Vector Calculus Summary

Calculus 3 · Axiom Academy

A wrap-up of how vector calculus extends calculus to fields and flows — line integrals, surface integrals, and the great theorems that connect them. A vector field attaches a vector to every point of space — a force, a velocity, a flow — and everything in this unit is a way to measure it. Line integrals add a quantity along a curve: for mass, for work or circulation. Surface integrals add over a surface: for mass, and the flux for how much of a field flows through. A field is conservative exactly when (on a simply connected region) — then work is path-independent and equals a change in potential. The three big theorems — Green's, Stokes', Divergence — all say the same thing: an integral over a region equals an integral over its boundary. A vector field assigns a vector to each point in space, modeling forces, velocities, or flows across a whole region. Visualize: arrow plots show direction and magnitude; streamlines trace particle paths. Conservative: for a potential f ; test on a simply connected region. Integrate along a curve C . The scalar version totals mass of a wire of density f ; the vector version gives work by a force field, or circulation around a closed loop. How to evaluate: parametrize C as , then substitute; for ds use . Watch out for: for a conservative field, work depends only on the endpoints — not the path. Core Concept Surface Integrals & Flux

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