z = f(x, y): reading a surface
What you'll learn
See a two-variable function as a landscape — and read the same landscape from its flat contour map.
A function of one variable draws a curve. Give it a second input and it draws a landscape: z = f(x, y) assigns a height to every point of the floor. Welcome to the part of the course you can walk on.
From rule to terrain
Take z = x² + 2y². Every floor point (x, y) gets a height: the origin sits at 0, (1, 0) at height 1, (0, 1) at height 2. Assemble every reading and you get a bowl — steeper in the y-direction because of that 2.
Spin the real thing in the 3D surface explorer: eight families — bowls, saddles, cones, ripples — with the calculus read out live. This whole module is the theory behind that tool.
The contour map: 3D on a flat page
Cartographers solved "how do I draw a mountain on paper" centuries ago: slice the landscape at constant heights and draw the cut lines from above.
Reading contours is a skill worth having in the fingers:
- Along a contour: height constant — the flat way around.
- Across contours: climbing or descending.
- Contours crowded together: steep ground (heights change fast over a short walk).
- Closed rings shrinking to a point: a summit or a pit.
Level sets are everywhere
A contour is a level set: all inputs producing one output. You've met them outside math class — isobars on a weather map (equal pressure), isotherms (equal temperature), indifference curves in economics (equal utility). Same object, different landscape.
Why this matters
The two views — surface in 3D, contours in 2D — are the same information, and fluency means moving between them without friction. The next two lessons put slopes on the landscape, and the contour map is where the gradient will become obvious.
Check your understanding
Question 1 of 2
Walking exactly along one contour line of z = f(x, y), your height: