Parametrization Techniques

Effective computation of line and surface integrals requires choosing appropriate parametrizations adapted to the geometry of the curve or surface.

Parametrizing Curves

Definition

Common parametrizations for curves in $\mathbb{R}^3$ :

Line segment from $\mathbf{a}$ to $\mathbf{b}$ : $\mathbf{r}(t) = (1-t)\mathbf{a} + t\mathbf{b}$ , $0 \leq t \leq 1$
Circle of radius $R$ in the $xy$ -plane: $\mathbf{r}(t) = (R\cos t, R\sin t, 0)$ , $0 \leq t \leq 2\pi$
Graph $y = f(x)$ : $\mathbf{r}(t) = (t, f(t))$ , $a \leq t \leq b$
Helix: $\mathbf{r}(t) = (R\cos t, R\sin t, ct)$

ExampleArc length of a helix

For $\mathbf{r}(t) = (\cos t, \sin t, t)$ over $[0, 2\pi]$ : $L = \int_0^{2\pi} \|\mathbf{r}'(t)\|\,dt = \int_0^{2\pi} \sqrt{\sin^2 t + \cos^2 t + 1}\,dt = \int_0^{2\pi} \sqrt{2}\,dt = 2\pi\sqrt{2}$

Parametrizing Surfaces

Definition

Standard parametrizations for common surfaces:

Sphere $x^2+y^2+z^2=R^2$ : $\mathbf{r}(\phi,\theta) = (R\sin\phi\cos\theta, R\sin\phi\sin\theta, R\cos\phi)$
Cylinder $x^2+y^2=R^2$ : $\mathbf{r}(\theta,z) = (R\cos\theta, R\sin\theta, z)$
Cone $z=\sqrt{x^2+y^2}$ : $\mathbf{r}(r,\theta) = (r\cos\theta, r\sin\theta, r)$
Graph $z=f(x,y)$ : $\mathbf{r}(x,y) = (x, y, f(x,y))$
Surface of revolution ( $y = f(x)$ around $x$ -axis): $\mathbf{r}(x,\theta) = (x, f(x)\cos\theta, f(x)\sin\theta)$

ExampleNormal vector for a graph

For the graph $z = f(x,y)$ with parametrization $\mathbf{r}(x,y) = (x, y, f(x,y))$ : $\mathbf{r}_x \times \mathbf{r}_y = (1, 0, f_x) \times (0, 1, f_y) = (-f_x, -f_y, 1)$ The upward-pointing unit normal is $\hat{\mathbf{n}} = \frac{(-f_x, -f_y, 1)}{\sqrt{1+f_x^2+f_y^2}}$ .

Reparametrization Invariance

Theorem11.1Reparametrization Invariance

Scalar line integrals $\int_C f\,ds$ and scalar surface integrals $\iint_S f\,dS$ are independent of the parametrization. Vector line integrals $\int_C \mathbf{F} \cdot d\mathbf{r}$ and flux integrals $\iint_S \mathbf{F} \cdot d\mathbf{S}$ are independent of the parametrization but depend on the orientation: reversing orientation changes their sign.

RemarkChoosing good parametrizations

The key to simplifying integral computations is choosing a parametrization that aligns with the symmetry of the problem. If the integrand or domain has circular symmetry, use polar/spherical parametrizations. If the surface is given as a graph, use $(x, y)$ as parameters. The right choice can turn an intractable integral into a routine computation.