Curves and Surfaces

5.1 Parametric Curves

A parametric curve in $\mathbb{R}^3$ is a $C^1$ function $\mathbf{r} : [a, b] \to \mathbb{R}^3$ Written $\mathbf{r}(t) = (x(t),\, y(t),\, z(t))$.

Definition. The arc length of $\mathbf{r}$ over $[a, b]$ is

$L = \int_a^b \lVert \mathbf{r}"(t) \rVert\, dt = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}\, dt$

Proposition 5.1. The arc length function $s(t) = \int_a^t \lVert \mathbf{r}'(\tau) \rVert\, d\tau$ Satisfies $\frac{ds}{dt} = \lVert \mathbf{r}'(t) \rVert$And reparametrising by arc length gives a Unit-speed curve: $\lVert \frac{d\mathbf{r}}{ds} \rVert = 1$.

Proof. By the Fundamental Theorem of Calculus, $\frac{ds}{dt} = \lVert \mathbf{r}'(t) \rVert$. If we reparametrise by $s$I.e., write $\mathbf{r}(s) = \mathbf{r}(t(s))$Then by the chain rule $\frac{d\mathbf{r}}{ds} = \mathbf{r}'(t) \cdot \frac{dt}{ds}$So $\lVert \frac{d\mathbf{r}}{ds} \rVert = \lVert \mathbf{r}'(t) \rVert \cdot \left\lvert \frac{dt}{ds} \right\rvert = 1$. $\blacksquare$

Problem. Find the arc length of the curve $\mathbf{r}(t) = (e^t \cos t,\, e^t \sin t,\, e^t)$ for $0 \leq t \leq \ln 2$.

Problem. Find the arc length of the helix $\mathbf{r}(t) = (\cos t,\, \sin t,\, t)$ for $0 \leq t \leq 4\pi$.

5.2 Curvature and Torsion

Definition. Let $\mathbf{r}(s)$ be a unit-speed curve ($\lVert \mathbf{r}'(s) \rVert = 1$). Define:

• Unit tangent vector: $\mathbf{T}(s) = \mathbf{r}'(s)$
• Curvature: $\kappa(s) = \lVert \mathbf{T}'(s) \rVert = \lVert \mathbf{r}''(s) \rVert$
• Principal normal: $\mathbf{N}(s) = \frac{\mathbf{T}'(s)}{\lVert \mathbf{T}'(s) \rVert}$ (when $\kappa \neq 0$)
• Binormal: $\mathbf{B}(s) = \mathbf{T}(s) \times \mathbf{N}(s)$
• Torsion: $\tau(s) = -\mathbf{B}'(s) \cdot \mathbf{N}(s)$

The vectors $\mathbf{T}$, $\mathbf{N}$, $\mathbf{B}$ form the Frenet—Serret frame, an orthonormal Basis that moves with the curve.

Theorem 5.2 (Frenet—Serret Formulas).

$\mathbf{T}' = \kappa\, \mathbf{N}, \quad \mathbf{N}' = -\kappa\, \mathbf{T} + \tau\, \mathbf{B}, \quad \mathbf{B}' = -\tau\, \mathbf{N}$

Proof. Since $\mathbf{T}$ is a unit vector, $\mathbf{T} \cdot \mathbf{T} = 1$So $\mathbf{T}' \cdot \mathbf{T} = 0$. Therefore $\mathbf{T}'$ is orthogonal to $\mathbf{T}$So $\mathbf{T}'$ is parallel to $\mathbf{N}$ (when $\kappa \neq 0$). This gives $\mathbf{T}' = \kappa\,\mathbf{N}$.

Similarly, $\mathbf{B} = \mathbf{T} \times \mathbf{N}$ is a unit vector, so $\mathbf{B}' \cdot \mathbf{B} = 0$. Also $\mathbf{B} \cdot \mathbf{T} = 0$So $\mathbf{B}' \cdot \mathbf{T} + \mathbf{B} \cdot \mathbf{T}' = 0$ Giving $\mathbf{B}' \cdot \mathbf{T} = -\mathbf{B} \cdot \kappa\,\mathbf{N} = 0$. So $\mathbf{B}'$ is Parallel to $\mathbf{N}$Giving $\mathbf{B}' = -\tau\,\mathbf{N}$.

For $\mathbf{N}'$: since $\{\mathbf{T}, \mathbf{N}, \mathbf{B}\}$ is an orthonormal basis, $\mathbf{N}' = (\mathbf{N}' \cdot \mathbf{T})\,\mathbf{T} + (\mathbf{N}' \cdot \mathbf{N})\,\mathbf{N} + (\mathbf{N}' \cdot \mathbf{B})\,\mathbf{B}$. From $\mathbf{N} \cdot \mathbf{T} = 0$: $\mathbf{N}' \cdot \mathbf{T} = -\mathbf{N} \cdot \mathbf{T}' = -\kappa$. From $\mathbf{N} \cdot \mathbf{N} = 1$: $\mathbf{N}' \cdot \mathbf{N} = 0$. From $\mathbf{N} \cdot \mathbf{B} = 0$: $\mathbf{N}' \cdot \mathbf{B} = -\mathbf{N} \cdot \mathbf{B}' = \tau$. This gives $\mathbf{N}' = -\kappa\,\mathbf{T} + \tau\,\mathbf{B}$. $\blacksquare$

Intuition. The curvature $\kappa$ measures how sharply the curve bends (deviation from a straight line). The torsion $\tau$ measures how sharply the curve twists out of the osculating plane (deviation from a Plane curve). A curve lies in a plane if and only if $\tau = 0$ everywhere.

