Line Integrals

Motivation

Recall that one interpretation of the integral \( \int_a^bf(x)\,dx \) for \( f(x)\geq 0 \) was the area below the graph of the function \( y=f(x) \) between the points \( a \) and \( b \). This is the area of the wall whose base is the line segment between \( (a,0) \) and \( (b,0) \) and whose height above any point \( x\in[a,b] \) is \( f(x) \).

We want to generalize this concept to allow us to compute the areas of those ``walls’’ whose base is a curve, and not just a line segment.

Line integrals of functions in two dimensions

Integral sums

Assume that \( \gamma \) is a curve given by parametric equations: \( \overrightarrow r(t)=\langle \phi(t),\psi(t)\rangle \), as parameter \( t \) ranges from \( a \) to \( b \). Assume that \( f \) is a function defined on the curve \( \gamma \), i.e. \( f:\gamma\to\mathbb R \).

Consider the sequence \( a=t_0\leq t_1\leq t_2\leq\cdots\leq t_n=b \). This sequence of numbers from \( [a,b] \) corresponds to the sequence of points \( \gamma_i=(\phi(t_i),\psi(t_i)) \) on the curve \( \gamma \) (\( 0\leq i\leq n \)). This sequence of points generates the partition of \( \gamma \). The distance between two consecutive points \( \gamma_{i-1} \) and \( \gamma_i \) in the partition is equal to \[ l_i=\sqrt{\left(\phi(t_i)-\phi(t_{i-1})\right)^2+\left(\psi(t_i)-\psi(t_{i-1})\right)^2}.\] Consider an arbitrary sequence of numbers \( c_i\in [t_i,t_{i+1}] \), for \( 0\leq i\leq n-1 \). This sequence of numbers corresponds to the following sequence of points on the curve \( \gamma \): \[ P_0=(\phi(c_0),\psi(c_0)),\;\;\; P_1=(\phi(c_1),\psi(c_1)),\;\;\; P_2=(\phi(c_2),\psi(c_2)),\;\;\;\dots,\;\;\; P_{n-1}=(\phi(c_{n-1}),\psi(c_{n-1})).\] The integral sum corresponding to the partition \( t_0 \), \( t_1 \), \( \dots \), \( t_n \) with the points \( P_0 \), \( P_1 \), \( \dots \), \( P_{n-1} \) is defined as: \[ S(f,t_0,t_1,\dots, t_n,c_0,c_1,\dots, c_{n-1})=f(P_0)\cdot l_0+f(P_1)\cdot l_1+\cdots+ f(P_{n-1})\cdot l_{n-1}.\] The diameter of the partition \( t_0 \), \( t_1 \), \( \dots \), \( t_n \) is defined to be: \[ \delta(t_0,t_1,\dots, t_n)=\max\left\{l_0,l_1,\dots, l_{n-1}\right\}.\]

Definition: Line integral of a function (in 2 dimensions)
 
Let \( \gamma \) be the curve with the parametrization \( x=\phi(t) \), \( y=\psi(t) \), \( a\leq t\leq b \). The function \( f:\gamma\to\mathbb R \) has a line integral and its integral over the curve \( \gamma \) is equal to \( I \) if for each \( \varepsilon> 0 \) there exists \( \delta> 0 \) such that for every partition \( \{t_0,\dots, t_{n}\} \) with \( \delta(t_0,\dots, t_{n})< \delta \) we have \[ \left|S(f,t_0,\dots, t_{n},c_0,\dots, c_{n-1})-I\right|< \varepsilon\] for every choice of \( c_0\in [t_0,t_1] \), \( c_1\in [t_1,t_2] \), \( \dots \), \( c_{n-1}\in [t_{n-1},t_n] \).

In the case that the line integral from the previous definition exists, we write \begin{eqnarray*}I=\int_{\gamma} \mbox{ }f\,ds.\end{eqnarray*}

We will now find a way to calculate line integrals using the integrals from single variable calculus. This reduction is possible in the case when the curve \( \gamma \) is smooth, i.e. when the functions \( \phi \) and \( \psi \) are differentiable and their derivatives, \( \phi^{\prime} \) and \( \psi^{\prime} \) are continuous.

Theorem 1
 
Let \( \gamma \) be the curve with the parametrization \( x=\phi(t) \), \( y=\psi(t) \), \( a\leq t\leq b \). Assume that \( \phi:[a,b]\to\mathbb R \) and \( \psi:[a,b]\to\mathbb R \) are differentiable functions such that \( \phi^{\prime} \) and \( \psi^{\prime} \) are continuous on \( [a,b] \). Assume that \( f:\gamma\to\mathbb R \) is a continuous function. Then \[ \int_{\gamma}f\,ds=\int_a^b f(\phi(t),\psi(t))\cdot\sqrt{\phi^{\prime}(t)^2+\psi^{\prime}(t)^2}\,dt.\]

Remark: The previous theorem holds under weaker assumptions. One useful generalization is obtained when the assumption smoothness of \( \gamma \) is replaced by piecewise smoothness. This means that there are number \( a\leq z_0\leq z_1\leq\cdots \leq z_m=b \) such that \( \gamma \) is smooth on each of the intervals \( [z_i,z_{i+1}] \).

Line integrals of functions in three dimensions

The definition of the line integral in three dimensions is analogous to the one in two dimensions. The definition of the partition and the sequence of chosen points in the partition is precisely the same.

Definition: Line integral of a function
 
Let \( \gamma \) be the curve with the parametrization \( x=\phi(t) \), \( y=\psi(t) \), \( z=\theta(t) \), \( a\leq t\leq b \). The function \( f:\gamma\to\mathbb R \) has a line integral and its integral over the curve \( \gamma \) is equal to \( I \) if for each \( \varepsilon> 0 \) there exists \( \delta> 0 \) such that for every partition \( \{t_0,\dots, t_{n}\} \) with \( \delta(t_0,\dots, t_{n})< \delta \) we have \[ \left|S(f,t_0,\dots, t_{n},c_0,\dots, c_{n-1})-I\right|< \varepsilon\] for every choice of \( c_0\in [t_0,t_1] \), \( c_1\in [t_1,t_2] \), \( \dots \), \( c_{n-1}\in [t_{n-1},t_n] \).

As in two dimensions we have the theorem that establishes the relation between the line integral and the definite integral from single-variable calculus.

Theorem 2
 
Let \( \gamma \) be the curve with the parametrization \( x=\phi(t) \), \( y=\psi(t) \), \( z=\theta(t) \), \( a\leq t\leq b \). Assume that \( \phi:[a,b]\to\mathbb R \), \( \psi:[a,b]\to\mathbb R \), and \( \theta:[a,b]\to\mathbb R \) are differentiable functions such that \( \phi^{\prime} \), \( \psi^{\prime} \), and \( \theta^{\prime} \) are continuous on \( [a,b] \). Assume that \( f:\gamma\to\mathbb R \) is a continuous function. Then \[ \int_{\gamma}f\,ds=\int_a^b f(\phi(t),\psi(t),\theta(t))\cdot\sqrt{\phi^{\prime}(t)^2+\psi^{\prime}(t)^2+ \theta^{\prime}(t)^2}\,dt.\]

Example
 
Assume that \( \gamma \) is the upper semi-circle with center \( (1,0) \) and radius \( 2 \). Find the integral \[ \int_{\gamma} x^2y\,ds.\]

Line integrals of vector fields

Assume that \( \overrightarrow{F}:\gamma\to\mathbb R^3 \) is a vector field defined on the curve \( \gamma \). Let \( a=t_0\leq t_1\leq \cdots \leq t_{n-1}\leq t_n=b \) be the partition of the interval \( [a,b] \). Assume that \( \overrightarrow r(t)=\langle \phi(t),\psi(t),\theta(t)\rangle \) is the parametrization of \( \gamma \) for \( a\leq t\leq b \). The partition of the interval \( [a,b] \) generates the sequence of points \( E_0=(\phi(t_0),\psi(t_0),\theta(t_0)) \), \( E_1=(\phi(t_1),\psi(t_1),\theta(t_1)) \), \( \dots \), \( E_n=(\phi(t_n),\psi(t_n),\theta(t_n)) \), and these points define the vectors \[ \overrightarrow{r_0}=\overrightarrow{E_0E_1}=\overrightarrow r(t_1)-\overrightarrow r(t_0), \;\;\; \overrightarrow{r_1}=\overrightarrow{E_1E_2}=\overrightarrow r(t_2)-\overrightarrow r(t_1), \;\;\; \overrightarrow{r_{n-1}}=\overrightarrow{E_{n-1}E_n}=\overrightarrow r(t_n)-\overrightarrow r(t_{n-1}).\]

