Foundations — Parametric surfaces — tangent planes, surface area

This page assumes you have seen none of the notation in Parametric surfaces — tangent planes, surface area. We build every symbol from the ground up, in an order where each piece leans only on the pieces before it.

0. A point in 3D and how we write it

Before anything else: a point in space needs three numbers — how far right ($x$), how far forward ($y$), how far up ($z$).

The picture: start at the origin (the corner where the three axes meet) and step $x$ along the first axis, $y$ along the second, $z$ along the third. The tip is your point $(x,y,z)$; the whole journey-arrow is $⟨x,y,z⟩$.

Why the topic needs it: a surface lives in 3D, so every point on it, and every arrow we draw on it, is one of these triples.

1. The parameter plane $(u,v)$ and the map $\mathbf{r}$

Here is the single most important idea to picture.

The picture: on the left, a flat piece of graph paper — the $(u,v)$ plane, a region we call $D$. On the right, 3D space. The map $\mathbf{r}$ picks up each dot of the flat paper and glues it somewhere in space. The whole sheet, once glued, is the surface.

2. Grid lines: $u$-curves and $v$-curves

On plain graph paper there are two families of lines: the horizontal ones (fix $v$, let $u$ change) and the vertical ones (fix $u$, let $v$ change). When we glue the paper into space, those straight grid lines become curved lines on the surface.

The picture: latitude and longitude lines on a globe. Longitude lines are one family, latitude lines the other; every point sits where one of each crosses.

Why the topic needs it: the directions these grid curves head in are the two arrows from which the entire theory (planes and area) is built. That is the next section.

3. The tangent arrow — "which way is the curve heading?"

If you walk along a curve, at each instant you are pointing somewhere. That instantaneous direction-and-speed is the tangent vector. We build this notion right here, from scratch, before using it.

The picture: a bead sliding along a bent wire; the little arrow taped to the bead, always aimed straight ahead along the wire, never sideways off it. That arrow lies flat against the curve.

This idea is developed further in Tangent Lines and Velocity Vectors, and the length of such a moving arrow, added up, is Arc Length — but you need nothing from those pages to follow this one. Why the topic needs it: the two grid curves each have a tangent arrow, and those two arrows will pin down the tangent plane.

4. The derivative and the partial derivative $\partial$

The limit above — "chord over step, as the step shrinks to zero" — is exactly the tool called the derivative.

Now, our map $\mathbf{r}(u,v)$ has two dials. When we differentiate we must say which dial we're nudging while holding the other still. That is exactly what the curly $\partial$ marks.

Why $\partial$ and not the ordinary $d$? The plain $\frac{d}{dt}$ is for a function of one dial (like the curve above). When there are two or more, we need to declare which one moves — the curly $\partial$ is that declaration. It answers the precise question "rate of change in the $u$-direction alone."

Prerequisites here also connect to Double Integrals and, later, Change of Variables and the Jacobian.

5. Two arrows span a plane — and the dot product

Two arrows starting at the same point, as long as they don't lie along the same line, sweep out a flat sheet: a plane. The $u$-tangent and $v$-tangent do exactly this.

Why the topic needs it: "a point lies in the plane" will be phrased as "an arrow inside the plane is perpendicular to the normal," and perpendicular = dot product zero. That single equation is the tangent-plane equation.

6. The cross product $\times$ — the star tool

This is the tool the whole topic hinges on. It takes two arrows and produces a third arrow.

The picture below shows all three gifts at once: the flat parallelogram, the arrow rising perpendicular from it, and the right-hand twist from $\mathbf{a}$ to $\mathbf{b}$.

Full details of computing it live in Cross Product. Note the length bars vs no bars: ${\mathbf{r}}_u\times {\mathbf{r}}_v$ (no bars) is the arrow — used as the normal; $|{\mathbf{r}}_u\times {\mathbf{r}}_v|$ (with bars) is its length — used as the area factor.

7. The catch: when the two arrows collapse (regularity)

Everything above quietly assumed the two tangent arrows genuinely span a plane. They might not — and that is a real case we must confront.

8. Measuring the flat patch: $dA=dudv$, and summing with $\iint$

The final symbol: to get total area we chop the surface into countless tiny patches, measure each, and add them all up. That infinite sum is an integral.

Why the topic needs it: the area of the surface $S$ (the glued sheet from §1) is $A(S)={\iint }_D|{\mathbf{r}}_u\times {\mathbf{r}}_v|dA={\iint }_D|{\mathbf{r}}_u\times {\mathbf{r}}_v|dudv$ — every symbol is now something you have met: $A(S)$ (the area of the finished surface $S$), $\iint$ (add up patches), $D$ (parameter region), $|\times |$ (area of one stretched patch), $dA=dudv$ (size of one tile on the flat paper). And thanks to §7 the integrand is nonzero wherever the surface is regular.

The same normal arrow ${\mathbf{r}}_u\times {\mathbf{r}}_v$, kept with its sign, later powers Surface Integrals and Flux.

Prerequisite map

Each lettered box below is one idea from this page; the label after the dash names the section it lives in. Arrows mean "feeds into."

Equipment checklist

Test yourself — cover the right side and answer aloud.