# 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:

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

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

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

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

$v=\frac{\mathrm{d}x&space;}{\mathrm{d}t&space;}$

The lenticity W is is defined by the matrix:

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

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

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

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

$w=\frac{\mathrm{d}z&space;}{\mathrm{d}s&space;}$

Velocity may be weighted with mass m for momentum:

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

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

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

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

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

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

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

Acceleration can be weighted with mass m for force:

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

Relentation B is defined by the matrix

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

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

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

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

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