I am reading Roger Penrose's road to reality. On page 209 ,210 he introduces the idea of Clifford algebra and how we can get back quarternion algebra by consideration of the 'second order' quantities in it. Right after that, something called Grassmann algebra is introduced in the following way:
Finally, let me turn to Grassmann algebra. From the point of view of the above discussion, we may think of Grassmann algebra as a kind of degenrate case of Clifford algebra, where we have basic anticommuting generating elements $\eta_1, \eta_2, \eta_3...,\eta_n$ similar to $ \gamma_1, \gamma_2...,\gamma_n$ of the Clifford algebra, where each $\eta_s$ squares to zero, rather than to the $-1$ we have in the Clifford case
The anticommutation law $\eta_p \eta_q = - \eta_q \eta_p$ holds as before , except that the Grassmann algebra is now more 'systematic' than the Clifford algebra, because we do not have to specify '$p \neq q$ in this equation.
What are the practical use of such Grassman algebra? At best, I am looking for illustrations where the above algebra laws can be used to solve problems in geometry or physics that the Gibbs school of mathematics (e.g., determinants) cannot or that the Gibbs school solves significantly more contortedly/incompletely.
In comment, some user had explained that it extends cross products and determinants. But, doesn't determinant itself already capture everything you need to capture with the determinant? What is use of another way of describing the same thing?
