Video 12 - Jacobian Examples

This paragraph introduces the concept of the Jacobian matrix within the context of tensor calculus. The video script begins with an overview of the Jacobian, explaining its role in transforming between different coordinate systems. It uses affine coordinates as an example, illustrating how the Jacobian can be represented as a matrix with entries that correspond to partial derivatives. The script then demonstrates how to calculate the Jacobian for affine coordinates, showing the process of taking partial derivatives with respect to different variables and organizing them into a matrix format. The inverse Jacobian is also discussed, highlighting its importance in transforming coordinates back to their original system.

The second paragraph delves into the process of deriving covariant basis vectors and expressing affine coordinates in terms of Cartesian coordinates. It explains the covariant transformation, which involves the use of the Jacobian with a free index at the bottom, and how this transformation can be expanded to include summation over dummy indices. The script provides explicit formulas for deriving the basis vectors in the affine coordinate system and demonstrates how to apply these formulas to obtain the basis vectors as functions of Cartesian coordinates. The significance of the Jacobian, inverse Jacobian, and basis vectors being constant throughout the entire space in affine coordinates is also highlighted, contrasting this feature with curvilinear coordinates.

This paragraph focuses on calculating the Jacobian for plane polar and cylindrical polar coordinates. It begins with the Jacobian for plane polar coordinates, detailing the partial derivatives of x and y with respect to r and theta, and then moves on to the inverse Jacobian. The process involves taking partial derivatives of the radial and angular coordinates with respect to Cartesian coordinates and vice versa. The script then transitions to cylindrical polar coordinates, which introduces a third dimension. The Jacobian for cylindrical coordinates is shown to be an extension of the plane polar case, with additional terms for the z-component, which does not depend on the radial or angular coordinates.

The fourth paragraph discusses the Jacobian for spherical polar coordinates, which adds another layer of complexity due to the three-dimensional nature of the coordinate system. The script outlines the partial derivatives of x, y, and z with respect to the spherical coordinates r, theta, and phi. It then provides a brief overview of how to find the inverse Jacobian, although it acknowledges the tedious nature of the calculations and opts to present the results directly. The basis vectors in spherical polar coordinates are also derived, showing how each column of the Jacobian matrix can be multiplied by the respective Cartesian basis vectors to obtain the spherical basis vectors.

The final paragraph serves as a summary, aiming to reinforce the viewer's understanding and appreciation of the Jacobian and covariant transformations. It encapsulates the essence of the video series on tensor calculus, emphasizing the importance of the Jacobian in coordinate transformations and the process of deriving covariant basis vectors. The paragraph concludes with an encouragement for viewers to practice these concepts to solidify their grasp of tensor calculus.