Lecture 20

Surfaces & Areas

APMA E2000

Drew Youngren dcy2@columbia.edu

$\gdef\RR{\mathbb{R}}$ $\gdef\vec{\mathbf}$ $\gdef\bv#1{\begin{bmatrix} #1 \end{bmatrix}}$ $\gdef\proj{\operatorname{proj}}$ $\gdef\comp{\operatorname{comp}}$

Announcements

  • HW 12 due Tuesday
  • Quiz 8 this week
    • FTLI
  • Recitation as usual

1-minute review

Green's Theorem

\[\oint\limits_{\partial D} P\,dx + Q\,dy = \iint\limits_{D} \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right)\,dA \]
where:
  • $D$ is a (simply connected) region in the plane,
  • $\partial D$ is an counterclockwise-oriented, simple, closed curve forming the boundary of $D$, and
  • $\vec F(x,y) = \langle P(x,y),Q(x,y) \rangle$ is a continuously differentiable vector field.

Parametric Surfaces

Parametrization
Curves
\[ \vec r: \RR \to \RR^3 \]
\[ \vec r(t) = \bv{ x(t) \\ y(t) \\ z(t)} \]
Surfaces
\[ \vec r: \RR^2 \to \RR^3 \]
\[ \vec r(u,v) = \bv{ x(u,v) \\ y(u,v) \\ z(u,v)} \]
Parametrization
Curves
\[ \vec r: \RR \to \RR^n \]
Surfaces
\[ \vec r: \RR^2 \to \RR^n \]

Kinds of Surfaces

Graphs of Functions

The graph of a function $z = f(x,y)$ over domain $D$ can be parametrized by

\[\vec r(u,v) = \bv{ u \\ v \\ f(u,v) } \] with $(u,v) \in D$.
Surfaces of Revolution

A surface of revolution of a graph $y = g(x), a \leq x \leq b,$ revolved about the $x$-axis is given by

\[\vec r(u,v) = \bv{ u \\ g(u) \cos(v) \\ g(u) \sin(v) } \] with $(u,v) \in [a,b]\times [0, 2\pi]$.
Piece of Sphere

We can use spherical coordinates to parametrize a piece of a sphere

\[\vec r(u,v) = \bv{ R \sin(u) \cos(v) \\ R \sin(u) \sin(v) \\ R \cos(u) } \] with angles $u,v$ restricted appropriately.
Exercise

Find a parametrization to model a chocolate kiss.

Heshey's Kiss
Exercise

Find a parametrization to model a corrugated roof.

Hut with corrugated roof

Surface Area

Recall

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.

Piece of tangent

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 \[\Delta S = |\vec r_u \times \vec r_v| \, \Delta u \,\Delta v \]

Surface Area

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 \[\text{SA} = \iint\limits_\Omega dS = \iint\limits_D |\vec r_u \times \vec r_v|\,du\,dv \]

Example

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 $0\leq u \leq 2\pi, 0\leq v\leq h$. \[ \vec r_u \times \vec r_v = \begin{bmatrix} \vec i & \vec j & \vec k \\ -R \sin u & R \cos u & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} = \bv{ R\cos u\\ R\sin u \\ 0}\]

\[ \iint_\Omega dS = \int_0^{2\pi} \int_0^h R\,dv\,du = 2\pi R h \]

Example

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 $. \[ \vec r_u \times \vec r_v = \begin{bmatrix} \vec i & \vec j & \vec k \\ \cos v & \sin v & \cos v \\ -u \sin v & u \cos v & -u \sin v \\ \end{bmatrix} = \bv{ -u \\ 0 \\ u}\]

\[ \iint_\Omega dS = \int_0^{2\pi} \int_0^1 u\sqrt2\,du\,dv = \pi\sqrt2 \]

Surface Integrals

Other Questions

Besides area, what might we ask about surfaces?

  • What is the average temperature on the earth's surface?
  • What is the probability that a raindrop in Spain falls (mainly) on the plain?
  • What is the moment of inertia of a radar dish spinning on its axis?

All of these require integrating a scalar function over a surface.

Definition

A surface integral of a function $f$ over a parametric surface $\Omega$ with respect to surface area is written \[ \iint\limits_\Omega f\,dS\]

  • Where? on a parametric surface $\Omega$
  • What? scalar field $f$ with $\Omega \subset \text{dom }f$
  • How? w/r/t surface area $dS$

Formula

\[ \iint\limits_\Omega f\,dS = \iint\limits_D f(\vec r(u,v))\,|\vec r_u \times \vec r_v|\,du\,dv \] where $\vec r(u,v)$ is a parametrization of $\Omega$ over domain $D\subset \RR^2$.

Example

Find the moment of inertia of a hollow sphere of radius $R$ with mass $M$ and constant density about a central axis.

Answer. $\frac23 M R^2$.