Heat Distribution

Calculus 3 · Axiom Academy

Heat Distribution: Reaching Steady State Heat a metal bar in the middle and it spreads out — smoothing, settling, until it stops changing. One equation, u ₓ = k ∇²u, runs all of it. A metal bar sits hot in the middle , its ends held cold. Heat always flows from warm to cool, so the profile keeps changing — until it reaches a steady state that no longer moves. Three moves take you from watch it spread to why the answer is a straight line. Drag time forward. The bar starts with a sharp hot bump in the middle; as time runs, heat flows to the cold ends and the whole profile smooths and flattens — that s u ₓ = k ∇²u playing out. The dashed line is where it s heading: the steady state. What does ∇²u actually measure? Grab the middle point and drag its temperature. The Laplacian ∇²u compares it to the average of its two neighbors : colder than them and it s about to warm up (∇²u > 0), hotter and it ll cool (∇²u < 0). That sign is exactly which way u ₓ pushes it. Steady state is a straight line Once nothing changes, u ₓ = 0, so ∇²u = 0 everywhere inside. In one dimension that forces the profile straight: the temperature just interpolates linearly between the two ends. Drag the end temperatures and the steady state is always the line joining them.

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