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/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.

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 is mentioned
• 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.