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''line integral''
 
''line integral''
  
An integral along a curve. In <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274101.png" />-dimensional Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274102.png" /> consider a given rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274104.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274105.png" /> is the arc length; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274106.png" /> be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274107.png" />. The curvilinear integral
+
An integral along a curve. In $ n $-dimensional Euclidean space $ \mathbb{R}^{n} $ consider a given rectifiable curve $ \gamma = \{ x = x(s) \mid 0 \leq s \leq S \} $, $ x = (x_{1},\ldots,x_{n}) $, where $ s $ is the arc-length; let $ F = F(x(s)) $ be a function defined on $ \gamma $. The curvilinear integral
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274108.png" /></td> </tr></table>
+
\int_{\gamma} F(x) ~ \mathrm{d}{s}
 
+
$$
 
is defined by the equality
 
is defined by the equality
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c0274109.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
\int_{\gamma} F(x) ~ \mathrm{d}{s} \stackrel{\text{df}}{=} \int_{0}^{s} F(x(s)) ~ \mathrm{d}{s} \qquad (1)
 
+
$$
(the integral on the right is an integral over a real interval), and is called a line integral of the first kind, or a line integral with respect to arc length. It is the limit of suitable integral sums, which can be described in terms related to the curve. For example, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741010.png" /> is Riemann-integrable (see [[Riemann integral|Riemann integral]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741011.png" /> is a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741013.png" /> is its mesh, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741015.png" /> is the length of the section of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741016.png" /> from the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741017.png" /> to the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741019.png" />, and
+
(the integral on the right is an integral over a real interval), and is called a '''line integral of the first kind''', or a '''line integral with respect to arc-length'''. It is the limit of suitable integral sums, which can be described in terms related to the curve. For example, if $ F(x(s)) $ is Riemann-integrable (see [[Riemann integral|Riemann integral]]), $ \tau = (s_{i})_{i = 0}^{m} $ is a partition of $ [0,S] $, $ \delta_{\tau} = \max_{1 \leq i \leq m} (s_{i} - s_{i - 1}) $ is its mesh, $ \xi_{i} \in [s_{i - 1},s_{i}] $ is a sample point, $ \Delta s_{i} = s_{i} - s_{i - 1} $ is the length of the section of $ \gamma $ from the point $ x(s_{i - 1}) $ to the point $ x(s_{i}) $, where $ i \in \{ 1,\ldots,m \} $, and
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741020.png" /></td> </tr></table>
+
\sigma_{\tau} \stackrel{\text{df}}{=} \sum_{i = 1}^{m} F(x(\xi_{i})) \Delta s_{i},
 
+
$$
 
then
 
then
 +
$$
 +
\int_{\gamma} F(x) ~ \mathrm{d}{s} \stackrel{\text{df}}{=} \lim_{\delta_{\tau} \to 0} \sigma_{\tau}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741021.png" /></td> </tr></table>
+
If the rectifiable curve $ \gamma $ is given parametrically by $ x = x(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $, where $ a \leq t \leq b $, and $ F = F(x(t)) $ is a function defined on $ \gamma $, then the integral
 
+
$$
If the rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741022.png" /> is given parametrically by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741025.png" /> is a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741026.png" />, then the integral
+
\int_{\gamma} F(x) ~ \mathrm{d}{x_{k}}, \qquad k \in \{ 1,\ldots,n \}
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741027.png" /></td> </tr></table>
 
 
 
 
is defined by
 
is defined by
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741028.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
\int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} \stackrel{\text{df}}{=} \int_{a}^{b} F(x(t)) ~ \mathrm{d}{{\phi_{k}}(t)} \qquad (2)
 
