Transvector Algebra and Analysis Operate at Singularities

We show that transcomplex numbers can be modelled by real, homogeneous co-ordinates with arbitrary Cartesian elements and a non-negative homogeneous element. Hence we develop transvectors as tuples of real, homogeneous co-ordinates. We show how to map transvectors with strictly transreal components onto vectors that have only real components. Thus we obtain transvectors that are total over all transreal components but which are expressed with real components. This allows us to extend vector algebra and vector analysis so that they operate on division by zero. We state, but do not solve, Maxwell's differential equations in this form.