Subsubsection 6.2.2.1 Line Integrals in the Plane with Respect to the Coordinates, \(x\) and \(y\)
In some situations, the quantity to be summed is \(f(x,y) \Delta x\) or \(f(x,y) \Delta y\text{:}\) for example if \(f\) is a force acting on an object in the \(x\) direction, \(f(x,y) \Delta x\) is the work done by that force when the object moves distance \(\Delta x\) in that direction (no work is associated with movement perpendicular to the force).
Limits of sums give the work done by such a force in the \(x\)-direction as an object moves along curve \(C\) to be
\begin{equation}
\int_C f(x,y) \, dx = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*,y_i^*) \Delta x_i\tag{6.2.8}
\end{equation}
called the line integral of \(f\) along \(C\) with respect to \(x\). Likewise the line integral of \(f\) along \(C\) with respect to \(y\) is
\begin{equation}
\int_C f(x,y) \, dy = \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*,y_i^*) \Delta y_i\tag{6.2.9}
\end{equation}
Much as with Equation
(6.2.5), these line integrals can be computed with
\begin{align}
\int_C f(x,y) \ dx
\amp= \int_{t=a}^b f(x(t),y(t)) \frac{dx}{dt} dt\tag{6.2.10}\\
\int_C f(x,y) \ dy
\amp= \int_{t=a}^b f(x(t),y(t)) \frac{dy}{dt} dt\tag{6.2.11}
\end{align}
However, the absence of the square root term can make these new integrals easier to work with. It also allows the increments
\(\Delta x\) and
\(\Delta y\) to be negative depending on the direction of motion along the curve (and so intuitively, also the differentials
\(dx\) and
\(dy\)), which leads to an important difference in the behavior of these integrals.
Subsubsection 6.2.2.2 Line Integrals in Space Coordinates
Line integrals with respect to each of the space coordinate variables \(x\text{,}\) \(y\) and \(z\) can be defined similarly, so that for example
\begin{equation}
\int_C f(x,y,z) dz
= \lim_{n \to \infty} \sum_{i=1}^n f(x_i^*,y_i^*,z_i^*) \Delta z_i
= \int_a^b f(x(t),y(t),z(t)) \frac{dz}{dt}\ dt\tag{6.2.12}
\end{equation}
Subsubsection 6.2.2.3 Paired (and Tripled) Line Integrals and Vector Line Integrals
One often gets a sum of line integrals in the plane in both \(x\) and \(y\text{,}\) and these can be abbreviated
\begin{equation}
\int_C P(x,y) \, dx + \int_C Q(x,y) \, dy = \int_C P(x,y) \, dx + Q(x,y) \, dy\tag{6.2.13}
\end{equation}
This combination has a concise vector form, called a Vector Line Integral:
\begin{align}
\int_C P(x,y) \, dx + Q(x,y)\ dy
\amp= \int_C \vector{P(x,y), Q(x,y)} \cdot \vector{dx,dy}\tag{6.2.14}\\
\amp= \int_C \vec{F} \cdot d\vec{r}\tag{6.2.15}
\end{align}
where \(\vec{F}= \vector{P(x,y), Q(x,y)}\) and for the position vector \(\vec{r} = \vector{x,y}\text{,}\) we use the short-hand
\begin{equation*}
d\vec{r} = d\vector{x,y} = \vector{dx,dy}\text{.}
\end{equation*}
When \(P\) and \(Q\) are the components in the \(x\) and \(y\) directions of a force vector \(\vec{F}\text{,}\) this gives the total work done by the force as the object traverses curve \(C\text{.}\)
Likewise, integrals of the form
\begin{equation}
\int_C P(x,y,z)\ dx + \int_C Q(x,y,z)\ dy + \int_C R(x,y,z)\ dz = \int_C P\ dx + Q\ dy + R\ dz\tag{6.2.16}
\end{equation}
arise—in particular for the example of the work done by a force—and they have the same vector form
(6.2.15), now with
\(\vec{F} = \vector{P,Q,R}\text{.}\)
For a more completely vector notation form, describe the path
\(C\) as
\(\vector{r}(t),\ a \le t \le b\) and express each line integral above in terms of this: The integral in Equation
(6.2.16) has the form
\begin{equation}
\int_C P\ dx + Q\ dy + R\ dz
= \int_a^b \left[ P(\vec{r}(t)) \frac{dx}{dt} + Q(\vec{r}(t))
\frac{dy}{dt} + R(\vec{r}(t)) \frac{dz}{dt} \right] dt
= \int_a^b \vec{F}(\vec{r}(t)) \cdot \frac{d\vec{r}}{dt}\ dt\tag{6.2.17}
\end{equation}
and in practice evaluation of \(\int_C \vec{F} \cdot d\vec{r}\) is done via this form, with the formal substutution \(\ds d\vec{r} = \frac{d\vec{r}}{dt}\ dt\text{:}\) this is sometimes called the line integral of \(\vec{F}\) along \(C\).
This integral \(\ds \int_C \vec{F} \cdot d\vec{r}\) can instead be defined rigorously by the familiar process of approximation by a sum of terms \(\vec{F}(\vec{r}(t_i^*)) \Delta \vec{r}_i\) and taking a suitable limit.
