Video 94 - Sample Embedded Curve

This paragraph introduces the topic of tensor calculus, specifically focusing on the example of a circle embedded in a spherical surface. The circle is positioned at a certain angle theta from the north pole, and the coordinate value for the curve is given by phi, which varies from 0 to 2 pi. The ambient space's coordinates are s1 and s2, which are the spherical surface's coordinates. The task is to find a parametrization of these coordinates as a function of phi, leading to the discovery of the shift tensor, which is essential for further calculations in the video.

The paragraph delves into the calculation of the shift tensor and covariant basis vector for the curve embedded in the spherical surface. The shift tensor is found by taking partial derivatives of the ambient coordinates with respect to the curve coordinates, resulting in s11 being 0 and s21 being 1. The covariant basis vector is then derived using the relationship between the shift tensor and the surface's covariant basis vector, which is obtained from previous analysis in video 61.

The focus of this paragraph is on finding the covariant metric tensor for the curve, which involves calculating the dot product of the covariant basis vector with itself. The result is a single component tensor, which also represents the determinant and volume element of the curve. The contravariant metric tensor is then derived as the reciprocal of the covariant metric tensor, and the contravariant basis vector is calculated using a specific relationship involving the covariant basis vector.

The paragraph discusses the calculation of the inverse shift tensor, which involves expanding and simplifying expressions using the known values from the shift tensor and the surface's covariant metric tensor. The Christoffel symbols are also explored, with the conclusion that they are zero for the given curve, simplifying the calculation of the vector curvature normal.

This paragraph explains the process of finding the vector curvature normal by taking the partial derivative of the covariant basis vector with respect to the curve coordinate. The resulting vector is then used to calculate the curvature magnitude, which is shown to be the reciprocal of the curve's radius of curvature. The principal normal is derived by dividing the vector curvature normal by its magnitude, confirming its unit vector nature.

The paragraph covers the derivation of the curve normal using the Levi-Civita symbols and the volume elements for both the surface and the curve. The components of the curve normal are calculated, and the vector itself is expressed in terms of the surface's covariant basis vectors. The surface projection is then explored, involving the calculation of the covariant derivative of the shift tensor and the Christoffel symbols, leading to the geodesic curvature tensor.

The final paragraph wraps up the video with a demonstration using graphing software to visually represent the concepts discussed in the previous paragraphs. It shows the decomposition of the vector curvature normal into surface and normal projections, and the significance of these components in understanding the curvature of the curve on the spherical surface. The special case of a geodesic curve, which has no surface projection and only a normal component, is highlighted, and the video concludes with a promise of further analysis of geodesic curves in the next installment.