Cartesian

Examples of Cartesian in the following topics:

• The Cartesian System

  • The Cartesian coordinate system is used to specify points on a graph by showing their absolute distances from two axes.
  • A Cartesian coordinate system specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances from the point to the two axes.
  • The usefulness of Cartesian coordinates can be shown with a series of points showing a relationship.  
  • The four quadrants of a Cartesian coordinate system.
  • The Cartesian coordinate system with 4 points plotted, including the origin $(0,0)$.

• Converting Between Polar and Cartesian Coordinates

  • Polar and Cartesian coordinates can be interconverted using the Pythagorean Theorem and trigonometry.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • Derive and use the formulae for converting between Polar and Cartesian coordinates

• Three-Dimensional Coordinate Systems

  • Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters $x$, $y$, and $z$.
  • Often, you will need to be able to convert from spherical to Cartesian, or the other way around.
  • This is a three dimensional space represented by a Cartesian coordinate system.
  • The cylindrical coordinate system is like a mix between the spherical and Cartesian system, incorporating linear and radial parameters.

• Polar Coordinates

  • This is called the Cartesian coordinate system.
  • The $xy$ or Cartesian coordinate system is not always the easiest system to use for every problem.
  • Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
  • The $x$ Cartesian coordinate is given by $r \cos \theta$ and the $y$ Cartesian coordinate is given by $r \sin \theta$.

• Cylindrical and Spherical Coordinates

  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • For the conversion between cylindrical and Cartesian coordinate co-ordinates, it is convenient to assume that the reference plane of the former is the Cartesian $xy$-plane (with equation $z = 0$), and the cylindrical axis is the Cartesian $z$-axis.
  • Then the $z$ coordinate is the same in both systems, and the correspondence between cylindrical $(\rho,\varphi)$ and Cartesian $(x,y)$ are the same as for polar coordinates, namely $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
  • The spherical coordinates (radius $r$, inclination $\theta$, azimuth $\varphi$) of a point can be obtained from its Cartesian coordinates ($x$, $y$, $z$) by the formulae:

• Vectors in Three Dimensions

  • In the Cartesian coordinate system, a vector can be represented by identifying the coordinates of its initial and terminal point.
  • Typically in Cartesian coordinates, one considers primarily bound vectors.
  • A vector in the 3D Cartesian space, showing the position of a point $A$ represented by a black arrow.

• Inverse Trigonometric Functions: Differentiation and Integration

  • The usual principal values of the $\text{arctan}(x)$ and $\text{arccot}(x)$ functions graphed on the Cartesian plane.
  • Principal values of the $\text{arcsec}(x)$ and $\text{arccsc}(x)$ functions graphed on the Cartesian plane.
  • The usual principal values of the $\arcsin(x)$ and $\arccos(x)$ functions graphed on the Cartesian plane.

• Double Integrals in Polar Coordinates

  • In $R^2$, if the domain has a cylindrical symmetry and the function has several particular characteristics, you can apply the transformation to polar coordinates, which means that the generic points $P(x, y)$ in Cartesian coordinates switch to their respective points in polar coordinates.
  • The polar coordinates $r$ and $\varphi$ can be converted to the Cartesian coordinates $x$ and $y$ by using the trigonometric functions sine and cosine:
  • The Cartesian coordinates $x$ and $y$ can be converted to polar coordinates $r$ and $\varphi$ with $r \geq 0$ and $\varphi$ in the interval $(−\pi, \pi]$:
  • This figure illustrates graphically a transformation from cartesian to polar coordinates

• Graphing Equations

  • Here we will use the Cartesian plane, where the         $x$-axis will be a horizontal line, the $y$-axis a vertical line, and where the two cross is called the origin.
  • How to graph an equation in two variables in the Cartesian plane
  • After creating a few $x$ and $y$ ordered pairs, plot them on the Cartesian plane and connect the points.

• Real Numbers, Functions, and Graphs

  • In particular, if $x$ is a real number, "graph" means the graphical representation of this collection, in the form of a line chart, a curve on a Cartesian plane, together with Cartesian axes, etc.
  • Graphing on a Cartesian plane is sometimes referred to as curve sketching.