+
$$
(the integral on the right is a [[Stieltjes integral|Stieltjes integral]]), and is called a line integral of the second kind or a line integral with respect to the coordinate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741029.png" />. It is also the limit of suitably constructed Riemann sums: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741030.png" /> is a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741034.png" />, and
+
(the integral on the right is a [[Stieltjes integral|Stieltjes integral]]), and is called a line integral of the second kind or a line integral with respect to the coordinate $ x_{k} $. It is also the limit of suitably constructed Riemann sums: If $ \tau = (t_{i})_{i = 0}^{m} $ is a partition of $ [a,b] $, $ \eta_{i} \in [t_{i - 1},t_{i}] $ is a sample point, $ \Delta x_{k i} = {\phi_{k}}(t_{i}) - {\phi_{k}}(t_{i - 1}) $, where $ i \in \{ 1,\ldots,m \} $, and
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741035.png" /></td> </tr></table>
+
\widetilde{\sigma}_{\tau} \stackrel{\text{df}}{=} \sum_{i = 1}^{m} F(x(\eta_{i})) \Delta x_{k i},
 
+
$$
 
then
 
then
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741036.png" /></td> </tr></table>
+
\int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} \stackrel{\text{df}}{=} \lim_{\delta_{\tau} \to 0} \widetilde{\sigma}_{\tau}.
 
+
$$
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741037.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741038.png" />, then the curvilinear integrals (1) and (2) always exist. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741039.png" /> is the initial point and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741040.png" /> the end point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741041.png" />, then the curvilinear integrals (1) and (2) are denoted by
+
If $ F $ is a continuous function on $ \gamma $, then the curvilinear integrals (1) and (2) always exist. If $ A $ is the initial point and $ B $ the end point of $ \gamma $, then the curvilinear integrals (1) and (2) are denoted by
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741042.png" /></td> </tr></table>
+
\int_{\widehat{AB}} F(x) ~ \mathrm{d}{s} \qquad \text{and} \qquad \int_{\widehat{AB}} F(x) ~ \mathrm{d}{x_{k}}
 
+
$$
 
respectively.
 
respectively.
  
 
Line integrals of the first kind are independent of the orientation of the curve:
 
Line integrals of the first kind are independent of the orientation of the curve:
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741043.png" /></td> </tr></table>
+
\int_{\widehat{BA}} F(x) ~ \mathrm{d}{s} = \int_{\widehat{AB}} F(x) ~ \mathrm{d}{s}
 
+
$$
 
but line integrals of the second kind change sign when the orientation is reversed:
 
but line integrals of the second kind change sign when the orientation is reversed:
 +
$$
 +
\int_{\widehat{BA}} F(x) ~ \mathrm{d}{x_{k}} = - \int_{\widehat{AB}} F(x) ~ \mathrm{d}{x_{k}}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741044.png" /></td> </tr></table>
+
If $ \gamma $ is a continuously differentiable curve given by a continuously differentiable representation $ x(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $, where $ a \leq t \leq b $, and $ F $ is a continuous function on $ \gamma $, then
 
+
\begin{align}
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741045.png" /> is a continuously differentiable curve given by a continuously differentiable representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741047.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741048.png" /> is a continuous function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741049.png" />, then
+
\int_{\gamma} F(x) ~ \mathrm{d}{s}
 
+
& = \int_{a}^{b} F(x(t)) \underbrace{\sqrt{\sum_{k = 1}^{n} [{\phi_{k}'}(t)]^{2}}}_{> 0} ~ \mathrm{d}{t}, \\
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741050.png" /></td> </tr></table>
+
\int_{\gamma} F(x) ~ \mathrm{d}{x_{k}}
 
+
& = \int_{a}^{b} F(x(t)) {\phi_{k}'}(t) ~ \mathrm{d}{t}, \qquad k \in \{ 1,\ldots,n \},
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741051.png" /></td> </tr></table>
+
\end{align}
 
+
and hence the integrals on the right of these equalities are independent of the choice of the parameter on $ \gamma $. If $ \tau = (\cos(\alpha_{1}),\ldots,\cos(\alpha_{n})) $ is a unit tangent vector to the curve $ \gamma $, then the line integral of the second kind may be expressed in terms of a line integral of the first kind via the formula
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741052.png" /></td> </tr></table>
+
$$
 
