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

The relationship between surface integrals and surface area is much the same
as the relationship between line integrals and arc length.

\subsubsection{Surface Integral of a Scalar Function}

\begin{itemize}
\item  Let $S$ denote a surface given by the vector equation 
\[
\mathbf{r}\left( u,v\right) =x\left( u,v\right) \mathbf{i}+y\left(
u,v\right) \mathbf{j}+z\left( u,v\right) \mathbf{k}\qquad \left( u,v\right)
\in D\text{.}
\]
\ Let $f(x,y,z)$ be a function defined at each point of the surface $S$. \
The \textbf{surface integral of }$\mathbf{f}$\textbf{\ over the surface }$%
\mathbf{S}$ is defined by the formula 
\[
\iint_{S}f\left( x,y,z\right) dS=\lim_{\left\| P\right\| \rightarrow
0}\sum_{i=1}^{n}\sum_{j=1}^{n}f\left( x_{i},y_{i},z_{i}\right) \Delta S_{ij}
\]
In this formula $s$ denotes arc length along the curve, $P$ denotes a
partition of the surface into $n^{2}$ \ patches, and $\left\| P\right\| $is
the area of the largest patch. $\left( x_{i},y_{i},z_{i}\right) $ is a point
on the $\left( i,j\right) ^{th}$ patch. \ (The definition of an ordinary
double integral $\int_{c}^{d}\int_{a}^{b}g\left( x,y\right) dxdy$ is defined
by a special case of this process, in which the surface $S$ is a rectangle
in the plane.) \ In practice, this limiting process is rarely carried out,
since 
\[
\iint_{S}f\left( x,y,z\right) dS=\iint_{D}f\left( \mathbf{r}\left(
u,v\right) \right) \left| \mathbf{r}_{u}\mathbf{\times r}_{v}\right| dA
\]
where the integral on the right is an ordinary double integral integral.
\end{itemize}

\subsubsection{Surface Integral of a Vector Function}

\begin{itemize}
\item  Let $S$ denote an oriented surface given by the vector equation 
\[
\mathbf{r}\left( u,v\right) =x\left( u,v\right) \mathbf{i}+y\left(
u,v\right) \mathbf{j}+z\left( u,v\right) \mathbf{k}\qquad \left( u,v\right)
\in D\text{.}
\]
\ Let $\mathbf{F}=P\mathbf{i}+Q\mathbf{j}+R\mathbf{k}$ denote a vector field
for which the partial derivatives of \ $P$, $Q$, and $R$ are all continuous.
\ The \textbf{surface integral of }$\mathbf{F}$\textbf{\ over }$\mathbf{S}$
is defined by 
\[
\iint_{S}\mathbf{F}\cdot d\mathbf{S}=\iint_{D}\mathbf{F}\cdot \left( \mathbf{%
r}_{u}\mathbf{\times r}_{v}\right) dA
\]
This integral is also called the \textbf{flux} of $\mathbf{F}$ across $S$. \
\hfill 
\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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