a regular surface
Think of the skin of an orange, the surface of a soap bubble, or a saddle on a horse. Up close, each looks like a slightly warped sheet of paper — at any one spot you could lay a tiny flat tile against it without leaving a gap or a crease. A regular surface is the precise version of that idea: a two-dimensional shape sitting in ordinary three-dimensional space that is smooth everywhere, with no sharp edges, no corners, and no places where it crosses through itself or pinches to a point.
Here is the working definition. A subset S of space is a regular surface if, around every point of S, you can find a small patch and a map x(u, v) = (x(u,v), y(u,v), z(u,v)) from a flat region of the (u, v)-plane onto that patch, such that three things hold: (1) the map is smooth (you can differentiate it as often as you like); (2) it is a homeomorphism onto the patch (it bends the flat region onto S without tearing or gluing, and the inverse is continuous); and (3) at every point the two partial-derivative vectors x_u and x_v are linearly independent — equivalently their cross product x_u x x_v is never the zero vector. Condition (3) is the 'regularity': it guarantees the patch genuinely has two independent directions, so it spreads out into a real two-dimensional sheet rather than collapsing to a line or a point.
Why insist on all this? Because regularity is exactly what lets calculus work on the surface — it is the entrance ticket to a well-defined tangent plane, normal vector, area, and curvature. The cone z = sqrt(x^2 + y^2) is a cautionary example: it is a perfectly good shape except at its tip, where x_u x x_v degenerates, so the tip is NOT a regular point and the cone (with its tip) is not a regular surface. A sphere, a plane, a torus (the surface of a doughnut), and a cylinder are all regular surfaces. One honest subtlety: a single patch usually cannot cover an entire curved surface — a sphere needs at least two — so the definition only asks for a patch around EACH point, and the patches overlap to cover the whole.
The unit sphere S^2: all points with x^2 + y^2 + z^2 = 1. Around the 'north pole' region you can parametrize it by x(u, v) = (u, v, sqrt(1 - u^2 - v^2)) for u^2 + v^2 < 1 — a smooth map whose partials x_u and x_v stay independent, so this is a valid patch. But this single patch cannot reach the equator or below; covering the whole sphere takes several overlapping patches. The sphere is regular at every point — there is no tip, edge, or self-crossing anywhere.
A patch covers part of the sphere; the whole sphere needs several overlapping patches.
Regularity is a LOCAL condition checked patch by patch, not a global one. A surface can be regular everywhere yet still be globally interesting (closed like a sphere, or have a hole like a torus); regularity only rules out tips, creases, and self-intersections, not topology.