+
\int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} = \int_{\gamma} F(x) \cos(\alpha_{k}) ~ \mathrm{d}{s}
and hence the integrals on the right of these equalities are independent of the choice of the parameter on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741053.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741054.png" /> is a unit tangent vector to the curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741055.png" />, then the line integral of the second kind may be expressed in terms of a line integral of the first kind via the formula
+
$$
 
+
If $ \gamma $ is given in vector notation $ \mathbf{r}(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $ and $ \mathbf{a}(x(t)) = ({a_{1}}(x(t)),\ldots,{a_{n}}(x(t))) $ is a vector function defined on $ \gamma $, then, by definition,
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741056.png" /></td> </tr></table>
+
$$
 
+
\int_{\gamma} \mathbf{a}(x) ~ \mathrm{d}{\mathbf{r}}
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741057.png" /> is given in vector notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741059.png" /> is a vector function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741060.png" />, then, by definition,
+
\stackrel{\text{df}}{=} \int_{\gamma} \langle \mathbf{a},\mathbf{r} \rangle ~ \mathrm{d}{s}
 
+
= \sum_{k = 1}^{n} \int_{\gamma} {a_{k}}(x) ~ \mathrm{d}{x_{k}}.
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741061.png" /></td> </tr></table>
+
$$
 
 
 
The relationship between line integrals and integrals of other types is established by the [[Green formulas|Green formulas]] and the [[Stokes formula|Stokes formula]].
 
The relationship between line integrals and integrals of other types is established by the [[Green formulas|Green formulas]] and the [[Stokes formula|Stokes formula]].
  
Line integrals may be used to calculate the area of plane domains: If a finite plane domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741062.png" /> is bounded by a simple rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741063.png" />, then its area is
+
Line integrals may be used to calculate the area of plane domains: If a finite plane domain $ G $ is bounded by a simple rectifiable curve $ \gamma $, then its area is
 +
$$
 +
\operatorname{mes}(G)
 +
= \int_{\gamma} x_{1} ~ \mathrm{d}{x_{2}}
 +
= - \int_{\gamma} x_{2} ~ \mathrm{d}{x_{1}}
 +
= \frac{1}{2} \int_{\gamma} (x_{1} ~ \mathrm{d}{x_{2}} - x_{2} ~ \mathrm{d}{x_{1}}),
 +
$$
 +
where the contour $ \gamma $ is oriented in the counter-clockwise sense.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741064.png" /></td> </tr></table>
+
If $ M $ is a mass distributed over $ \gamma $ with linear density $ \rho(x) $, then
 +
$$
 +
M = \int_{\gamma} \rho(x) ~ \mathrm{d}{s}.
 +
$$
  
where the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741065.png" /> is oriented in the counter-clockwise sense.
+
If $ \mathbf{F}(x) $ is the intensity of a force field (i.e., the force acting on a unit mass), then
 +
$$
 +
\int_{\gamma} \mathbf{F}(x) \bullet \mathrm{d}{\mathbf{r}}
 +
$$
 +
is equal to the work performed by the field in moving a unit mass along $ \gamma $.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741066.png" /> is a mass distributed over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741067.png" /> with linear density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741068.png" />, then
+
Line integrals are used in the theory of vector fields. If $ \mathbf{a} = \mathbf{a}(x) = ({a_{1}}(x),\ldots,{a_{n}}(x)) $ is a continuous vector field defined on some $ n $-dimensional domain $ G $, where $ n > 1 $, then the following three properties are equivalent:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741069.png" /></td> </tr></table>
+
1) For any closed rectifiable curve $ \gamma \subseteq G $,
 +
$$
 +
\int_{\gamma} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = 0.
 +
$$
 +
(a vector field possessing this property is called a '''potential field''').
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741070.png" /> is the intensity of a force field (i.e. the force acting on a unit mass), then
+
2) For any pair of points $ A,B \in G $ and any two rectifiable curves $ \left( \widehat{AB} \right)_{1},\left( \widehat{AB} \right)_{2} $ with initial point $ A $ and end point $ B $:
 +
$$
 +
\int_{\left( \widehat{AB} \right)_{1}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = \int_{\left( \widehat{AB} \right)_{2}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}}.
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741071.png" /></td> </tr></table>
+
3) There exists in $ G $ a function $ u(x) $ (called a potential function of the field $ \mathbf{a}(x) $), such that $ {\nabla u}(x) = \mathbf{a}(x) $, i.e. $ \dfrac{\partial u(x)}{\partial x_{k}} = {a_{k}}(x) $, where $ k \in \{ 1,\ldots,n \} $, and moreover, for any $ A,B \in G $ and any curve $ \widehat{AB} \subseteq G $,
 +
$$
 +
\int_{\widehat{AB}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = u(B) - u(A).
 +
$$
  
