It really is useful even at the advanced level. Following it far enough leads to the Atiyah-Singer Index Theorem and Hodge Theory. The advantage over the exteri

It really is useful even at the advanced level. Following it far enough leads to the Atiyah-Singer Index Theorem and Hodge Theory. The advantage over the exterior algebra is that you have that and an interior algebra, which leads to many formulas in differential geometry becoming very natural (like Cartan's magic formula).