Chapter 3: Surfaces, Limits, and Linearity

A function of two variables assigns a height to every point of the plane, and its graph is a landscape. This chapter builds the three reading skills that all of multivariable differential calculus rests on. The first is cartographic: a contour map is the surface disassembled into level curves, and once you can fuse map and terrain into a single mental object — crowded curves meaning steep ground — you can read a function's behavior at a glance.

The second skill is skepticism about limits. In one variable a point can be approached from only two sides; in the plane there are infinitely many roads into any point, and a limit exists only if every road agrees. We build a surface where different straight roads arrive at genuinely different heights, so the failure of a two-variable limit is something you can walk along and see, not just a counterexample on paper.

The third skill is the heart of the differential calculus: near a point where a function is differentiable, its curved graph is indistinguishable from a plane. We first slice a surface into one-variable curves to define partial derivatives, then assemble those slopes into the tangent plane and test the claim of local flatness the only fair way — by zooming in until curve and plane cannot be told apart. Everything later — gradients, optimization, the machinery of Chapter 4 — is built on this local linearity.