Conservative Vector Fields

Calculus 3 · Axiom Academy

LESSON Conservative Vector Fields Gradient fields, path independence, and the curl test — what makes a field conservative, and how to spot one. 1. What Makes a Field Conservative? A field is conservative when it is assembled from a single scalar potential f by taking its gradient. The gradient points in the direction f increases fastest — straight uphill on the potential's surface — so the field runs perpendicular to the level curves of f and grows steeper exactly where f does. Take , a round bowl. Its gradient points radially outward — uphill, away from the low point at the origin. We use this same field throughout the lesson. The signature property of a conservative field: the line integral of from to is the same for every path between them. The route drops out — only the endpoints matter. With and , the work from to is — the same total for two very different routes, as the running readouts converge. How do you test conservativeness without hunting for a potential? Measure the field's curl — its local rotation, the tendency to spin a tiny paddle wheel. On a simply connected region (one with no holes), zero curl is exactly the condition. 4. Finding the Potential Function Once the curl test passes, recover the potential by reversing the gradient — integrate, then match the leftover pieces against the remaining components. Differentiate in y and match Q . Differentiate in z and match R . Solve for the unknown functions

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