  In 1960-62, E. K\"ahler developed what looks as a generalization of the
exterior calculus, which he based on Clifford rather than exterior algebra. The
role of\ the exterior derivative, $du$, was taken by the more comprehensive
derivative $\partial u$ ($\equiv dx^{\mu }\vee d_{\mu }u$), where `$\vee $'
stands for Clifford product. The $d_{\mu }u$ represents a set of quantities to
which he referred as covariant derivative, and for which he gave a long, ad hoc
expression. We provide the geometric foundation for this derivative, based on
Cartan's treatment of the structure of a Riemannian differentiable manifold
without resort to the concept of the so called affine connections. Buried at
advanced points in his presentations is the implied statement that $\partial
u=du+\ast ^{-1}d$ $u\ast $, the sign at the front of the coderivative term is a
matter of whether we include the unit imaginary or not in the definition of
Hodge dual, $\ast $. We extract and put together the pieces of theory that go
into his derivation of that statement, which seems to have gone unnoticed in
spite of its relevance for a quick understanding of what his `K\"ahler
calculus'. K\"ahler produced a most transparent, compelling and clear
formulation of relativistic quantum mechanics (RQM) as a virtual concomitant of
his calculus. We shall enumerate several of its notable features, which he
failed to emphasize. The exterior calculus in K\"ahler format thus reveals
itself as the computational tool for RQM, making the Dirac calculus unnecessary
and its difficulties spurious.
