coordinates

Examples of coordinates in the following topics:

• Converting Between Polar and Cartesian Coordinates

  • When given a set of polar coordinates, we may need to convert them to rectangular coordinates.
  • To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships.
  • Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in Figure.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.
  • A right triangle with rectangular (Cartesian) coordinates and equivalent polar coordinates.

• Coordinating Conjunctions

• Cylindrical and Spherical Coordinates

  • Cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • 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:
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Polar Coordinates

  • This is called the Cartesian coordinate system.
  • Such definitions are called polar coordinates.
  • Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
  • A set of polar coordinates.
  • The $x$ Cartesian coordinate is given by $r \cos \theta$ and the $y$ Cartesian coordinate is given by $r \sin \theta$.

• The Cartesian System

  • 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.
  • Any point in the first quadrant has both positive x and y coordinates.
  • Points in the second quadrant have negative x and positive y coordinates.
  • The third quadrant has both negative x and y coordinates.
  • The four quadrants of a Cartesian coordinate system.

• Three-Dimensional Coordinate Systems

  • The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth ($x$, $y$, and $z$).
  • 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 $z$.
  • 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.
  • Identify the number of parameters necessary to express a point in the three-dimensional coordinate system

• Increasing Coordination

  • Increasing coordination helps organizations to maintain efficient operations through communication and control.
  • Coordination is the act of organizing and enabling different people to work together to achieve an organization's goals.
  • Coordination is simply the managerial ability to maintain operations and ensure they are properly integrated with one another; therefore, increasing coordination is closely related to improving managerial skills.
  • The management team must pay special attention to issues related to coordination and governance and be able to improve upon coordination through effective management.
  • In practice, coordination involves a delicate balance between centralization and decentralization.

• Introduction to the Polar Coordinate System

  • The polar coordinate system is an alternate coordinate system where the two variables are $r$ and $\theta$, instead of $x$ and $y$.
  • When we think about plotting points in the plane, we usually think of rectangular coordinates $(x,y)$ in the Cartesian coordinate plane.
  • The distance from the pole is called the radial coordinate or radius, and the angle is called the angular coordinate, polar angle, or azimuth.
  • The radial coordinate is often denoted by $r$ or $ρ$ , and the angular coordinate by $ϕ$, $θ$, or $t$.
  • In green, the point with radial coordinate $3$ and angular coordinate $60$ degrees or $(3,60^{\circ})$.

• Reactions of Coordination Compounds

  • Many metal-containing compounds consist of coordination complexes.
  • The central atom or ion, together with all ligands, comprise the coordination sphere.
  • The central atoms or ion and the donor atoms comprise the first coordination sphere.
  • Coordination refers to the coordinate covalent bonds (dipolar bonds) between the ligands and the central atom.
  • As applied to coordination chemistry, this meaning has evolved.

• 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 $(−\pi, \pi]$:
  • In general, the best practice is to use the coordinates that match the built-in symmetry of the function.
  • This figure illustrates graphically a transformation from cartesian to polar coordinates