4.6 Line Integrals Of Vector Fields

Suppose a force field \(\mathbf F\) moves a particle along a smooth curve \(C\). Over a short arc of length \(\Delta s\) the particle moves approximately in the direction of the unit tangent vector \(\mathbf T\), and the work done is the tangential component of the force times the distance, \((\mathbf F\cdot \mathbf T)\,\Delta s\). Adding and passing to the limit, the work done along \(C\) is \begin {equation} W = \int _C \mathbf F\cdot \mathbf T\,ds , \label {eq:work} \end {equation} the line integral with respect to arc length of the tangential component of the force. If \(C\) is given by \(\mathbf r(t) = x(t)\,\mathbf i + y(t)\,\mathbf j + z(t)\,\mathbf k\), \(a\leq t\leq b\), then \(\mathbf T(t) = \frac {\mathbf r'(t)}{|\mathbf r'(t)|}\) and \(ds = |\mathbf r'(t)|\,dt\), so (6) becomes \[W = \int ^b_a \left [\mathbf F(\mathbf r(t)) \cdot \frac {\mathbf r'(t)}{|\mathbf r'(t)|}\right ]|\mathbf r'(t)|\,dt = \int ^b_a\mathbf F(\mathbf r(t))\cdot \mathbf r'(t)\,dt .\]

Definition 4.6.1. Let \(\mathbf F\) be a continuous vector field defined on a smooth curve \(C\) given by a vector function \(\mathbf r(t)\), \(a\leq t \leq b\). The line integral of \(\mathbf F\) along \(C\) is \[\int _C \mathbf F\cdot d\mathbf r = \int ^b_a \mathbf F(\mathbf r(t))\cdot \mathbf r'(t)\, dt = \int _C \mathbf F\cdot \mathbf T\,ds .\] If \(\mathbf F\) is the velocity field of a fluid, it is called the flow along \(C\), and if \(C\) is closed, the circulation around \(C\).

In components, with \(\mathbf F = P\,\mathbf i + Q\,\mathbf j + R\,\mathbf k\) and \(d\mathbf r = \mathbf r'(t)\,dt = \langle dx, dy, dz\rangle \), \[\int _C \mathbf F\cdot d\mathbf r = \int _C P\,dx + Q\,dy + R\,dz ,\] so the line integrals with respect to the coordinates are line integrals of vector fields.

Note. Although \(\int _C \mathbf F\cdot d\mathbf r = \int _C \mathbf F\cdot \mathbf T\, ds\) is a line integral with respect to arc length, it is still true that \(\int _{-C}\mathbf F\cdot d\mathbf r = -\int _C \mathbf F\cdot d\mathbf r\), because the unit tangent vector \(\mathbf T\) is replaced by its negative when \(C\) is replaced by \(-C\).

Example 4.6.2. Find the work done by the force \[\mathbf F(x,y,z) = (x^2 -y)\,\mathbf i + (y^2 - z)\,\mathbf j + (z^2-x)\,\mathbf k\] along each of the following curves from \((0,0,0)\) to \((1,1,1)\):

1.: \(\mathbf r(t) = t\,\mathbf i + t \,\mathbf j + t\,\mathbf k\), \(0\leq t\leq 1\);
2.: \(\mathbf r(t) = t\,\mathbf i + t^2\,\mathbf j + t^3\,\mathbf k\), \(0\leq t \leq 1\).

Solution.

1.: Here \(\mathbf r'(t) = \mathbf i + \mathbf j + \mathbf k\) and \(\mathbf F(\mathbf r(t)) = (t^2 - t)(\mathbf i + \mathbf j + \mathbf k)\), so \[W = \int ^1_0 \mathbf F(\mathbf r(t))\cdot \mathbf r'(t)\,dt = \int _0^1 (3t^2 - 3t)\,dt = 1 - \frac 32 = -\frac {1}{2} .\]
2.: Here \(\mathbf r'(t) = \mathbf i + 2t\,\mathbf j + 3t^2\,\mathbf k\) and \[\mathbf F(\mathbf r(t)) = (t^2 -t^2)\,\mathbf i + (t^4 - t^3)\,\mathbf j + (t^6 -t)\,\mathbf k ,\] so \[W = \int ^1_0 \left [2t(t^4-t^3) + 3t^2(t^6-t)\right ]dt = \int ^1_0 ( 2t^5 - 2t^4 + 3t^8 - 3t^3) \,dt = \frac 13 - \frac 25 + \frac 13 - \frac 34 = -\frac {29}{60} .\]

The work depends on the path, so \(\mathbf F\) cannot be conservative; indeed \(P_y = -1\neq 0 = Q_x\).

Example 4.6.3. Find the circulation of \(\mathbf F = -y\,\mathbf i + x\,\mathbf j\) around the circle of radius \(a\) about the origin, traversed counter-clockwise.

Solution. With \(\mathbf r(t) = \langle a\cos t, a\sin t\rangle \), \(0\leq t\leq 2\pi \), \(\mathbf F(\mathbf r(t))\cdot \mathbf r'(t) = \langle -a\sin t, a\cos t\rangle \cdot \langle -a\sin t, a\cos t\rangle = a^2\), so the circulation is \(2\pi a^2\). The field always points along the circle, so every bit of the path contributes positively.

Some Important Definitions

: Simple curve. The image of a piecewise \(C^1\) map \(\mathbf r: I \to \mathbb {R}^3\) that is one-to-one on the interval \(I\): a curve that does not cross itself.
: Closed curve. A curve whose terminal point coincides with its initial point, \(\mathbf r(a) = \mathbf r(b)\).
: Simple closed curve. The image of a piecewise \(C^1\) map \(\mathbf r: [a,b] \to \mathbb {R}^3\) that is one-to-one on \([a,b)\) and satisfies \(\mathbf r(a) = \mathbf r(b)\).
: Open region. A region \(D\) is open if every point \(P\) of \(D\) is the centre of a disc lying entirely in \(D\).
: Connected region. A region \(D\) is connected if any two points of \(D\) can be joined by a path that lies in \(D\).
: Simply connected region. A plane region \(D\) is simply connected if it is connected and every simple closed curve in \(D\) encloses only points of \(D\): the region has no holes. In space, the condition is that every closed curve in \(D\) can be shrunk to a point without leaving \(D\).

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