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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Surface Area">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 14:20:51 1997}
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\section{Surface Area \label{Surface Area}}

\subsubsection{Areas of Parametric Surfaces}

\begin{itemize}
\item  Just as a space curve is described by a vector function $\mathbf{r}%
\left( t\right) $ of one parameter, a surface is described by a vector
function of two parameters. Let $S$ denote a surface given by the vector
equation 
\[
\mathbf{r}\left( u,v\right) =x\left( u,v\right) \mathbf{i}+y\left(
u,v\right) \mathbf{j}+z\left( u,v\right) \mathbf{k}\qquad \left( u,v\right)
\in D\text{.}
\]
Assume that the partial derivatives of $x$, $y$, and $z$ with respect to $u$
and $v$ are all continuous.

\item  If you fix one of the parameters and let the other vary, you get a
space curve that lies in the surface. \ The vectors\quad\ 
\[
\mathbf{r}_{u}=\frac{\partial x}{\partial u}\mathbf{i}+\frac{\partial y}{%
\partial u}\mathbf{j}+\frac{\partial y}{\partial u}\mathbf{j\quad }\text{and}%
\quad \mathbf{r}_{v}=\frac{\partial x}{\partial v}\mathbf{i}+\frac{\partial y%
}{\partial v}\mathbf{j}+\frac{\partial y}{\partial v}\mathbf{j}
\]
are tangents to these ``coordinate curves'', so both lie in the tangent
plane to the surface. \ If you evaluate $\mathbf{r}_{u}\mathbf{\ }$and $%
\mathbf{r}_{v}$ at some point on the surface and calculate the cross product 
$\mathbf{r}_{u}\mathbf{\times r}_{v}$ you will get a vector \textbf{normal}
(i.e. perpendicular) to the surface at that point.

\item  If you assume that each point of the surface corresponds to a
distinct point of $D$, the formula for the \textbf{area of the} \textbf{%
surface} is 
\[
A\left( S\right) =\iint_{D}\left| \mathbf{r}_{u}\mathbf{\times r}_{v}\right|
dA
\]
In the special case in which $D$ is a domain in the $x$-$y$ plane and the
surface is given explicitly by the formula $z=f(x,y)$, the parametrized
description of the surface is 
\[
\mathbf{r}\left( x,y\right) =x\mathbf{i}+y\mathbf{j}+z\left( x,y\right) 
\mathbf{k}\qquad \left( x,y\right) \in D\text{.}
\]
and the formula for surface area in this case is 
\[
A\left( S\right) =\iint_{D}\sqrt{1+\left( \frac{\partial z}{\partial x}%
\right) ^{2}+\left( \frac{\partial z}{\partial y}\right) ^{2}}dxdy
\]
\end{itemize}

\subsection{\ Related Scientific Notebook Topics}

{\small \hyperref{Cross product}{}{}{../../Help/DM9-1.tex#Cross product}}

{\small \hyperref{Iterated integrals}{}{}{../../Help/DM7-8.tex#Iterated
integrals}}\hfill 

\noindent \hrulefill

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