Almost all mathematicians will teach curvilinear coordinates in the sense of: \begin{eqnarray} x&=r\cos\theta\\ y&=r\sin\theta \end{eqnarray} for polar coordinates, for example. However, almost no math courses address the curvilinear basis vectors such as $\rhat$ and $\that$ that are adapted to these coordinate systems. In fact, many mathematicians have not heard of these basis vectors. Therefore the following activity is really important. It's also quick and fun. Try it

At some stage, you will want to address the question of how to write the position vector using curvilinear basis vectors. This makes a great small whiteboard question. Unfortunately, understanding the answer is very difficult for many students. Therefore, it is best not to bring this up at the time at which you first introduce curvilinear basis vectors; let the students gain some experience first. The question will come up naturally when the students first need to add or subtract two position vectors at different points. (For us, the question typically comes up when the students are doing the activity to find the electric field due to a ring of charge where the numerator has a factor of $\rr-\rrp$.) We suggest that you wait to assign the section of the book on The Position Vector until after this question has arisen in class.

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