Let \( c_0 \), \( c_1 \), \( \dots \), \( c_{n-1} \) be a sequence of real numbers such that \( c_i\in[t_i,t_{i+1}] \). This sequence of real numbers generates the sequence of points on \( \gamma \): \[ P_0=(\phi(c_0),\psi(c_0),\theta(c_0)),\;\;\; P_1=(\phi(c_1),\psi(c_1),\theta(c_1)),\;\;\; \cdots,\;\;\; P_{n-1}=(\phi(c_{n-1}),\psi(c_{n-1}),\theta(c_{n-1})).\] The integral sum corresponding to the partition \( t_0 \), \( t_1 \), \( \dots \), \( t_{n} \) and the sequence of numbers \( c_0 \), \( \dots \), \( c_{n-1} \) is defined as \[ S(\overrightarrow{F},t_0, \dots, t_n, c_0,\dots, c_{n-1})=\overrightarrow{F}(P_0)\cdot \overrightarrow{r_0}+\overrightarrow F(P_1)\cdot\overrightarrow{r_1}+\cdots+\overrightarrow F(P_{n-1})\cdot \overrightarrow{r_{n-1}}.\]

The diameter of the partition \( \delta(t_0,\dots, t_n) \) is defined as above to be the maximum of \( |t_i-t_{i-1}| \).

Definition: Line integral of a vector field
 
Assume that \( \gamma \) is a curve in space and that \( \overrightarrow F:\gamma\to\mathbb R^3 \) (or \( \overrightarrow F:\gamma\to\mathbb R^2 \)) is a vector field. We say that the vector field \( \overrightarrow F \) is integrable over the curve \( \gamma \) and its integral is equal to \( I \) if for every \( \varepsilon> 0 \) there exists \( \delta> 0 \) such that \( \delta(t_0,\dots, t_n)< \delta \) implies that \[ \left|I-S(\overrightarrow{F},t_0,\dots, t_n,c_0,\dots, c_n)\right|\leq \varepsilon\] whenever \( c_0 \), \( \dots \), \( c_{n-1} \) is a sequence of numbers such that \( c_i\in[t_i,t_{i+1}] \).

The value \( I \) is called the integral of \( \overrightarrow F \) with respect to \( \gamma \) and is denoted as \[ I=\int_{\gamma} \overrightarrow F\cdot d\overrightarrow r.\] Line integral of the vector field \( \overrightarrow F=\langle P, Q, R\rangle \) over the curve \( \gamma \) is often denoted using one of the following two expressions: \[ I=\int_{\gamma}\overrightarrow F\cdot \overrightarrow T\,ds=\int_{\gamma}P\,dx+Q\,dy+R\,dz.\]

The following theorem establishes the relation between line integrals of vector fields and definite integrals from single-variable calculus.

Theorem 3
 
Assume that \( \gamma \) is a curve defined with parametric equations \( x=\phi(t) \), \( y=\psi(t) \), \( z=\theta(t) \), for \( a\leq t\leq b \). Assume that the functions \( \phi \), \( \psi \), and \( \theta \) are differentiable and that \( \phi^{\prime} \), \( \psi^{\prime} \), and \( \theta^{\prime} \) are continuous on \( [a,b] \). Let \( \overrightarrow F:\gamma\to\mathbb R^3 \) be a vector field that is integrable with respect to \( \gamma \). Then \[ \int_{\gamma} \overrightarrow F\cdot d\overrightarrow r=\int_a^b \overrightarrow F\left(\phi(t),\psi(t),\theta(t)\right)\cdot \overrightarrow{r^{\prime}}(t)\,dt.\]

Notice that if \( \rho \) is a curve with the same range as \( \gamma \) but with opposite orientation, then \( \int_{\gamma} \overrightarrow{F}\cdot d\overrightarrow r=-\int_{\rho}\overrightarrow F\cdot d\overrightarrow r \).

Example
 
Let \( \overrightarrow F=\langle y,2x\rangle \), and let \( \gamma \) be the circle of radius \( 3 \) parametrized counter-clockwise. Evaluate \[ \int_{\gamma} \overrightarrow F\cdot d\overrightarrow r.\]

Remark. If we are dealing with and integral in two dimensions over the curve \( \gamma \) that is closed (i.e. the starting point is the same as the ending point), we will often write \( \oint_{\gamma}\overrightarrow F\cdot d\overrightarrow r \) to emphasize that the curve is parametrized counter-clockwise.

Fundamental theorem for line integrals

Theorem (Fundamental theorem for line integrals)
 
Assume that \( \gamma \) is a smooth curve parametrized by \( \overrightarrow r(t) \) for \( a\leq t\leq b \). Assume that \( D \) is a domain in \( \mathbb R^2 \) that contains the curve \( \gamma \) in its interior. Assume that \( f:D\to\mathbb R \) is a differentiable function. Then the following equality holds: \[ \int_{\gamma}\nabla f\cdot d\overrightarrow r=f(\overrightarrow r(b))-f(\overrightarrow r(a)).\]


2005-2017 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us