We now study integrals along curves of vector valued functions. If \(\vect f\) is a function defined on \(D\subset\mathbb R^N\) to \(\mathbb R^N\) we can “attach” the image \(\vect f(\vect x)\) of every point \(\vect x\in D\) to \(\vect x\text{.}\) As a result we get what we call a vector field. There are many physical interpretations of vector fields. For instance \(\vect f(\vect x)\) is the velocity of a fluid at the point \(\vect x\text{,}\) or the force acting on a particle. In the latter case we talk about a `force field.' Examples of force fields are the electric, magnetic or gravitational fields. As an example of a vector field consider \(\vect f(x,y):=(xy,x^2-y^2)\) shown in Figure 7.13.
Figure7.13.The vector field \(\vect f(x,y):=(xy,x^2-y^2)\) in \(\mathbb R^2\text{.}\)
To motivate the definition of line integrals we consider \(\vect f\) to be a force field. We want to estimate the work done moving along the curve from one end to the other. Quite obviously it matters in which direction we move. Hence we look at a positively oriented curve as defined at the end of Section 7.1, and assume that a particle travels in the positive direction from one end of the curve to the other. Recall that in physics work is defined by the force times the distance travelled. Only the component in the direction travelled counts. To estimate the work done moving along \(C\) we partition \(C\) by \(n\) points \(P_0\) to \(P_n\text{.}\) Denote the distance along \(C\) from \(P_{i-1}\) to \(P_i\) by \(\Delta s_i\text{.}\) If \(\vect f\) is a continuous vector field, then \(\vect f(P)\) is almost constant between \(P_{i-1}\) and \(P_i\) provided \(\Delta s_i\) is sufficiently small. Moreover, the segment of the curve between \(P_{i-1}\) and \(P_i\) is almost on the tangent to \(C\) at \(P_{i-1}\text{.}\) The situation is depicted in Figure 7.14.
Figure7.14.Work done moving along a small segment of the curve \(C\text{.}\)
The component of \(\vect f(P_{i-1})\) into the direction of the tangent is given by
where \(\vect\tau(P_{i-1})\) denotes the unit tangent vector at \(P_{i-1}\) to \(C\) as shown in Figure 7.14 we can rewrite this component as a scalar product \(\vect f(P_{i-1})\cdot\vect\tau(P_{i-1})\) (see Remark 1.10). Hence the work done travelling from \(P_{i-1}\) to \(P_i\) is approximately \(\vect f(P_{i-1})\cdot\vect\tau(P_{i-1})\Delta s_i\text{.}\) The total work moving along the curve is therefore close to
The above sum looks like a Riemann sum for the scalar valued function \(\vect f\cdot\vect\tau\) along the curve \(C\text{.}\) Using the theory from the previous section the limit of the sum as \(\max\Delta s_i\) tends to zero is \(\int_C\vect f\cdot\vect\tau\,ds\text{.}\) If \(\vect\gamma(t)\text{,}\)\(t\in [a,b]\text{,}\) is a regular parametrisation consistent with the orientation of \(C\) then the positive unit tangent vector \(\vect\tau(t)\) at \(\vect\gamma(t)\) is given by (7.1). Hence applying Definition 7.10 to \(\vect f\cdot\vect\tau\) we get
Setting \(x_i=\gamma_i(t)\) we can formally write \(dx_i=\gamma_i'(t)dt\text{.}\) Hence, if \(\vect x=(x_1,\dots,x_N)\) and \(\vect\gamma =(\gamma_1,\dots,\gamma_N)\) as usual, we see that formally
The last expression is obtained by formally taking the scalar product of \(\vect f(\vect x)\) with \(d\vect x:=(dx_1,\dots,dx_N)\text{.}\) The above motivates the following definition.
Definition7.15.Line integral of vector fields.
Let \(C\) be a smooth positively oriented curve, and \(\vect\gamma(t)\text{,}\)\(t\in[a,b]\text{,}\) a regular parametrisation of \(C\) consistent with the orientation of \(C\text{.}\) Then for every continuous vector field \(\vect f\) on \(C\) we define
\begin{align*}
\int_C\vect f\cdot\,d\vect x
\amp:=\int_C\vect f\cdot\vect\tau\,ds\\
\amp=\int_a^b\vect f(\vect\gamma(t))\cdot\vect\gamma'(t)\,dt
\end{align*}
The integral over a piecewise smooth curve is defined to be the sum over the integrals over the smooth parts of the curve.
As we integrate the tangential component along the curve we also talk about the circulation of \(\vect f\) along the curve \(C\).
Insight7.16.Physical interpretation.
If \(\vect f\) is a force field, then, as shown before, \(\int_C\vect f\cdot\,d\vect x\) is the work done by a particle moving along \(C\text{.}\) If \(\vect f\) describes the motion of a medium (fluid or gas) then the same arguments as above show that \(\int_C\vect f\cdot\,d\vect x\) is the net flow along the curve \(C\text{.}\) If \(C\) is a closed curve we talk about the circulation of the field along \(C\text{.}\)
In the special case of a function of three variables, \(x,y,z\) and a vector field \(\vect f=(P,Q,R)\) the above reads
if \((x(t),y(t),z(t))\text{,}\)\(t\in[a,b]\text{,}\) is a parametrisation of \(C\) consistent with the orientation of \(C\text{.}\)
Remark7.17.
We saw that it matters how the curve is oriented. As the opposite orientation of a curve \(C\) is given by reversing the sign of the unit tangent vectors we have that
\begin{equation*}
\int_{-C}\vect f\cdot\,d\vect x
=-\int_C\vect f\cdot\,d\vect x\text{.}
\end{equation*}
We compute the integral over the smooth parts, \(C_1\) and \(C_2\text{,}\) of \(C\) separately. We first look at \(C_1\text{.}\) The function \(\vect\gamma(t):=(t,t)\text{,}\)\(t\in[-1,1]\) is a parametrisation consistent with the orientation of \(C_1\text{.}\) Hence \(\vect\gamma'(t)=(1,1)\) and therefore
Next we compute the integral over \(C_2\text{.}\) A regular parametrisation, consistent with the orientation, is given by \(\vect\gamma(t):=\sqrt{2}(\cos t,\sin t)\text{,}\)\(t\in[\pi/4,5\pi/4]\text{.}\) We have \(\vect\gamma'(t)=\sqrt{2}(-\sin t,\cos t)\text{,}\) and so
