Partial Differential Equations

Calculus 3 · Axiom Academy

One idea runs all of it: every point drifts toward the average of its neighbors — and the Laplacian is the number that measures the gap. Heat a single spot on a metal rod, then wait. The warmth spreads into the cold regions on either side and the peak sinks, until the whole rod sits at one lukewarm temperature. The heat equation is just the calculus of that spreading. temperature u(x,t) · position x · time t · diffusivity k 2. The Laplacian Compares a Point to Its Neighbors Why does curvature drive the flow? Because curvature is comparison . The Laplacian at a point measures how that point's value stacks up against the average of its immediate neighbors — and heat always flows to close that gap. in 2D, the Laplacian sums the curvature in every direction . The point is colder than its surroundings, so u_t > 0 — it heats up . . The point is hotter than its surroundings, so u_t < 0 — it cools down . . The point already matches its neighbors, so nothing pushes it — u_t = 0 . . The flow speed is proportional to the neighbor gap. The Laplacian is the engine of the "Big Three" Diffusion is (heat), oscillation is (waves), and the steady state is (Laplace). Same operator, three different physical stories. 3. The Flow Stops at Equilibrium Run the clock forward. Every peak is pushed down, every dip is pushed up, and the bumps keep shrinking. The motion only halts when every point equals its neighbor average — that is exactly the condition , the steady state Laplace studied.

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