# Definitions with 3-dimensional time

In order to combine the three dimensions of length space and three dimensions of duration space in definitions for motion in six dimensions (3+3), it is necessary to use Jacobian matrices. The 3+1 and 1+3 dimensional definitions are simplifications of these.

Let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their magnitude without it.

Velocity = V, lenticity = W, displacement = x, distimement = z, independent distance = s, independent distime = t. The velocity V is defined by the matrix:

which is the Jacobian matrix of x(t) so that Vij = ∂xi/∂zj for i, j = 1, 2, 3. If z is replaced with its independent magnitude, t, then V becomes v, which is

If z is replaced with its independent magnitude, x, then Vv, which is

The lenticity W is is defined by the matrix:

which is the Jacobian matrix of t(s) so that Wij = ∂zi/∂xj for i, j = 1, 2, 3. If x is replaced by its independent magnitude, s, then W becomes w, which is

If z is replaced by its independent magnitude, z, then Ww, which is

Velocity may be weighted with mass m for momentum:

Lenticity may be weighted with vass n = 1/m for levamentum:

One may define other matrices similarly. Acceleration A is defined by the matrix:

which is the Jacobian matrix of v(t) so that Aij = ∂vi/∂tj for i, j = 1, 2, 3. If t is reduced to its magnitude, t, then A becomes a, which is

Acceleration can be weighted with mass m for force:

Relentation B is defined by the matrix

which is the Jacobian matrix of w(s) so that Bij = ∂wi/∂sj for i, j = 1, 2, 3. If s is reduced to its magnitude, s, then B becomes b, which is

Relentation can be weighted with vass n = 1/m for release: