\section{Algebra of Tensor}
Tensor units $\partial_t$ and $dt$ are reciprocal linear transformations
\[
\partial_{x(t)}=\partial_t\cdot\partial_{x} t,\quad dx(t)=dt\cdot\partial_tx
\]
and logically
\[
\partial=d,\quad \partial_t:=\frac{1}{\partial t}
\]
and traditionally it's written as below, inconsistently to the definitions herewith
\[
\partial_t f:=\partial f/\partial t
\]
This equation is transferred to the definition in Euclidean space
\[
\partial_{x^i}\cdot dx^j=1
\]
In Rieman space they are suitable to Lorentz transform,
\[
\partial_j^*\cdot dx^i=1
\]
\[
(t',ir')=(t,ir)\left[\begin{array}{cc}\cos(i\rho)& -\sin(i\rho)\\ \sin(i\rho)&\cos(i\rho)\end{array}\right].
\]
Space rotation causes no variance. Rieman measurement can be expressed in Euclidean space
\[
dx^i=g^{ij}\partial_j^*
\]
\[
dx^{i}\cdot dx^j=:g^{ij}
\]
It's noticed that $g^{ij}$ itself is a differential of Euclidean measure. \emph{There is an imaginary unit in the space coordinates.}
Hence
\[
p^j\partial_{x^j}^*\cdot P_{i}dx^{i}=p^iP_{i}
\]
and
\[
dx^i\cdot\partial_{x^i}
\]
are co-variant (or invariant).
\par
Harmonic wave phase and front are described by
\[
F(x^i(s^0)),\quad \partial_{s^0}:=\sum_i\partial_{x^i},\quad (x^i)=(t,il)
\]
\[
\partial s^0\frac{\partial F}{\partial s^0}=\partial x^i\frac{\partial F}{\partial x^i}
\]
The co-variance of the vectors are checked, for the movement of the wave front
\[
p^i=\partial x^i/\partial s^0=\partial_{s^0}/\partial_{x^i}
\]
and for the wave vector
\[
P_i=\partial s^0/ \partial x^i
\]
then
\[
p=\partial_i\cdot\partial x^i/\partial s^0,
\]
\[
P=dx^i\cdot\partial s^0/ \partial x^i
\]
One is co-variant, the other is contra-variant, in one dimension
\[
P=p,\quad mds^0=dF
\]
This result leads to the operators of momentum.
※ 修改:·supproton 于 Feb 24 21:34:19 2024 修改本文·[FROM: 117.155.182.*]
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