# Transformations with 3-dimensional time

Following the previous post here, we use Jacobian matrices to transform location and chronation vectors between inertial observers.

As before, let matrices be written with upper case boldface. Let vectors be written in lowercase boldface and their scalar magnitude without it. Velocity = V, lenticity = W, displacement = x, distimement = z, independent distance = s, independent distime = t.  Components are in lowercase italics with subscripts.

The expanded Galilean transformation for the length frame is:

$\mathbf{x'}=\mathbf{x}-\mathbf{V}&space;\mathbf{t}=&space;\begin{pmatrix}&space;\mathrm{d}&space;x_{1}&space;\\&space;\\&space;\mathrm{d}&space;x_{2}&space;\\&space;\\&space;\mathrm{d}&space;x_{3}&space;\end{pmatrix}&space;-&space;\begin{pmatrix}&space;\frac{\partial&space;x_{1}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{1}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{1}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;x_{2}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{2}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{2}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;x_{3}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{3}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{3}}{\partial&space;t_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;t_{1}&space;\\&space;\\&space;\mathrm{d}&space;t_{2}&space;\\&space;\\&space;\mathrm{d}&space;t_{3}&space;\end{pmatrix}$

The expanded Galilean transformation for the length frame is:

$\mathbf{z'}=\mathbf{z}-\mathbf{W}&space;\mathbf{s}=&space;\begin{pmatrix}&space;\mathrm{d}&space;z_{1}&space;\\&space;\\&space;\mathrm{d}&space;z_{2}&space;\\&space;\\&space;\mathrm{d}&space;z_{3}&space;\end{pmatrix}-&space;\begin{pmatrix}&space;\frac{\partial&space;z_{1}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{1}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{1}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;z_{2}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{2}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{2}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;z_{3}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{3}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{3}}{\partial&space;s_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;s_{1}&space;\\&space;\\&space;\mathrm{d}&space;s_{2}&space;\\&space;\\&space;\mathrm{d}&space;s_{3}&space;\end{pmatrix}$

Motion can be considered a transformation. For example, free fall with acceleration A is:

$\mathbf{x'}=\mathbf{x}-\mathbf{A}&space;\mathbf{t}^2=&space;\begin{pmatrix}&space;\mathrm{d}&space;x_{1}&space;\\&space;\\&space;\mathrm{d}&space;x_{2}&space;\\&space;\\&space;\mathrm{d}&space;x_{3}&space;\end{pmatrix}-\begin{pmatrix}&space;\frac{\partial&space;v_{1}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;v_{1}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;v_{1}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;v_{2}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;v_{2}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;v_{2}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;v_{3}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;v_{3}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;v_{3}}{\partial&space;t_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;t_{1}&space;\\&space;\\&space;\mathrm{d}&space;t_{2}&space;\\&space;\\&space;\mathrm{d}&space;t_{3}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;t_{1}&space;\\&space;\\&space;\mathrm{d}&space;t_{2}&space;\\&space;\\&space;\mathrm{d}&space;t_{3}&space;\end{pmatrix}$

$=\begin{pmatrix}&space;\mathrm{d}&space;x_{1}&space;\\&space;\\&space;\mathrm{d}&space;x_{2}&space;\\&space;\\&space;\mathrm{d}&space;x_{3}&space;\end{pmatrix}&space;-&space;\begin{pmatrix}&space;\frac{\partial&space;x_{1}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{1}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{1}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;x_{2}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{2}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{2}}{\partial&space;t_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;x_{3}}{\partial&space;t_{1}}&space;&&space;\frac{\partial&space;x_{3}}{\partial&space;t_{2}}&space;&&space;\frac{\partial&space;x_{3}}{\partial&space;t_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;t_{1}&space;\\&space;\\&space;\mathrm{d}&space;t_{2}&space;\\&space;\\&space;\mathrm{d}&space;t_{3}&space;\end{pmatrix}$

where the matrix is multiplied by the distime vector twice. With relentation B it is:

$\mathbf{z'}=\mathbf{z}-\mathbf{B}&space;\mathbf{s}^2=&space;\begin{pmatrix}&space;\mathrm{d}&space;z_{1}&space;\\&space;\\&space;\mathrm{d}&space;z_{2}&space;\\&space;\\&space;\mathrm{d}&space;z_{3}&space;\end{pmatrix}-\begin{pmatrix}&space;\frac{\partial&space;w_{1}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;w_{1}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;w_{1}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;w_{2}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;w_{2}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;w_{2}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;w_{3}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;w_{3}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;w_{3}}{\partial&space;s_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;s_{1}&space;\\&space;\\&space;\mathrm{d}&space;s_{2}&space;\\&space;\\&space;\mathrm{d}&space;s_{3}&space;\end{pmatrix}\begin{pmatrix}&space;\mathrm{d}&space;s_{1}&space;\\&space;\\&space;\mathrm{d}&space;s_{2}&space;\\&space;\\&space;\mathrm{d}&space;s_{3}&space;\end{pmatrix}$

$=\begin{pmatrix}&space;\mathrm{d}&space;z_{1}&space;\\&space;\\&space;\mathrm{d}&space;z_{2}&space;\\&space;\\&space;\mathrm{d}&space;z_{3}&space;\end{pmatrix}-\begin{pmatrix}&space;\frac{\partial&space;z_{1}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{1}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{1}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;z_{2}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{2}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{2}}{\partial&space;s_{3}}&space;\\&space;&&space;&&space;\\&space;\frac{\partial&space;z_{3}}{\partial&space;s_{1}}&space;&&space;\frac{\partial&space;z_{3}}{\partial&space;s_{2}}&space;&&space;\frac{\partial&space;z_{3}}{\partial&space;s_{3}}&space;\end{pmatrix}&space;\begin{pmatrix}&space;\mathrm{d}&space;s_{1}&space;\\&space;\\&space;\mathrm{d}&space;s_{2}&space;\\&space;\\&space;\mathrm{d}&space;s_{3}&space;\end{pmatrix}$

where the matrix is multiplied by the distance vector twice.