is equal to the work performed by the field in moving a unit mass along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741072.png" />.
+
If $ n = 2 $ or $ 3 $ and $ G $ is a simply-connected domain ($ n = 2 $) or a simply-connected surface ($ n = 3 $), while the field $ \mathbf{a}(x) $ is continuously differentiable, then the properties 1)–3) are equivalent to the following property:
  
Line integrals are used in the theory of vector fields. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741073.png" /> is a continuous vector field defined on some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741074.png" />-dimensional domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741075.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741076.png" />, then the following three properties are equivalent:
+
4) The rotation of the vector field vanishes in $ G $:
 +
$$
 +
\operatorname{rot} \mathbf{a}(x) = 0, \qquad x \in G.
 +
$$
  
1) For any closed rectifiable curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741077.png" />,
+
If $ G $ is not simply connected, then 4) need not be equivalent to 1)–3). For example, for the field
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741078.png" /></td> </tr></table>
+
\mathbf{a}(x_{1},x_{2}) = \left( - \frac{x_{2}}{x_{1}^{2} + x_{2}^{2}},\frac{x_{1}}{x_{1}^{2} + x_{2}^{2}} \right),
 
+
$$
(a vector field possessing this property is called a potential field).
+
defined on the plane punctured at the origin one has $ \operatorname{rot} \mathbf{a}(x) = 0 $ for $ x \neq 0 $, but
 
+
$$
2) For any pair of points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741079.png" /> and any two rectifiable curves <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741080.png" /> with initial point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741081.png" /> and end point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741082.png" />:
+
\int_{\| \mathbf{r} \| = 1} \mathbf{a} \bullet \mathrm{d}{\mathbf{r}} = 2 \pi \neq 0.
 
+
$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741083.png" /></td> </tr></table>
 
 
 
3) There exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741084.png" /> a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741085.png" /> (called a potential function of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741086.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741087.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741089.png" />, and moreover, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741090.png" /> and any curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741091.png" />,
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741092.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741093.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741094.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741095.png" /> is a simply-connected domain (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741096.png" />) or a simply-connected surface (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741097.png" />), while the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741098.png" /> is continuously differentiable, then the properties 1)–3) are equivalent to the following property:
 
 
 
4) The rotation of the vector field vanishes in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c02741099.png" />:
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410100.png" /></td> </tr></table>
 
 
 
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410101.png" /> is not simply connected, then 4) need not be equivalent to 1)–3). For example, for the field
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410102.png" /></td> </tr></table>
 
 
 
defined on the plane punctured at the origin one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410104.png" />, but
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410105.png" /></td> </tr></table>
 
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Il'in,   E.G. Poznyak,   "Fundamentals of mathematical analysis" , '''2''' , MIR (1982) (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> L.D. Kudryavtsev,   "A course in mathematical analysis" , '''2''' , Moscow (1981) (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> S.M. Nikol'skii,   "A course of mathematical analysis" , '''2''' , MIR (1977) (Translated from Russian)</TD></TR></table>
+
<table>
 
+
<TR><TD valign="top">[1]</TD> <TD valign="top"> V.A. Il’in, E.G. Poznyak, “Fundamentals of mathematical analysis”, '''2''', MIR (1982) (Translated from Russian).</TD></TR>
 
