Drew Youngren dcy2@columbia.edu
The graph of a function $z = f(x,y)$ over domain $D$ can be parametrized by
A surface of revolution of a graph $y = g(x), a \leq x \leq b,$ revolved about the $x$-axis is given by
We can use spherical coordinates to parametrize a piece of a sphere
Find a parametrization to model a chocolate kiss.
Find a parametrization to model a corrugated roof.
We defined $ds$ to measure arc length for curves. \[ ds = |\vec r'(t)| \,dt \]
We look for an analogous way to measure area along surfaces.
We can use a parametrization $\vec r(u,v)$ to approximate the surface with a piece of the tangent plane.
Tangent vectors \[\begin{align*}\vec r_u\,\Delta u &= \left\langle x_u, y_u, z_u \right\rangle\Delta u \\ \vec r_v\,\Delta v &= \left\langle x_v, y_v, z_v \right\rangle\Delta v \\ \end{align*} \]
The area of a parallelogram defined by these is \[\Delta S = |\vec r_u \times \vec r_v| \, \Delta u \,\Delta v \]
If $D\subset \RR^2$ is a domain in $uv$-space and $\vec r: D \to \RR^3$ a parametrization of a smooth surface $\Omega$, then \[\text{SA} = \iint\limits_\Omega dS = \iint\limits_D |\vec r_u \times \vec r_v|\,du\,dv \]
Find the surface area of a cylinder with height $h$ and radius $R$.
Solution. $\vec r(u,v) = \langle R\cos u, R \sin u, v\rangle$ for $0\leq u \leq 2\pi, 0\leq v\leq h$. \[ \vec r_u \times \vec r_v = \begin{bmatrix} \vec i & \vec j & \vec k \\ -R \sin u & R \cos u & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} = \bv{ R\cos u\\ R\sin u \\ 0}\]
\[ \iint_\Omega dS = \int_0^{2\pi} \int_0^h R\,dv\,du = 2\pi R h \]
Find the surface area of the piece of the plane $z = x + 1$ inside the cylinder $x^2 + y^2 = 1$.
Solution. $\vec r(u,v) = \langle u\cos v, u \sin v, u \cos v + 1\rangle$ for $0\leq u \leq 1, 0\leq v\leq 2\pi $. \[ \vec r_u \times \vec r_v = \begin{bmatrix} \vec i & \vec j & \vec k \\ \cos v & \sin v & \cos v \\ -u \sin v & u \cos v & -u \sin v \\ \end{bmatrix} = \bv{ -u \\ 0 \\ u}\]
\[ \iint_\Omega dS = \int_0^{2\pi} \int_0^1 u\sqrt2\,du\,dv = \pi\sqrt2 \]
Besides area, what might we ask about surfaces?
All of these require integrating a scalar function over a surface.
A surface integral of a function $f$ over a parametric surface $\Omega$ with respect to surface area is written \[ \iint\limits_\Omega f\,dS\]
\[ \iint\limits_\Omega f\,dS = \iint\limits_D f(\vec r(u,v))\,|\vec r_u \times \vec r_v|\,du\,dv \] where $\vec r(u,v)$ is a parametrization of $\Omega$ over domain $D\subset \RR^2$.
Find the moment of inertia of a hollow sphere of radius $R$ with mass $M$ and constant density about a central axis.
Answer. $\frac23 M R^2$.