text: Cartesian Coordinates
   Rene Descartes, 1596—1650.
  
  Let particle P be described by a chain of events
  
  $\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \; \ldots \; \sf{P}_{\it{k}} \; \ldots \; \right)$
  
  where each event is characterized by its displacement vector
  
  $d\bar{r} = \left( dx, dy, dz \right)$
  
  Definition: the abscissa of event $k$ is
  
  \begin{align} x_{k} \equiv x_{0} + \sum_{i=1}^{k} dx_{i} \end{align}
  
  where $x_{0}$ is arbitrary and often set to zero. The ordinate is
  
  \begin{align} y_{k} \equiv y_{0} + \sum_{i=1}^{k} dy _{i} \end{align}
  
  And the $z$-cooordinate or applicate of event $k$ is defined as
  
  \begin{align} z_{k} \equiv z_{0} + \sum_{i=1}^{k} dz_{i} \end{align}
  
  The $z$-component of the displacement $dz$ is a simple linear function of the wavelength. So if the particle is isolated then P moves in regular steps along the polar-axis as
  
  $z_{k} = z_{0} + k dz$
  
  The three numbers $x$, $y$ and $z$ are called the Cartesian coordinates of event $k$ after the work of More exactly, they are the rectangular Cartesian coordinates in a descriptive system that is centered on P. We use them to express the position of an event as
  
  $\overline{r} = ( x, y, z )$
  
  where $\overline{r}_{0} = ( x_{0}, y_{0}, z_{0} )$. WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So $N$, the number of quarks in a description, may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that $N$ is large enough to make an approximation to spatial continuity, then allowing the use of calculus.
  
  page revision: 200, last edited: 05 Nov 2019 12:38
explanation: The text extract provides a detailed explanation of Cartesian coordinates, specifically in the context of describing the position of events in a particle's chain. It defines the abscissa, ordinate, and z-coordinate (or applicate) of an event, and explains how these coordinates are used to express the position of an event. The text does not contain a specific question or a question and answer.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: Cartesian Coordinates
   Rene Descartes, 1596—1650.
  
  Let particle P be described by a chain of events
  
  $\Psi ^{\sf{P}} = \left( \sf{P}_{1}, \sf{P}_{2}, \sf{P}_{3} \; \ldots \; \sf{P}_{\it{k}} \; \ldots \; \right)$
  
  where each event is characterized by its displacement vector
  
  $d\bar{r} = \left( dx, dy, dz \right)$
  
  Definition: the abscissa of event $k$ is
  
  \begin{align} x_{k} \equiv x_{0} + \sum_{i=1}^{k} dx_{i} \end{align}
  
  where $x_{0}$ is arbitrary and often set to zero. The ordinate is
  
  \begin{align} y_{k} \equiv y_{0} + \sum_{i=1}^{k} dy _{i} \end{align}
  
  And the $z$-cooordinate or applicate of event $k$ is defined as
  
  \begin{align} z_{k} \equiv z_{0} + \sum_{i=1}^{k} dz_{i} \end{align}
  
  The $z$-component of the displacement $dz$ is a simple linear function of the wavelength. So if the particle is isolated then P moves in regular steps along the polar-axis as
  
  $z_{k} = z_{0} + k dz$
  
  The three numbers $x$, $y$ and $z$ are called the Cartesian coordinates of event $k$ after the work of More exactly, they are the rectangular Cartesian coordinates in a descriptive system that is centered on P. We use them to express the position of an event as
  
  $\overline{r} = ( x, y, z )$
  
  where $\overline{r}_{0} = ( x_{0}, y_{0}, z_{0} )$. WikiMechanics uses a finite categorical scheme of binary distinctions to describe sensation. So $N$, the number of quarks in a description, may be large but not infinite. In principle $N$ is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we may assume that $N$ is large enough to make an approximation to spatial continuity, then allowing the use of calculus.
  
  page revision: 200, last edited: 05 Nov 2019 12:38
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract provides a detailed explanation of Cartesian coordinates, specifically in the context of describing the position of events in a particle's chain. It defines the abscissa, ordinate, and z-coordinate (or applicate) of an event, and explains how these coordinates are used to express the position of an event. The text does not contain a specific question or a question and answer.
  
  Label: no-question