I'm writing a text and I would like to insert a meaningful example on concrete manipulation and evaluation of differential forms. The idea of the text is being abstract in stating the theorems, but really concrete and explanatory in the examples.
I was thinking about starting from one or two differential forms, then do a wedge product and finally do an exterior derivative (maybe of the wedge product). But I'd like to do it in a meaningful case or starting from an interesting form. This could be an idea but in this case should be elegant or meaningful.
Another idea could be to calculate exterior derivative of a 1-form and show concretely Cartan's Formula. I don't know... could you list and suggest some meaningful, simple and selfcontained example involving direct manipulation of differential form that could be pedagogically helpful? Thank you in advance!
Using Cartan structural equations to calculate the Riemann curvature and connection coefficients of 2D manifolds embedded in 3 space. Comparison with the classical (i.e. Gaussian) methods of calculation is particularly enlightening as to the elegance and convenience of the algebra and operations.