Multivariable Functions

Calculus 3 · Axiom Academy

LESSON Multivariable Functions z = f(x,y) hands every point of the plane one number — read those numbers as heights and the graph is a surface. 1. Every Point of the Plane Gets a Height Pick any point (x,y) on the floor. The rule f answers with exactly one number. Draw that number as a height above the point — do it for every point of the domain, and the heights sweep out a surface in space, exactly as the outputs of f(x)=x^2 sweep out a curve in the plane. 2. The Domain Is a Region; the Range Is an Interval The domain is the set of all input pairs (x,y) for which f(x,y) is defined. The range is the set of all output values z = f(x,y) . They are different kinds of object living in different places: the domain is a region of the xy -plane , while the range is a set of numbers on the z -axis . The square root needs , so D is the closed disk of radius 3 . The output is largest at the centre, f(0,0) = 3 , and falls to 0 on the rim, so . The only forbidden point is the origin, so D is every (x,y) with . Outputs are always positive and grow without bound near the origin, so . 3. Slice the Surface and You Are Back in Calculus I Freeze one input. Setting y = c slides a vertical wall across the surface, and where the wall meets the surface you get an ordinary single-variable curve z = f(x,c) . A surface of two variables is nothing more than a whole family of these familiar curves, stacked side by side.

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