--- title: "Divergence and Curl" description: "Divergence measures local net outward flux per unit volume. Curl measures local circulation and rotation. Compute both from partial derivatives and use curl to test conservative fields." canonical: https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE source: https://lightmysky.com/learn/mathematics/divergence-and-curl-mt_JbjW8WKOPE.md retrieved: 2026-11-12 --- > **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`. # Divergence and Curl Divergence measures local net outward flux per unit volume. Curl measures local circulation and rotation. Compute both from partial derivatives and use curl to test conservative fields. 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 what each means physically at a point - Justify zero curl for smooth gradient fields and apply the converse with its domain conditions ## Lesson: Outflow at a point, spin at a point For a differentiable field F = (P, Q, R), P_x means the partial derivative of P with respect to x. Other subscripts work the same way. Divergence is the scalar div F = P_x + Q_y + R_z. Differentiate each component in its matching coordinate, holding the other coordinates fixed, then add. A term independent of that coordinate contributes zero. Divergence is the limit of net outward flux through a shrinking closed surface divided by its enclosed volume. Positive divergence means local net outflow; negative means local net inflow. Zero means balance in this limit, not that the field is zero. For a velocity field, divergence describes local expansion or compression. It does not mean fluid is being created or destroyed at a point. *(drawing: The matching-coordinate derivatives add to 5. Divergence is a scalar.)* Curl is the vector curl F = (R_y - Q_z, P_z - R_x, Q_x - P_y). Find the six partial derivatives, then subtract in this order. For a plane field (P, Q, 0) independent of z, only the third component remains. It is Q_x - P_y, the expression you used in Green's circulation theorem. For a differentiable velocity field, half the curl is the angular-velocity vector of the local rotational part. Curl's magnitude is twice that angular speed. Its direction follows the right-hand rule: point your right thumb along curl and your curled fingers show the rotation. Positive z means counterclockwise when viewed from positive z toward the origin. Local rotation is distinct from a particle's path around a distant point. *(drawing: Curl is (0, 0, 6). The three matching-coordinate derivatives are zero, so divergence is zero.)* Divergence does not determine curl. For nonzero constants a and b, (ax, ay, az) has divergence 3a and zero curl; (-by, bx, 0) has zero divergence and curl (0, 0, 2b). A constant field has both zero. Differentiate sums term by term when combining these patterns. Arrow directions alone do not settle the calculation: parallel shear (y, 0, 0) has curl (0, 0, -1). **Example.** Differentiate before substituting a point. For F = (xy, y squared, z squared), the matching derivatives are y, 2y and 2z, so div F = 3y + 2z. At (0, 1, 2), this is 3 + 4 = 7. Hold the other coordinates fixed when differentiating products. If a coefficient is unknown, compute divergence first, set it equal to the required value and solve for the coefficient. A conservative field is a gradient F = (f_x, f_y, f_z). If f has continuous second partial derivatives, differentiating in either order gives equal mixed partials. A second subscript means differentiate again. Substitution into the curl formula makes every component a difference of equal mixed partials. Each difference is zero, proving that this gradient field has zero curl. For a field with continuous first partial derivatives, zero curl guarantees a potential on a connected open domain where every closed loop can shrink to a point within the domain. Without that domain condition, zero curl alone may not be sufficient. Check the domain as well as the derivatives: an excluded set can prevent loops from shrinking inside the domain. Outward flux adds the normal component across a closed surface. Circulation adds the tangential component around a closed curve. For smooth fields and suitable boundaries, the divergence theorem gives outward flux as the volume integral of divergence. Stokes' theorem gives circulation as the surface integral of curl dotted with the unit normal over a spanning surface. The surface normal and boundary direction must have matching orientations. **Recap.** Divergence measures local net outward flux per unit volume. Curl measures local circulation; for velocity fields, half the curl gives local angular velocity. *Lesson, questions, worked solutions and flashcards written for this dataset.* ## Practice 14 questions on this page, each with its working shown. ## Needs first - [Green's Theorem in the Plane](https://lightmysky.com/learn/mathematics/greens-theorem-in-the-plane-mt_UDgivmPVEJ) ## Optional connections - [The Cross Product and Oriented Area](https://lightmysky.com/learn/mathematics/the-cross-product-and-oriented-area-mt_QEBM_Ms-SK) ## 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)