+
<TR><TD valign="top">[2]</TD> <TD valign="top"> L.D. Kudryavtsev, “A course in mathematical analysis”, '''2''', Moscow (1981) (in Russian).</TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> S.M. Nikol’skii, “A course of mathematical analysis”, '''2''', MIR (1977) (translated from Russian).</TD></TR></table>
  
 
====Comments====
 
====Comments====
Line integrals are a special instance of integrals of differential forms over chains, namely integrals of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410106.png" />-form over a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027410/c027410107.png" />-chain (cf. [[Differential form|Differential form]]; [[Chain|Chain]] and especially [[Integration on manifolds|Integration on manifolds]]).
+
Line integrals are a special instance of integrals of differential forms over chains, namely integrals of a $ 1 $-form over a $ 1 $-chain (cf. [[Differential form|Differential form]]; [[Chain|Chain]] and especially [[Integration on manifolds|Integration on manifolds]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin,   "Principles of mathematical analysis" , McGraw-Hill (1976) pp. 107–108</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Spivak,   "Calculus on manifolds" , Benjamin/Cummings (1965)</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> W. Rudin, “Principles of mathematical analysis”, McGraw-Hill (1976), pp. 107–108.</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Spivak, “Calculus on manifolds”, Benjamin/Cummings (1965).</TD></TR>
 +
</table>

Latest revision as of 06:40, 1 December 2016

line integral

An integral along a curve. In $ n $-dimensional Euclidean space $ \mathbb{R}^{n} $ consider a given rectifiable curve $ \gamma = \{ x = x(s) \mid 0 \leq s \leq S \} $, $ x = (x_{1},\ldots,x_{n}) $, where $ s $ is the arc-length; let $ F = F(x(s)) $ be a function defined on $ \gamma $. The curvilinear integral $$ \int_{\gamma} F(x) ~ \mathrm{d}{s} $$ is defined by the equality $$ \int_{\gamma} F(x) ~ \mathrm{d}{s} \stackrel{\text{df}}{=} \int_{0}^{s} F(x(s)) ~ \mathrm{d}{s} \qquad (1) $$ (the integral on the right is an integral over a real interval), and is called a line integral of the first kind, or a line integral with respect to arc-length. It is the limit of suitable integral sums, which can be described in terms related to the curve. For example, if $ F(x(s)) $ is Riemann-integrable (see Riemann integral), $ \tau = (s_{i})_{i = 0}^{m} $ is a partition of $ [0,S] $, $ \delta_{\tau} = \max_{1 \leq i \leq m} (s_{i} - s_{i - 1}) $ is its mesh, $ \xi_{i} \in [s_{i - 1},s_{i}] $ is a sample point, $ \Delta s_{i} = s_{i} - s_{i - 1} $ is the length of the section of $ \gamma $ from the point $ x(s_{i - 1}) $ to the point $ x(s_{i}) $, where $ i \in \{ 1,\ldots,m \} $, and $$ \sigma_{\tau} \stackrel{\text{df}}{=} \sum_{i = 1}^{m} F(x(\xi_{i})) \Delta s_{i}, $$ then $$ \int_{\gamma} F(x) ~ \mathrm{d}{s} \stackrel{\text{df}}{=} \lim_{\delta_{\tau} \to 0} \sigma_{\tau}. $$

