Parametric surface

A parametric surface is a map $r(u, v)$ that sends each point $P = (u, v)$ of a flat rectangle of parameters to a point in space; the surface is the set of all the image points. Hold v fixed and let u run: $r$ traces a curve on the surface, a u-line, just like a parametric curve. Hold u fixed and you get a v-line. Together they weave the grid engraved on the glass, the image of the grid in the rectangle below it. On the sphere, u is the longitude and v is measured up from the south pole, so the u-lines are the circles of latitude.

In these five examples, different points inside the rectangle go to different points of the surface. The edges are another matter. On the cylinder, sphere and torus $r(0, v) = r(2\pi, v)$: the left and right edges land on the same curve, and the rectangle is glued into a tube. The torus glues its top and bottom edges as well. The Möbius strip glues its ends with a flip, $r(2\pi, v) = r(0, -v)$. On the sphere, the whole bottom edge goes to the south pole and the whole top edge to the north pole. The arrows on the rectangle's sides show which sides are glued, and which way.

The partial derivatives $r_u$ and $r_v$ are the velocities along the u-line and the v-line through P. When $r_u \times r_v \ne 0$, they span the tangent plane at $r(P)$, and $r_u \times r_v$ is a normal vector to it. Where $r_u \times r_v = 0$, the parametrization is singular: at the sphere's poles $r_u = 0$, because a whole edge was squeezed into a point. The sphere itself is perfectly smooth there and has a tangent plane like anywhere else; only this choice of parameters breaks down.

Near P, $r(u + \Delta u, v + \Delta v)$ $\approx r(u, v) + r_u\,\Delta u + r_v\,\Delta v$. So a small cell $\Delta u \times \Delta v$ becomes a patch close to the parallelogram with sides $r_u\,\Delta u$ and $r_v\,\Delta v$, drawn dashed beside it. A cross product's length is the area of the parallelogram its vectors span, so the patch's area is about $|r_u \times r_v|\,\Delta u\,\Delta v$. The area factor is $|r_u \times r_v| = |r_u|\,|r_v| \sin\theta$, where θ is the angle between the parameter lines; it equals $|r_u|\,|r_v|$ only where they cross at a right angle. Turn the cell size down and the readout "cell area ÷ ΔuΔv" closes in on the area factor as the cell shrinks to nothing. For a parametrization like these, which covers the surface once apart from its edges, adding up the cells gives the surface area $\iint_D |r_u \times r_v|\,du\,dv$, the start of surface integrals.