A bivector extended by a third vector, (a [and] b) [and] c, is a directed volume element called a trivector.

For example, for a bivector B = a ^ b, the rotation, B', can be expressed as

We take [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be a bivector space of bifunctions on [H.

2] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the inverse mapping of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the bivector space of K-valued bifunctions on [E.

A set of tensor fields located in an n-dimensional Riemannian space is known as a

bivector set, and its representation at a point is known as a local

bivector set.

Notice that in [7]and[8] various bispaces, such as bigroup, bisemigroup, biquasigroup, biloop, bigroupoid, biring, bisemiring,

bivector, bisemivector, binear-ring, .

Starting from generic bilinear Hamiltonians, constructed by covariant vector,

bivector or tensor fields, it is possible to derive a general symplectic structure which leads to holonomic and anholonomic formulations of Hamilton equations of motion directly related to a hydrodynamic picture.

We identify the time-like

bivector formed by the basis vectors of the Euclidean plan with the time-like basis vector of the Minkowskian plane and identify the space-like

bivector formed by the basis vectors of the Minkowskian plane with one of the space-like basis vectors of the Euclidean plane.

Before doing that we shall demonstrate that the derivative with respect to the

bivector coordinate [x.

Their outer product R(4) describes Minkowskian space-time, whose inner products form

bivectors, pseudo-vectors, and pseudo-scalars, describing spin, momentum-energy, and action, respectively, which automatically satisfy the Heisenberg Uncertainty Principle.

n]-tuples is split into subspaces corresponding to scalars (0-vectors), vectors (1-vectors),

bivectors (2-vectors), and so on.

The products of Dirac matrices form a graded Clifford algebra with one scalar, n vectors, n(n-1)/2

bivectors, and so on, up to n convectors (products of n-1 vectors), and one coscalar (the product of all n vectors).