Curves and surfaces

Vector calculus integrates over curves and surfaces, so we first describe them precisely.

1. Regular curves (Def. 1.1), arc length (Prop. 1.2) and reparametrisation.
2. Curvature (Def. 1.3, Prop. 1.4) and the Frenet frame with torsion (Prop. 1.5).
3. Parametrised surfaces (Def. 1.6), normals (Prop. 1.7) and the area element (Prop. 1.8).

A parametrised curve is a $C^1$ map $\mathbf{r}: [a, b]\to\mathbb{R}^3$; it is regular if $\mathbf{r}'(t)\ne\mathbf{0}$ for all $t$, and then $\mathbf{T} = \frac{\mathbf{r}'}{\|\mathbf{r}'\|}$ is the unit tangent. A reparametrisation is $\tilde{\mathbf{r}} = \mathbf{r}\circ\phi$ with $\phi$ a $C^1$ bijection with $\phi' > 0$ (orientation-preserving) or $\phi' < 0$ (reversing).

The arc length $L = \int_a^b\|\mathbf{r}'(t)\|\,\mathrm{d}t$ is unchanged by reparametrisation. A regular curve can be reparametrised by arc length $s(t) = \int_a^t\|\mathbf{r}'\|$, and then $\big\|\frac{\mathrm{d}\mathbf{r}}{\mathrm{d}s}\big\| = 1$.

For a regular curve parametrised by arc length, the curvature is $\kappa(s) = \big\|\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}\big\|$. Where $\kappa\ne0$, the principal normal is $\mathbf{N} = \frac1\kappa\frac{\mathrm{d}\mathbf{T}}{\mathrm{d}s}$, a unit vector perpendicular to $\mathbf{T}$, and $\rho = \frac1\kappa$ is the radius of curvature.

For any regular parametrisation, $\kappa = \frac{\|\mathbf{r}'\times\mathbf{r}''\|}{\|\mathbf{r}'\|^3}$. For a plane graph $y = f(x)$, $\kappa = \frac{|f''|}{(1 + f'^2)^{3/2}}$.

Where $\kappa\ne0$, let $\mathbf{B} = \mathbf{T}\times\mathbf{N}$ (the binormal). Then $(\mathbf{T}, \mathbf{N}, \mathbf{B})$ is an orthonormal frame and there is a function $\tau$ (the torsion) with

(derivatives with respect to $s$). A curve with $\kappa\ne0$ lies in a plane iff $\tau\equiv0$.

A parametrised surface is a $C^1$ map $\mathbf{r}: D\to\mathbb{R}^3$, $D\subseteq\mathbb{R}^2$, $(u, v)\mapsto\mathbf{r}(u, v)$; it is regular if $\mathbf{r}_u\times\mathbf{r}_v\ne\mathbf{0}$. Surfaces may also be given as graphs $z = f(x, y)$ (take $\mathbf{r} = (x, y, f(x, y))$) or as level sets $g(x, y, z) = c$ with $\nabla g\ne0$.

At a regular point, $\mathbf{r}_u$ and $\mathbf{r}_v$ span the tangent plane and $\mathbf{n} = \mathbf{r}_u\times\mathbf{r}_v$ is normal. For a graph $z = f(x, y)$, $\mathbf{n} = (-f_x, -f_y, 1)$. For a level set $g = c$, $\nabla g$ is normal.

The area of a regular parametrised surface is $A = \iint_D\|\mathbf{r}_u\times\mathbf{r}_v\|\,\mathrm{d}u\,\mathrm{d}v$, independent of the parametrisation; write $\mathrm{d}S = \|\mathbf{r}_u\times\mathbf{r}_v\|\,\mathrm{d}u\,\mathrm{d}v$. For a graph, $\mathrm{d}S = \sqrt{1 + f_x^2 + f_y^2}\,\mathrm{d}x\,\mathrm{d}y$; for a sphere of radius $a$, $\mathrm{d}S = a^2\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi$.

• Check regularity: $\mathbf{r}'\ne0$, $\mathbf{r}_u\times\mathbf{r}_v\ne0$ (degenerate points like the poles of a sphere are measure-zero and harmless for integrals).
• Units: $\kappa$ is 1/length, $\mathrm{d}S$ is length².
• Sanity: circle $\kappa = \frac1R$, sphere area $4\pi a^2$.
• Orientation: the order $\mathbf{r}_u\times\mathbf{r}_v$ fixes which side the normal points.