Integrating Vectors II

Section 8.4 Integrating Vectors II

In curvilinear coordinates, however, care must be taken to remember that the basis vectors are not constant, and must also be integrated. This is not always easy!

One example can be obtained by comparing the expressions for \(\rr\) and \(d\rr\) in polar coordinates, namely

\begin{align*} \rr \amp = s \shat\\ d\rr \amp = ds \,\shat + s\,d\phi \,\phat \end{align*}

from which it follows, using the product rule, that

\begin{equation} d\shat = d\phi \,\phat\tag{8.4.1} \end{equation}

which in turn implies that

\begin{equation} \int_{\phi_1}^{\phi_2} \phat \,d\phi = \shat(\phi_2)-\shat(\phi_1) .\tag{8.4.2} \end{equation}

Even in this relatively simple case, care must be taken when evaluating the final integral, as it requires comparing the directions of \(\shat\) at two different points.

Integration is possible in this case because the integrand can be recognized as the derivative of something — it can be obtained by “zapping” another vector with \(d\text{.}\) Recall that this is also the case when integrating a conservative vector field along a curve, in which case the integrand can be obtained by “zapping” the potential function with \(d\text{.}\)

Except in such simple cases, however, it is rarely possible to integrate a vector field directly using curvilinear basis vectors. It is therefore usually necessary to convert to rectangular basis vectors before integrating, even if you choose to keep the components in curvilinear coordinates.