If the rectifiable curve $ \gamma $ is given parametrically by $ x = x(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $, where $ a \leq t \leq b $, and $ F = F(x(t)) $ is a function defined on $ \gamma $, then the integral $$ \int_{\gamma} F(x) ~ \mathrm{d}{x_{k}}, \qquad k \in \{ 1,\ldots,n \} $$ is defined by $$ \int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} \stackrel{\text{df}}{=} \int_{a}^{b} F(x(t)) ~ \mathrm{d}{{\phi_{k}}(t)} \qquad (2) $$ (the integral on the right is a Stieltjes integral), and is called a line integral of the second kind or a line integral with respect to the coordinate $ x_{k} $. It is also the limit of suitably constructed Riemann sums: If $ \tau = (t_{i})_{i = 0}^{m} $ is a partition of $ [a,b] $, $ \eta_{i} \in [t_{i - 1},t_{i}] $ is a sample point, $ \Delta x_{k i} = {\phi_{k}}(t_{i}) - {\phi_{k}}(t_{i - 1}) $, where $ i \in \{ 1,\ldots,m \} $, and $$ \widetilde{\sigma}_{\tau} \stackrel{\text{df}}{=} \sum_{i = 1}^{m} F(x(\eta_{i})) \Delta x_{k i}, $$ then $$ \int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} \stackrel{\text{df}}{=} \lim_{\delta_{\tau} \to 0} \widetilde{\sigma}_{\tau}. $$ If $ F $ is a continuous function on $ \gamma $, then the curvilinear integrals (1) and (2) always exist. If $ A $ is the initial point and $ B $ the end point of $ \gamma $, then the curvilinear integrals (1) and (2) are denoted by $$ \int_{\widehat{AB}} F(x) ~ \mathrm{d}{s} \qquad \text{and} \qquad \int_{\widehat{AB}} F(x) ~ \mathrm{d}{x_{k}} $$ respectively.

Line integrals of the first kind are independent of the orientation of the curve: $$ \int_{\widehat{BA}} F(x) ~ \mathrm{d}{s} = \int_{\widehat{AB}} F(x) ~ \mathrm{d}{s} $$ but line integrals of the second kind change sign when the orientation is reversed: $$ \int_{\widehat{BA}} F(x) ~ \mathrm{d}{x_{k}} = - \int_{\widehat{AB}} F(x) ~ \mathrm{d}{x_{k}}. $$

If $ \gamma $ is a continuously differentiable curve given by a continuously differentiable representation $ x(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $, where $ a \leq t \leq b $, and $ F $ is a continuous function on $ \gamma $, then \begin{align} \int_{\gamma} F(x) ~ \mathrm{d}{s} & = \int_{a}^{b} F(x(t)) \underbrace{\sqrt{\sum_{k = 1}^{n} [{\phi_{k}'}(t)]^{2}}}_{> 0} ~ \mathrm{d}{t}, \\ \int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} & = \int_{a}^{b} F(x(t)) {\phi_{k}'}(t) ~ \mathrm{d}{t}, \qquad k \in \{ 1,\ldots,n \}, \end{align} and hence the integrals on the right of these equalities are independent of the choice of the parameter on $ \gamma $. If $ \tau = (\cos(\alpha_{1}),\ldots,\cos(\alpha_{n})) $ is a unit tangent vector to the curve $ \gamma $, then the line integral of the second kind may be expressed in terms of a line integral of the first kind via the formula $$ \int_{\gamma} F(x) ~ \mathrm{d}{x_{k}} = \int_{\gamma} F(x) \cos(\alpha_{k}) ~ \mathrm{d}{s} $$ If $ \gamma $ is given in vector notation $ \mathbf{r}(t) = ({\phi_{1}}(t),\ldots,{\phi_{n}}(t)) $ and $ \mathbf{a}(x(t)) = ({a_{1}}(x(t)),\ldots,{a_{n}}(x(t))) $ is a vector function defined on $ \gamma $, then, by definition, $$ \int_{\gamma} \mathbf{a}(x) ~ \mathrm{d}{\mathbf{r}} \stackrel{\text{df}}{=} \int_{\gamma} \langle \mathbf{a},\mathbf{r} \rangle ~ \mathrm{d}{s} = \sum_{k = 1}^{n} \int_{\gamma} {a_{k}}(x) ~ \mathrm{d}{x_{k}}. $$ The relationship between line integrals and integrals of other types is established by the Green formulas and the Stokes formula.

Line integrals may be used to calculate the area of plane domains: If a finite plane domain $ G $ is bounded by a simple rectifiable curve $ \gamma $, then its area is $$ \operatorname{mes}(G) = \int_{\gamma} x_{1} ~ \mathrm{d}{x_{2}} = - \int_{\gamma} x_{2} ~ \mathrm{d}{x_{1}} = \frac{1}{2} \int_{\gamma} (x_{1} ~ \mathrm{d}{x_{2}} - x_{2} ~ \mathrm{d}{x_{1}}), $$ where the contour $ \gamma $ is oriented in the counter-clockwise sense.

