Introduction
Multivector calculus is of interest for modelling fluidic flow, gravitational fields, and so forth. The intention here is to provide a quick inroad into the subject rather than a full and formal presentation. For a rigourous approach, see Hestenes and Sobcyk.
This document is still under revision. All suggestions, critique, or comment gratefully received.
This document assumes familiarity with Multivectors. Notations defined in that document are retained here. Note that we here use labels e1,e2,... to denote a typically fixed, "base", "universal", "fiducial" frame and hip to denote tangent vectors. In much of the literature, ei represent tangent or otherwise "motile" vectors while si or gi represent a "base frame" .
This document makes extensive use of subscripts and superscripts to indicate dependencies usually "dropped" in conventional treatments and is, in consequnce, theoretically ambiguous. Does vip , for example, mean that vi is defined over or dependant on p , or that v is a function of ip? In practice, meanings will be clear in context.
Tensors are traditionally a difficult concept but multivectors make them far easier to understand, manipulate, and generalise. They are fundamental to many applications so we address them here.
Notations
Symbols such as J., J.,J., J., and J. are used variously in the literature for various "differentiating" operators. We will introduce the unorthodox notation Ðxa for the "directed" derivative with regard to a multivector parameter x in a particular multivector "direction" a, and use Ñ to denote the un-directed ("splayed") derivatives traditionally denoted J. or J. . We will typically use d or d to denote a small scalar and d to denote a 1-vector interpreted as a (possibly large) "displacement". We will sometimes use dx to denote a a small change in multi-vector parameter x when ambiguity with multiplication by a scalar d cannot arise.
