ABSTRACT
A geometric answer can be constructed by drawing rˆ and φˆ at two
nearby points, then comparing them. If the points are separated radially,
it is clear that these basis vectors do not change as you move from point
to point. Thus, along such a path,
drˆ = 0 = dφˆ. (17.1)
What if you move through an angle dφ along a circle? It is now readily
apparent that each basis vector is rotated through the same angle. Both
of these possibilities are shown in Figure 17.1. Furthermore, the difference
between two vectors of the same magnitude is a secant line on the circle
connecting their tips, as shown in Figure 17.2 for rˆ; a similar construction
holds for φˆ. As the two vectors approach each other, this secant line
becomes a tangent line and is therefore perpendicular to the original vector.
(An algebraic proof of this result is given in the discussion of (17.9) below.)
The magnitude of the difference between the two vectors is, in the limit,
just the length of the arc between them, which, for unit vectors, is given
by the angle. Thus, along such a path,