For a curve parameterised by an arbitrary parameter $t$ (not necessarily unit-speed):

$\kappa = \frac{\lVert \mathbf{r}'(t) \times \mathbf{r}''(t) \rVert}{\lVert \mathbf{r}'(t) \rVert^3}$

$\tau = \frac{[\mathbf{r}'(t) \times \mathbf{r}''(t)] \cdot \mathbf{r}^{\prime\prime\prime}(t)}{\lVert \mathbf{r}'(t) \times \mathbf{r}''(t) \rVert^2}$

Problem. Find the curvature and torsion of the helix $\mathbf{r}(t) = (a\cos t,\, a\sin t,\, bt)$ where $a, b \gt 0$.

5.3 Parametric Surfaces

A parametric surface is a $C^1$ map $\mathbf{r} : D \subseteq \mathbb{R}^2 \to \mathbb{R}^3$ $\mathbf{r}(u, v) = (x(u,v),\, y(u,v),\, z(u,v))$.

The tangent plane at $\mathbf{r}(u_0, v_0)$ is spanned by the tangent vectors

$\mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u}, \quad \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}$

The unit normal to the surface is

$\mathbf{n} = \frac{\mathbf{r}_u \times \mathbf{r}_v}{\lVert \mathbf{r}_u \times \mathbf{r}_v \rVert}$

Examples of parametric surfaces:

• Sphere (spherical coordinates): $\mathbf{r}(\theta, \phi) = (\rho\sin\phi\cos\theta,\, \rho\sin\phi\sin\theta,\, \rho\cos\phi)$
• Cylinder: $\mathbf{r}(\theta, z) = (r\cos\theta,\, r\sin\theta,\, z)$
• Graph of $z = f(x,y)$: $\mathbf{r}(x, y) = (x,\, y,\, f(x,y))$

For the graph $z = f(x,y)$The normal is $\mathbf{n} = \frac{(-f_x,\, -f_y,\, 1)}{\sqrt{1 + f_x^2 + f_y^2}}$.

5.4 Surface Area

Definition. The area of a parametric surface $\mathbf{r} : D \to \mathbb{R}^3$ is

$A(S) = \iint_D \lVert \mathbf{r}_u \times \mathbf{r}_v \rVert\, du\, dv$

Derivation. Partition $D$ into small rectangles $D_{ij}$ of area $\Delta u\, \Delta v$. The image $\mathbf{r}(D_{ij})$ is approximately a parallelogram spanned by $\mathbf{r}_u\, \Delta u$ and $\mathbf{r}_v\, \Delta v$With area $\lVert \mathbf{r}_u \times \mathbf{r}_v \rVert\, \Delta u\, \Delta v$. Summing and taking the limit gives the formula. $\blacksquare$

Problem. Find the surface area of the part of the paraboloid $z = x^2 + y^2$ that lies below The plane $z = 4$.

5.5 Surface Integrals

Definition (Scalar surface integral). The integral of a scalar function $f$ over a parametric Surface $S$ is

$\iint_S f\, dS = \iint_D f(\mathbf{r}(u,v))\, \lVert \mathbf{r}_u \times \mathbf{r}_v \rVert\, du\, dv$

Definition (Vector surface integral / flux). The flux of a vector field $\mathbf{F}$ through an Oriented surface $S$ is

$\iint_S \mathbf{F} \cdot d\mathbf{S} = \iint_D \mathbf{F}(\mathbf{r}(u,v)) \cdot (\mathbf{r}_u \times \mathbf{r}_v)\, du\, dv$

Where the orientation is determined by the choice of normal $\mathbf{r}_u \times \mathbf{r}_v$ vs. $\mathbf{r}_v \times \mathbf{r}_u$.

Problem. Evaluate $\iint_S \mathbf{F} \cdot d\mathbf{S}$ where $\mathbf{F} = (x,\, y,\, z^2)$ and $S$ is the hemisphere $x^2 + y^2 + z^2 = 4$, $z \geq 0$With Upward orientation.

Problem. Evaluate $\iint_S z\, dS$ where $S$ is the part of the plane $2x + 2y + z = 4$ in the First octant.

5.6 Common Pitfalls

:::caution Common Pitfalls

• Parameterisation domain. Always verify that the parameterisation covers the entire surface and that the map is one-to-one (except possibly on the boundary).
• Normal orientation. The cross product $\mathbf{r}_u \times \mathbf{r}_v$ determines the orientation. Swapping the order changes the sign of the flux integral.
• Surface area vs. Flux. Surface area uses $\lVert \mathbf{r}_u \times \mathbf{r}_v \rVert$ (scalar), while flux uses $\mathbf{r}_u \times \mathbf{r}_v$ (vector, oriented).

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