Introduction to curvilinear coordinates and reciprocal frame vectors

This is the first 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/curv... (or the accompanying pdf.)

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

In this video,

Curvilinear coordinates are introduced as tangent vectors for parameterized curves, surfaces, volumes and hypervolumes
Reciprocal frame vectors are introduced, and described geometrically,
A brief mention about the use of reciprocal frame vectors for computing coordinates
The reciprocal vector application for lattice structure (solid state physics) is outlined, showing a non-algebraic use of this concept.
We compute curvilinear coordinates for circular, elliptical and spherical parameterizations, showing in each case how to compute the area and volume elements.
We compute the area of both the circle, and ellipse, using these volume elements.

In the next video, we will compute the curvilinear coordinates and area elements for a few non-Euclidean surface parameterizations.