16.6 Videos Guide

16.6a

• Description of and notation for parametric surfaces

  • $\mathbf{r}(u,\ v)=x(u,\ v)\mathbf{i}+y(u,\ v)\mathbf{j}+z(u,\ v)\mathbf{k}$

  • Parametric equations are $x=x(u,\ v)$, $y=y(u,\ v)$, $z=z(u,\ v)$

• Planes as parametric surfaces

  • A plane that contains a point $P_{0}$, which corresponds to $\mathbf{r}_{0}=\left\langle x_{0},\ y_{0},z_{0}\right\rangle$, and vectors $\mathbf{a}$ and $\mathbf{b}$ has equation $\mathbf{r}(u,\ v)=\mathbf{r}_{0}+u\mathbf{a}+v\mathbf{b}$

• Spheres as parametric surfaces

  • A sphere of radius $a$ and center $(0,\ 0,\ 0)$ is parameterized as
    $\mathbf{r}(\phi,\ \theta)=a\cos\theta\sin\phi\ \mathbf{i}+a\sin\theta\sin\theta\ \mathbf{j}+a\cos\phi\ \mathbf{k}$, $0\leq\phi\leq\pi$, $0\leq\theta\leq2\pi$

• Functions as parametric surfaces

  • A surface given by a function $z=f(x,\ y)$ is parametrized as
    $$\mathbf{r}(x,\ y)=x\ \mathbf{i}+y\ \mathbf{j}+f(x,\ y)\ \mathbf{k}$$

Exercises:

16.6b

• Identify the surface with the given vector equation.

  • $\mathbf{r}(u,\ v)=u^{2}\ \mathbf{i}+u\cos v\ \mathbf{j}+u\sin v\ \mathbf{k}$

  • $\mathbf{r}(s,\ t)=\left\langle 3\cos t,s,\ \sin t\right\rangle$, $-1\leq s\leq1$

16.6c

• Find a parametric representation for the surface.

  • The part of the part of the cylinder $x^{2}+z^{2}=9$ that lies above the $xy$-plane and between the planes $y=-4$ and $y=4$.

  • The part of the plane $z=x+3$ that lies inside the cylinder $x^{2}+y^{2}=1$.

16.6d

• Tangent planes

  • The tangent plane to a surface given by $\mathbf{r}(u,\ v)$ has normal vector $\mathbf{r}_{u}\times\mathbf{r}_{v}$

Exercise:

• Find an equation of the tangent plane to the given parametric surface at the specified point.
  $x=u^{2}+1$, $y=v^{3}+1$, $z=u+v$; $(5,\ 2,\ 3)$

16.6e

• Surface area

  • If a smooth parametric surface $S$ is given by $\mathbf{r}(u,\ v)$, $(u,\ v)\ \epsilon\ D$ and $S$ is covered exactly once as $(u,\ v)$ covers $D$, then $A(S)=\iint^{\ }_{D}{\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|\ dA}$

  • If $z=f(x,\ y)$, then $\left|\mathbf{r}_{u}\times\mathbf{r}_{v}\right|=\sqrt{1+\left(f_{x}\right)^{2}+\left(f_{y}\right)^{2}}$

  • For a surface of revolution obtained by rotating $f(x)$, $a\leq x\leq b$ about the $x$-axis, use $x=x$, $y=f(x)\cos\theta$, $z=f(x)\sin\theta$, $0\leq\theta\leq2\pi$, so that
    $A(S)=\int^{b}_{a}{\int^{2\pi}_{0}{\left|\mathbf{r}_{x}\times\mathbf{r}_{\theta}\right|\ dA}}$

16.6f

Exercise:

• Find the area of the part of the surface $z=4-2x^{2}+y$ that lies above the triangle with vertices $(0,\ 0)$, $(1,\ 0)$, and $(1,\ 1)$.