If $ M $ is a mass distributed over $ \gamma $ with linear density $ \rho(x) $, then $$ M = \int_{\gamma} \rho(x) ~ \mathrm{d}{s}. $$

If $ \mathbf{F}(x) $ is the intensity of a force field (i.e., the force acting on a unit mass), then $$ \int_{\gamma} \mathbf{F}(x) \bullet \mathrm{d}{\mathbf{r}} $$ is equal to the work performed by the field in moving a unit mass along $ \gamma $.

Line integrals are used in the theory of vector fields. If $ \mathbf{a} = \mathbf{a}(x) = ({a_{1}}(x),\ldots,{a_{n}}(x)) $ is a continuous vector field defined on some $ n $-dimensional domain $ G $, where $ n > 1 $, then the following three properties are equivalent:

1) For any closed rectifiable curve $ \gamma \subseteq G $, $$ \int_{\gamma} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = 0. $$ (a vector field possessing this property is called a potential field).

2) For any pair of points $ A,B \in G $ and any two rectifiable curves $ \left( \widehat{AB} \right)_{1},\left( \widehat{AB} \right)_{2} $ with initial point $ A $ and end point $ B $: $$ \int_{\left( \widehat{AB} \right)_{1}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = \int_{\left( \widehat{AB} \right)_{2}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}}. $$

3) There exists in $ G $ a function $ u(x) $ (called a potential function of the field $ \mathbf{a}(x) $), such that $ {\nabla u}(x) = \mathbf{a}(x) $, i.e. $ \dfrac{\partial u(x)}{\partial x_{k}} = {a_{k}}(x) $, where $ k \in \{ 1,\ldots,n \} $, and moreover, for any $ A,B \in G $ and any curve $ \widehat{AB} \subseteq G $, $$ \int_{\widehat{AB}} \mathbf{a}(x) \bullet \mathrm{d}{\mathbf{r}} = u(B) - u(A). $$

If $ n = 2 $ or $ 3 $ and $ G $ is a simply-connected domain ($ n = 2 $) or a simply-connected surface ($ n = 3 $), while the field $ \mathbf{a}(x) $ is continuously differentiable, then the properties 1)–3) are equivalent to the following property:

4) The rotation of the vector field vanishes in $ G $: $$ \operatorname{rot} \mathbf{a}(x) = 0, \qquad x \in G. $$

If $ G $ is not simply connected, then 4) need not be equivalent to 1)–3). For example, for the field $$ \mathbf{a}(x_{1},x_{2}) = \left( - \frac{x_{2}}{x_{1}^{2} + x_{2}^{2}},\frac{x_{1}}{x_{1}^{2} + x_{2}^{2}} \right), $$ defined on the plane punctured at the origin one has $ \operatorname{rot} \mathbf{a}(x) = 0 $ for $ x \neq 0 $, but $$ \int_{\| \mathbf{r} \| = 1} \mathbf{a} \bullet \mathrm{d}{\mathbf{r}} = 2 \pi \neq 0. $$

References

[1] V.A. Il’in, E.G. Poznyak, “Fundamentals of mathematical analysis”, 2, MIR (1982) (Translated from Russian).
[2] L.D. Kudryavtsev, “A course in mathematical analysis”, 2, Moscow (1981) (in Russian).
[3] S.M. Nikol’skii, “A course of mathematical analysis”, 2, MIR (1977) (translated from Russian).

Comments

Line integrals are a special instance of integrals of differential forms over chains, namely integrals of a $ 1 $-form over a $ 1 $-chain (cf. Differential form; Chain and especially Integration on manifolds).

References

[a1] W. Rudin, “Principles of mathematical analysis”, McGraw-Hill (1976), pp. 107–108.
[a2] M. Spivak, “Calculus on manifolds”, Benjamin/Cummings (1965).
How to Cite This Entry:
Curvilinear integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Curvilinear_integral&oldid=15118
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article