## Abstract

As noted in the previous chapter (Section II-2), the Ricci calculus makes no use of the cotangent space **T** _{P} ^{*} , nor does it identify the basis of covectors in this space. For Cartan this omission is the starting point of his theory, i.e. the basis of **T** _{P} ^{*} is taken to be {*dx*^{ r }} in the coordinate-based representation or more generally {*θ*_{ a }} where, as in I-(3.19), these Pfaffians are linear combinations of the *dx*^{ r }. Moreover, products of such elements are skew-symmetric and involve serious consideration of the exterior product space of **T** _{P} ^{*} with itself; this is a subspace of the tensor product spaces ⊗^{2}**T** _{P} ^{*} , or more generally of ⊗^{3}**T** _{P} ^{*} . Hence, whereas the material in Section I-3 on the abstract notion of a tensor is somewhat of a luxury for the Ricci calculus, it now plays a central role in the Cartan calculus. Consequently, any attempt to understand the Cartan theory must be rooted in terms of an abstract setting. This has seriously impeded the dissemination of it among non-mathematicians and in elementary mathematical texts the theory is usually presented in an unpalatable and abbreviated form which conceals as much as it reveals to the reader. Unfortunately, the same criticism can probably be said of the presentation given in this chapter.

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