Temperature Maps

Calculus 3 · Axiom Academy

A weather map is a function of two variables in disguise. Give it a spot, it gives you back a number — and its curves tell the whole story. One number for every point on the map Everything you have graphed so far took one input and gave one output — a curve y = f(x) . But the world rarely lines up on a single axis. Temperature depends on where you stand: both how far east and how far north. That is a function of two variables, T(x,y) , and you already know how to read one — it is a weather map. Watch the flat map lift. Every point (x,y) carries a temperature, and here that temperature becomes a height : the warm center rises into a peak, the cool edges stay low. The surface you get is the graph of z = T(x,y) — the same idea as y=f(x) , one dimension up. Height above a spot = the temperature there. That surface is z = T(x,y) , and its formula here is T = 30 - (x^2 + y^2) . Trace one temperature: the isotherm Instead of the whole surface, slice it at one height. Pick a temperature k and mark every spot that reads exactly k — that curve is an isotherm , a level curve of T . Drag the dial and watch the isotherm move: because T = 30 - (x^2+y^2) , the equation T = k becomes x^2 + y^2 = 30 - k , a circle of radius . Hotter dial → smaller circle, tucked near the peak; cooler dial → a wide circle out at the edges. Same surface, sliced at different heights. Read the map like a meteorologist

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