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:

$\mathbf{V} = \begin{pmatrix} \frac{\partial x_{1}}{\partial z_{1}} & \frac{\partial x_{1}}{\partial z_{2}} & \frac{\partial x_{1}}{\partial z_{3}} \\ & & \\ \frac{\partial x_{2}}{\partial z_{1}} & \frac{\partial x_{2}}{\partial z_{2}} & \frac{\partial x_{2}}{\partial z_{3}} \\ & & \\ \frac{\partial x_{3}}{\partial z_{1}} & \frac{\partial x_{3}}{\partial z_{2}} & \frac{\partial x_{3}}{\partial z_{3}} \end{pmatrix}$

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

$\mathbf{v} = \begin{pmatrix} \frac{\mathrm{d} x_{1}}{\mathrm{d} t} & \frac{\mathrm{d} x_{2}}{\mathrm{d} t} & \frac{\mathrm{d} x_{3}}{\mathrm{d} t} \end{pmatrix}^{\textup{T}}$

If z is replaced with its independent magnitude, x, then V → v, which is

$v=\frac{\mathrm{d}x }{\mathrm{d}t }$

The lenticity W is is defined by the matrix:

$\mathbf{W} = \begin{pmatrix} \frac{\partial z_{1}}{\partial x_{1}} & \frac{\partial z_{1}}{\partial x_{2}} & \frac{\partial z_{1}}{\partial z_{3}} \\ & & \\ \frac{\partial z_{2}}{\partial x_{1}} & \frac{\partial z_{2}}{\partial x_{2}} & \frac{\partial z_{2}}{\partial x_{3}} \\ & & \\ \frac{\partial z_{3}}{\partial x_{1}} & \frac{\partial z_{3}}{\partial x_{2}} & \frac{\partial z_{3}}{\partial x_{3}} \end{pmatrix}$

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

$\mathbf{w} = \begin{pmatrix} \frac{\mathrm{d} z_{1}}{\mathrm{d} s} & \frac{\mathrm{d} z_{2}}{\mathrm{d} s} & \frac{\mathrm{d} z_{3}}{\mathrm{d} s} \end{pmatrix}^{\textup{T}}$

If z is replaced by its independent magnitude, z, then W → w, which is

$w=\frac{\mathrm{d}z }{\mathrm{d}s }$

Velocity may be weighted with mass m for momentum:

$\mathbf{P}=m\mathbf{V},\, \mathbf{p}=m\mathbf{v},\,\textup{and} \, p=mv$

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

$\mathbf{Q}=n\mathbf{W},\, \mathbf{q}=n\mathbf{w},\,\textup{and} \, q=nw$

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

$\mathbf{A} = \begin{pmatrix} \frac{\partial v_{1}}{\partial t_{1}} & \frac{\partial v_{1}}{\partial t_{2}} & \frac{\partial v_{1}}{\partial t_{3}} \\ & & \\ \frac{\partial v_{2}}{\partial t_{1}} & \frac{\partial v_{2}}{\partial t_{2}} & \frac{\partial v_{2}}{\partial t_{3}} \\ & & \\ \frac{\partial v_{3}}{\partial t_{1}} & \frac{\partial v_{3}}{\partial t_{2}} & \frac{\partial v_{3}}{\partial t_{3}} \end{pmatrix}$

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

$\mathbf{a} = \begin{pmatrix} \frac{\mathrm{d} v_{1}}{\mathrm{d} t} & \frac{\mathrm{d} v_{2}}{\mathrm{d} t} & \frac{\mathrm{d} v_{3}}{\mathrm{d} t} \end{pmatrix}^{\textup{T}}$

Acceleration can be weighted with mass m for force:

$\mathbf{F}=m\mathbf{A},\, \mathbf{f}=m\mathbf{a},\,\textup{and} \, f=ma$

Relentation B is defined by the matrix

$\mathbf{B} = \begin{pmatrix} \frac{\partial w_{1}}{\partial s_{1}} & \frac{\partial w_{1}}{\partial s_{2}} & \frac{\partial w_{1}}{\partial s_{3}} \\ & & \\ \frac{\partial w_{2}}{\partial s_{1}} & \frac{\partial w_{2}}{\partial s_{2}} & \frac{\partial w_{2}}{\partial s_{3}} \\ & & \\ \frac{\partial w_{3}}{\partial s_{1}} & \frac{\partial w_{3}}{\partial s_{2}} & \frac{\partial w_{3}}{\partial s_{3}} \end{pmatrix}$

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

$\mathbf{b} = \begin{pmatrix} \frac{\mathrm{d} w_{1}}{\mathrm{d} s} & \frac{\mathrm{d} w_{2}}{\mathrm{d} s} & \frac{\mathrm{d} w_{3}}{\mathrm{d} s} \end{pmatrix}^{\textup{T}}$

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

$\mathbf{R}=n\mathbf{B},\, \mathbf{r}=n\mathbf{b},\,\textup{and} \, r=nb$

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