Flow Patterns

Calculus 3 · Axiom Academy

Release tracer dye into a vector field and let it drift. The shape it draws — bursting out, caving in, or spinning — is the field's divergence and curl made visible. A field you can't see — until something drifts in it A vector field parks an arrow at every point of the plane: at position (x,y) it hands you a velocity . You can't see the arrows themselves — but drop a speck of dye in and it obeys them, tracing out the flow. The path each speck follows is called a streamline , and the collective pattern reveals everything. Watch a ring of tracers released near the origin in the field . Every arrow points straight away from the center, so the specks stream outward and the cloud spreads. That net outward rush is exactly what divergence measures. The specks never circle — they flee the origin. A field whose flow bursts outward everywhere has positive divergence: this one is a pure source . Divergence: does the flow rush out, or cave in? Switch between a source and a sink , then release tracers. Divergence, , is the net flow out of a tiny region: positive where fluid is created, negative where it drains away. Watch the sign — and the specks — flip. Same radial shape, opposite sign. Positive divergence blows the tracers apart; negative divergence sucks them into the drain.

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