Flux Through Surfaces

Calculus 3 · Axiom Academy

How much of a flow actually passes THROUGH a surface — the fluid crossing a membrane, the heat leaving a wall, the field piercing a shell. A field F streams through space; a surface S sits in it. Flux asks one question: how much of F crosses S ? Three moves answer it — feel it , sum it , compute it . Only the part along the normal counts A uniform flow streams to the right. Tilt the membrane and watch the flux: face it head-on and every arrow crosses; turn it edge-on and the flow slides right past. That's F·n = |F| cos θ. Real surfaces curve, so the normal keeps turning. Chop the surface into patches, take F·n on each little piece, and add them all up. Crank the patch count and watch the sum settle onto the integral ∬ F·n dS. A real flux, two ways that agree Take the radial field F = ⟨x, y, z⟩ pushing straight out of a sphere of radius R. Compute the flux the direct way (F·n × area) and by the divergence theorem (∭ 3 dV) — drag R and watch both land on the exact same number, 4πR³. One integral, three moves: feel it (only F·n gets through), sum it (∬ F·n dS adds it over the surface), compute it (direct or by divergence). Anywhere something streams across a boundary — fluid through a membrane , heat out of a wall , electric field through a shell — this same flux ∬ S F·n dS is what you're measuring.

This is the written version of the interactive lesson above. See the full Calculus 3 course.