4.6 Line Integrals Of Vector Fields
The work done by the vector force field \(F\) in moving a particle along a curve \(C\) given by \[W = \int _C F(x,y,z)\cdot T(x,y,z)\,dS = \int _C F\cdot T\,dS\hspace {1.5cm}*\] where \(T(x,y,z)\) is the unit tangent vector on the curve at the point \((x,y,z)\). In other words, work is the line integral with respect to arc length of the tangential component of the force \(F\). If the curve \(C\) is given by \[\overline {r}(t) = x(t)\textbf {i} + y(t)\textbf {j} + z(t)\textbf {k},\] then \(\displaystyle {T(t) = \frac {\overline {r}'(t)}{ \left |\overline {r}'(t)\\\right | }}\) so then equation \(*\) can be written as \begin {align*} W & = \int ^b_a \Bigg [F\big (\overline {r}(t)\big ) \cdot \frac {\overline {r}'(t)}{\left |\overline {r}'(t)\\\right | }\Bigg ]\left |\overline {r}'(t)\\\right |\,dt\\\\ & = \int ^b_aF\big [\overline {r}(t)\big ]\cdot \overline {r}'(t)\,df\\ \end {align*}
Let \(F\) be continuous vector field defined on a smooth curve \(C\) given by a vector function \(\overline {r}(t),\hspace {0.4cm} a\leq t \leq b\). Then the line integral of \(F\) along \(C\) is \[\int _C F\cdot d \overline {r} = \int ^b_a F\big [\overline {r}(t)\big ]\cdot \overline {r}'(t)\, dt = \int _C F\cdot T\,dS\]
This is called the flow integral if \(F\) is the velocity field of a fluid flowing through a region in space. If \(C\) is closed, it is called the circulation around the curve.
Note. Although \(\displaystyle {\int _C F\cdot d \overline {r} = \int _C F\cdot T\, dS}\) line integral with respect to arclength, it is still true that \(\displaystyle {\int _{-C}F\cdot d\overline {r} = -\int _C F\cdot d\overline {r}}\) because the unit tangent vector is \(T\) is replaced by its negative when \(C\) is replaced by \(-C\).
- 1.
- Find the work done by the force
\[F(x,y,z) = \big (x^2 -y\big )\textbf {i} + \big (y^2 - z\big )\textbf {j} + \big (z^2-x\big )\textbf {k}\]
along the curve from \((0,0,0)\) to \((1,1,1)\) given by
- (a)
- \(\overline {r}(t) = t\textbf {i} + t \textbf {j} + t\textbf {k}\)
- (b)
- \(\overline {r}(t) = t\textbf {i} + t^2\textbf {j} + t^3\textbf {k}\hspace {1cm} 0\leq t \leq 1\)
Solution.
\[W = \int _C F\big [\overline {r}(t)\big ]\cdot \overline {r}'(t)\, dt\]
- (a).
- \(\overline {r}(t) = t\textbf {i} + t\textbf {j} + t\textbf {k}\hspace {0.2cm} , \hspace {0.5cm} \overline {r}'(t) = \textbf {i} + \textbf {j} + \textbf {k}\)
\[F\big [\overline {r}(t)\big ] =\big (t^2 - t\big ) \textbf {i} + \big (t^2 - t\big ) \textbf {j} + \big (t^2 - t\big ) \textbf {k}\]
\begin {align*} \implies \hspace {1cm} F\big [\overline {r}(t)\big ]\cdot \overline {r}'(t) & = t^2 - t + t^2 - t + t^2 - t\\ & = 3t^2 - 3t\\ \end {align*}
\[W = \int ^1_0 \big (3t^2 - 3t\big ) \,dt = \frac {-1}{2}\]
\[d\overline {r} = x'(t)\,dt\,\textbf {i} + y'(t)\,dt\, \textbf {j} + z'(t)\,dt\,\textbf {k}\]
- (b).
- \(\overline {r}(t) = t\textbf {i} + t^2 \textbf {j} + t^3\textbf {k}\hspace {0.2cm}, \hspace {0.5cm} 0\leq t \leq 1\)
\[W = \int _C F\cdot d\overline {r}\]
\[x(t) = t \hspace {0.3cm} , \hspace {1cm} y(t) = t^2\hspace {0.3cm} ,\hspace {1cm} z(t) = t^3\]
\[x'(t) = 1 \hspace {0.3cm} , \hspace {1cm} y'(t) = 2t\hspace {0.3cm} ,\hspace {1cm} z'(t) = 3t^2\]
\[d\overline {r} = x'(t)\,dt\,\textbf {i} + y'(t)\,dt\, \textbf {j} + z'(t)\,dt\,\textbf {k}\]
\begin {align*} F\big [\overline {r}(t)\big ] & = \big (t^2 -t^2\big )\textbf {i} + \big (t^4 - t^3\big )\textbf {j} + \big (t^6 -t\big )\textbf {k}\\\\ & = \big (t^4-t^3\big )\textbf {j} + \big (t^6 - t\big )\textbf {k} \end {align*}
\[F\cdot d\overline {r} = 0 + 2t\big (t^4-t^3\big ) dt + 3t^2\big (t^6-t\big )dt\]
\[W = \int ^1_0 \big ( 2t^5 - 2t^4 + 3t^8 - 3t^3\big ) \,dt = \dfrac {-29}{60}\]
Some Important Definitions
- Simple Curve: This is the image of a piecewise \(C'\) map \(\overline {r}: I \longrightarrow \mathbb {R}^3\) that is one\(-\) to \(-\) one on an interval \(I\).
- Closed Curve: A curve is called closed if its terminal point coincides with its initial point i.e \(\overline {r}(a) = \overline {r}(b)\).
- Simple Closed Curve: This is the image of a piecewise \(C'\) map \(\overline {r}: [a,b] \longrightarrow \mathbb {R}^3\) that is one \(-\) to \(-\) one on \([a,b]\) and satisfies \(\overline {r}(a) = \overline {r}(b).\)
- Open Closed: A region \(D\) is called open if for every point \(P\) in \(D\) there is a disc with centre \(P\) that lies entirely in \(D\).
- Connected Region: A region \(D\) is called connected if any two point in \(D\) can be joined by a path that lies in \(D\).
- Simply Connected Region: A region \(D\) is simply connected if it is connected and every simple closed curve in \(D\), encloses only points that are in \(D\). We can say that such a region has no hole.
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