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%TCIDATA{<META NAME="Title" CONTENT="Reference/Mathematics/Line Integrals">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 14:27:02 1997}
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\section{Line Integrals\label{Line Integrals}}

\subsubsection{2D Line Integral}

Let $C$ denote a plane curve given by the parametric equations 
\[
x=x\left( t\right) \qquad y=y\left( t\right) \qquad a\leq t\leq b 
\]
or equivalently by the vector equation $\mathbf{r}\left( t\right) =x\left(
t\right) \mathbf{i}+y\left( t\right) \mathbf{j}$. \ Assume the curve is 
\textbf{smooth}, which means that the tangent vector $\mathbf{r}\prime =%
\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}$ is continuous and never the
zero vector. \ Let $f(x,y)$ be a function defined at each point of the curve 
$C$. \ The line integral of $f$ along $C$ is defined by the formula 
\[
\int_{C}f\left( x,y\right) ds=\lim_{\left\| P\right\| \rightarrow
0}\sum_{i=1}f\left( x_{i},y_{i}\right) \Delta s_{i} 
\]
In this formula $s$ denotes arc length along the curve, $P$ denotes a
partition of the curve into $n$ \ pieces, and $\left\| P\right\| $is the
length of the longest piece. $\left( x_{i},y_{i}\right) $ is a point on the $%
i^{th}$ piece. \ (The definition of an ordinary integral $%
\int_{a}^{b}g\left( x\right) dx$ is defined by a special case of this
process, in which the curve $C$ is the segment of the x-axis between $a$ and 
$b$.) \ In practice, this limiting process is rarely carried out, since 
\[
\int_{C}f\left( x,y\right) ds=\int_{a}^{b}f\left( x(t),y(t)\right) \frac{ds}{%
dt}dt 
\]
and the integral on the right is an ordinary integral. \ Recall that $ds=%
\sqrt{dx^{2}+dy^{2}}$.

\subsubsection{3D Line Integral}

Let $C$ denote a space curve given by the parametric equations 
\[
x=x\left( t\right) \qquad y=y\left( t\right) \qquad z=z(t)\qquad a\leq t\leq
b 
\]
or equivalently by the vector equation $\mathbf{r}\left( t\right) =x\left(
t\right) \mathbf{i}+y\left( t\right) \mathbf{j}+z\left( t\right) \mathbf{k}$%
. \ Assume the curve is \textbf{smooth}, which means that the tangent vector 
$\mathbf{r}\prime =\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}+\frac{dz}{%
dt}\mathbf{j}$ is continuous and never the zero vector. \ Let $f(x,y,z)$ be
a function defined at each point of the curve $C$. \ The line integral of $f$
along $C$ is defined by the formula 
\[
\int_{C}f\left( x,y,z\right) ds=\lim_{\left\| P\right\| \rightarrow
0}\sum_{i=1}f\left( x_{i},y_{i},z_{i}\right) \Delta s_{i}\text{.} 
\]
Here, $s$ denotes arc length along the curve, $P$ denotes a partition of the
curve into $n$ \ pieces, and $\left\| P\right\| $is the length of the
longest piece. The point $\left( x_{i},y_{i},z_{i}\right) $ is a point on
the $i^{th}$ piece. \ In practice, this limiting process is rarely carried
out, since 
\[
\int_{C}f\left( x,y,z\right) ds=\int_{a}^{b}f\left( x(t),y(t),z(t)\right) 
\frac{ds}{dt}dt 
\]
and the integral on the right is an ordinary integral. \ In this case $ds=%
\sqrt{dx^{2}+dy^{2}+dz^{2}}$.

\subsubsection{Line Integral of a Vector Field \hfill}

Let $\mathbf{F}$ be a continuous vector field defined on a smooth curve $C$
specified by a vector function $\mathbf{r}(t),\quad a\leq t\leq b$. \ Then
the line integral of $\mathbf{F}$ along $C$ is defined by 
\[
\int \mathbf{F}\cdot d\mathbf{r}=\int \mathbf{F}\left( \mathbf{r}\left(
t\right) \right) \cdot \mathbf{r}^{\prime }\left( t\right) dt 
\]

\subsubsection{\ The Fundamental Theorem for Line Integrals \hfill}

\begin{theorem}
\vspace{1pt}Let $C$ be a smooth curve of two or three variables given by the
vector function \ $\mathbf{r}\left( t\right) $ for $a\leq t\leq b$. \ Let $f$
be a differentiable function of two or three variables whose gradient vector 
$\nabla f$ is continuous on $C$. Then 
\[
\int_{C}\nabla f\cdot d\mathbf{r}=f\left( \mathbf{r}\left( b\right) \right)
-f\left( \mathbf{r}\left( a\right) \right) 
\]
\end{theorem}

This theorem can be thought of as an analog of the fundamental theorem of
calculus, which for ordinary integrals asserts that $\int_{a}^{b}f^{\prime
}\left( x\right) dx=f(b)-f(a)$. \ Note also that if $\mathbf{r}\left(
a\right) $ is the point $\left( x_{1},y_{1},z_{1}\right) $ and $\mathbf{r}%
\left( b\right) $ is the point $\left( x_{2},y_{2},z_{2}\right) $

\[
\int_{C}\nabla f\cdot d\mathbf{r}=f\left( x_{2},y_{2},z_{2}\right) -f\left(
x_{1},y_{1},z_{1}\right) 
\]
One says that $\int_{C}\nabla f\cdot d\mathbf{r}$ is \textbf{path independent%
}, since the value of the integral depends only on the endpoints of the path 
$C$, and therefore will have the same value for any curve connecting those
endpoints. \ An equivalent statement is that for any closed curve $C$ (which
begins and ends at the same point) $\int_{C}\nabla f\cdot d\mathbf{r}=0$.

The fundamental theorem for line integrals embodies not only the fact that
gradient vector fields are path independent, but also the fact that the 
\emph{only} continuous vector fields which are path independent are the
gradient vector fields.\hfill

\noindent \hrulefill

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\noindent \hrulefill

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