Run the following code to initialize $\LaTeX$ output and set up toroidal
coordinates,
either by clicking on "Evaluate" or by typing Shift+Enter.
Express toroidal frame vector $\hat{e}_i$ in terms of rectangular frame $\hat{u}_j$:
Express toroidal dual basis vector $q_i=\sigma_i$ in terms of rectangular coordinates.
Print the attitude matrix:
Print the dual basis in terms of toroidal coordinates:
(Here and below, use "Show" rather than "show" to expand in dual basis.)
Compute the connection 1-forms "w(i,j)" (using the attitude matrix):
Compute the connection 1-forms "om(i,j)" (using Sage; differs by sign):
Check the first structure equation:
Check the second structure equation:
Compute the exterior derivative of a function $f$:
Compute the exterior derivative of a 1-form $\eta$:
Type your own code here (or in any of the other boxes):