  This article provides a pedagogically oriented introduction to geometric
(Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to
the topic, which rely on embedding the geometric algebra either within a tensor
algebra or within a vector manifold framework, here we define geometric
calculus directly, by elementary methods. In particular we use an axiomatic
approach that directly parallels textbook introductions to general relativity
and pseudo-Riemannian geometry, so that no structure outside the metrical
Clifford bundle of the manifold need be introduced. On this basis we develop
the full theory of differential calculus for vector, multivector, and tensor
fields.
