest a sink pattern. Nonzero arrows tangent to circles around a point show a rotational pattern. These describe the sampled directions. A zero vector has no direction. Arrow lengths compare speeds only when the drawing uses a common scale. **Recap.** Evaluate the field on the curve. For force-field work, integrate F dot dr. For wire mass, integrate linear density against arc length ds. *This lesson, practice questions and flashcards were written for this dataset.* ## Practice 13 questions on this page, each with its working shown. ## Needs first - [Triple Integrals and Coordinates for Solids](https://lightmysky.com/learn/mathematics/triple-integrals-and-coordinates-for-solids-mt_RFeK8LD_jT) ## Optional connections - [Vector-Valued Functions and Motion Along a Curve](https://lightmysky.com/learn/mathematics/vector-valued-functions-and-motion-along-a-curve-mt_Xcjt6exMy3) ## Opens up - [Conservative Fields and Path Independence](https://lightmysky.com/learn/mathematics/conservative-fields-and-path-independence-mt_5djKpI5F33) - [Contour Integrals Along Parametrised Paths](https://lightmysky.com/learn/mathematics/contour-integrals-along-parametrised-paths-mt_CaJ7ruAv8P)