Reciprocal Frame Vectors: Geometric algebra form

This is third video in a series about curvilinear coordinates and reciprocal frame vectors. This series will cover a subset of the material from the blog post http://peeterjoot.com/2022/04/05/curvilinear-coordinates-and-reciprocal-frames/ (or the accompanying pdf.)

This video series will make use geometric algebra concepts, including grade selection, bivectors, trivectors, and wedge products.

Previously,

• Curvilinear coordinates were introduced as tangent vectors for parameterized curves, surfaces, volumes and hypervolumes,
• Reciprocal frame vectors were introduced, and described geometrically,
• A brief mention was made about the use of reciprocal frame vectors for computing coordinates,
• The reciprocal vector application for lattice structure (solid state physics) was outlined, showing a non-algebraic use of this concept.
• We computed curvilinear coordinates for circular, elliptical and spherical parameterizations, showing in each case how to compute the area and volume elements,
• We computed the (pseudoscalar) areas of the circle, and ellipse, using these volume elements.
• We computed some relativistic curvilinear coordinates (Lorentz boost and scale, lightlike basis vectors, independent boost and rotate, ...), and their corresponding hypervolume elements.

In this video we

• review our definition for curvilinear bases associated with a parameterized position vector,
• review how to compute coordinates given an orthonormal bases, and discuss how that generalizes, first to mutually perpendicular (but not normal) unit vector bases, and then to arbitrary bases.
• We show how the concept of the reciprocal frame can be used to compute coordinates in the general case,
• then derive the form of the reciprocal frame for a 2D case,
• then show how to simplify that result and obtain the geometric algebra form for the 2D reciprocal frame vectors.
• Finally, we show how that GA form generalizes to higher dimensions.

We could have just started at the end result, and showed how that worked, but that is a little mean to start off with such an abstract quantity. Instead, here we show how the abstraction follows directly, as a result of simplification.

The remainder of the videos will cover

• Some examples, for both Euclidean vector spaces and STA (i.e. the Dirac basis: Space Time Algebra).
• A pure matrix representation for the reciprocal vectors. This will include the general case where the number of parameters is less than the dimension of the underlying vector space, and also the special case where the number of parameters equals the dimension of the space.
• The gradient representation of the reciprocal frame vectors. We will see that this representation has a simple matrix representation (the special case above.)