Video 61 - Sample Metric Objects

In this video, the presenter begins by introducing the concept of deriving basic metric objects for sample surfaces. The focus is on the cylindrical surface, where the surface coordinates are Theta and z, with Theta representing the angle from the x-axis and z representing the vertical distance. The Cartesian coordinates x, y, and z are used as ambient coordinates. The parameter equations are given as functions of Theta and z. The first step is to derive the shift tensor, which consists of partial derivatives of the ambient coordinates with respect to the surface coordinates. The presenter then explains how to derive the surface covariant basis vectors by contracting the shift tensor with the covariant basis vectors in the ambient space.

The presenter continues with the derivation process of the shift tensor for a cylindrical surface, providing the specific partial derivatives for x, y, and z with respect to Theta and z. The resulting shift tensor is displayed, and the process to derive the surface covariant basis vectors is explained using the formula involving the contraction of the shift tensor with the ambient covariant basis vectors. The presenter then moves on to calculate the surface covariant metric tensor by taking the dot products of the covariant basis vectors, resulting in a diagonal matrix with specific terms. The magnitudes of the vectors are determined by the square roots of the diagonal elements, and the volume element is found by taking the square root of the determinant of this tensor.

The presenter explains the inversion of the covariant metric tensor to obtain the contravariant metric tensor, which is simplified due to the diagonal nature of the matrix. The contravariant basis vectors are then calculated by contracting the contravariant metric tensor with the covariant basis vectors. The magnitudes of these vectors are found using the square roots of the diagonal elements. The presenter also discusses finding the inverse of the shift tensor using a formula derived in a previous video, which involves contracting the shift tensor with the covariant and contravariant metric tensors. The resulting inverse shift tensor is shown to have elements that match the components of the contravariant basis vectors.

The video script shifts focus to the spherical surface, where surface coordinates are Theta and Phi, with Theta being the angle from the Z-axis and Phi being the angle with the x-axis. The Cartesian coordinates x, y, and z are again used as ambient coordinates. The presenter outlines the process of deriving the shift tensor for the spherical surface by taking partial derivatives with respect to Theta and Phi. The covariant basis vectors are then derived from the components of the shift tensor. The covariant metric tensor is calculated using dot products, resulting in a diagonal matrix. The volume element is determined by the square root of the determinant of this matrix, and the covariant metric tensor is inverted to obtain the contravariant metric tensor. The contravariant basis vectors are then found by contracting the contravariant metric tensor with the covariant basis vectors.

The presenter concludes the video script with an analysis of the torus surface, where surface coordinates are again Theta and Phi, but with different interpretations related to the torus's geometry. The shift tensor for the torus is derived by taking partial derivatives of the parametric relationships with respect to Theta and Phi. The covariant basis vectors are directly obtained from the components of the shift tensor. The covariant metric tensor is formed by calculating the dot products of the basis vectors, resulting in a diagonal matrix. The volume element is found by taking the square root of the determinant of this matrix. The covariant metric tensor is inverted to get the contravariant metric tensor, and the contravariant basis vectors are derived by contracting the contravariant metric tensor with the covariant basis vectors. The magnitudes of these vectors are calculated, and the inverse shift tensor is trivially obtained from the components of the contravariant basis vectors.