Functions of Several Variables

Everything so far took one number in and gave one number out. Real problems rarely do. Temperature depends on latitude and longitude; the cost of a component depends on material and labour; a neural network's loss depends on thousands of weights at once.

The object

Two inputs, one output. The graph lives in three dimensions: a surface sitting above the $xy$-plane, with height $z$ at each point.

The problem with surfaces

You cannot draw four dimensions, and even three on paper is awkward. So we do what mapmakers do — slice.

Level curves

Fix the output at a constant $c$ and ask which inputs produce it:

The result is a curve in the plane: a level curve. Draw several for different $c$ and you have a contour map — the surface described entirely in two dimensions.

This is not an analogy with maps. It is the same construction. On an Ordnance Survey map, height is $f(x,y)$ and the brown contour lines are its level curves.

The standard three

$f(x,y) = x^2 + y^2$ — a paraboloid, a bowl.

Level curves: $x^2 + y^2 = c$, which are circles of radius $\sqrt{c}$. For $c = 1, 4, 9$ the radii are $1, 2, 3$.

Notice the spacing: as $c$ rises in equal steps the circles bunch closer together, because the bowl steepens as you climb. Contours crowding together means a steep surface. That reading is the whole point of a contour map.

$f(x,y) = x^2 - 5y^2$ — a saddle.

Level curves are hyperbolas. Along the $x$-axis the surface rises; along the $y$-axis it falls. There is a point that is simultaneously a minimum in one direction and a maximum in another, which single-variable calculus has no vocabulary for.

$f(x,y) = \dfrac{xy}{x^2+y^2}$ — undefined at the origin, and badly behaved near it in a way the next topic makes precise.

Reading a contour map

Contours for different values can never cross. A single point cannot have two different heights.