Here I’d like to explore some ideas from  where curvilinear coordinates, manifolds, and the vector derivative are introduced.
For simplicity, let’s consider the concrete example of a 2D manifold, a surface in an dimensional vector space, parameterized by two variables
Note that the indices here do not represent exponentiation. We can construct a basis for the manifold as
On the manifold we can calculate a reciprocal basis , defined by requiring, at each point on the surface
Associated implicitly with this basis is a curvilinear coordinate representation defined by the projection operation
(sums over mixed indexes are implied). These coordinates can be calculated by taking dot products with the reciprocal frame vectors
Let’s pause for a couple examples that have interesting aspects.
Example: Circular coordinates on a disk
Consider an infinite disk at height , with the origin omitted, parameterized by circular coordinates as in fig. 1.1.
Points on this surface are
The manifold basis vectors, defined by eq. 1.2.2 are
By inspection, the reciprocal basis is
The first thing to note here is that we cannot reach the points of eq. 1.3.6 by linear combination of these basis vectors. Instead these basis vectors only allow us to reach other points on the surface, when already there. For example we cannot actually write
unless . This is why eq. 1.2.4 was described as a projective operation (and probably deserves an alternate notation). To recover the original parameterized form of the position vector on the surface, we require
The coordinates follow by taking dot products
Therefore, a point on the plane, relative to the origin of the plane, in this case, requires just one of the tangent plane basis vectors
Example: Circumference of a circle
Now consider a circular perimeter, as illustrated in fig. 1.2, with the single variable parameterization
Our tangent space basis is
with, by inspection, a reciprocal basis
Here we have a curious condition, since the tangent space basis vector is perpendicular to the position vector for the points on the circular surface. So, should we attempt to calculate coordinates using eq. 1.2.4, we just get zero
It’s perhaps notable that a coordinate representation using the tangent space basis is possible, but we need to utilize a complex geometry. Assuming
and writing for the pseudoscalar, we can write
so that, by inversion, the coordinate is
Example: Surface of a sphere
It is also clear that any parameterization that has radial symmetry will suffer the same issue. For example, for a radial surface in 3D with radius we have
The reciprocals here were computed using the mathematica reciprocalFrameSphericalSurface.nb notebook.
Do we have a bivector parameterization of the surface using the tangent space basis? Let’s try
Wedging with and , and writing , respectively yields
However, substitution back into eq. 1.0.22 shows either pair parameterizes the radial position vector
It is interesting that duality relationships seem to naturally arise attempting to describe points on a surface using the tangent space basis for that surface.
 A. Macdonald. Vector and Geometric Calculus. CreateSpace Independent Publishing Platform, 2012.