coordinate system

Examples of coordinate system in the following topics:

• 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

• Cylindrical and Spherical Coordinates

  • While Cartesian coordinates have many applications, cylindrical and spherical coordinates are useful when describing objects or phenomena with specific symmetries.
  • A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis, the direction from the axis relative to a chosen reference direction, and the distance from a chosen reference plane perpendicular to the 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 $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
  • A spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• The Cartesian System

  • The Cartesian coordinate system is used to visualize points on a graph by showing the points' distances from two axes.
  • A Cartesian coordinate system is used to graph points.
  • The Cartesian coordinate system is broken into four quadrants by the two axes.
  • The four quadrants of theCartesian coordinate system.
  • The Cartesian coordinate system with 4 points plotted, including the origin, at $(0,0)$.

• Polar Coordinates

  • We use coordinate systems every day, even if we don't realize it.
  • For example, if you walk 20 meters to the right of the parking lot to find the car, you are using a coordinate system.
  • The coordinate system you are most likely familiar with is the $xy$-coordinate system, where locations are described as horizontal ($x$) and vertical ($y$) distances from an arbitrary point.
  • This is called the Cartesian coordinate system.
  • The $xy$ or Cartesian coordinate system is not always the easiest system to use for every problem.

• 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 $y$.
  • In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction.
  • However, there are other ways of writing a coordinate pair and other types of grid systems.
  • The reference point (analogous to the origin of a Cartesian system) is called the pole, and the ray from the pole in the reference direction is the polar axis.
  • Points in the polar coordinate system with pole $0$ and polar axis $L$.

• Vectors in Three Dimensions

  • The mathematical representation of a physical vector depends on the coordinate system used to describe it.
  • Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.
  • 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 bound vector is determined by the coordinates of the terminal point, its initial point always having the coordinates of the origin $O = (0,0,0)$.

• Other Curves in Polar Coordinates

  • Some curves have a simple expression in polar coordinates, whereas they would be very complex to represent in Cartesian coordinates.
  • To graph in the rectangular coordinate system we construct a table of $x$ and $y$  values.
  • To graph in the polar coordinate system we construct a table of $r$ and $\theta$ values.

• Functions of the Nervous System

  • The primary function of the nervous system is to coordinate and control the various functions of our body.
  • The nervous system has three overlapping functions.
  • The nervous system is a highly integrated system.
  • Nervous system processes and interprets sensory input and decides what should be done at the each moment.
  • The nervous system activates effector organs such as muscles and glands to cause a response called the motor input.

• Organization of the Nervous System

  • The nervous system is a network of cells called neurons that coordinate actions and transmit signals between different parts of the body.
  • A nervous system is what allows us to react to the changing environment around us.
  • The nervous system is an organ system that coordinates our actions by transmitting signals between different parts of our bodies.
  • Along with neurons, the nervous system relies on the function of other specialized cells called glial cells, or glia, that provide structural and metabolic support to the nervous system.
  • Gross organization of the nervous system, with the peripheral nervous system, the spinal, and the cortical levels.

• Reactions of Coordination Compounds

  • Many metal-containing compounds consist of coordination complexes.
  • 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.
  • In electron transfer, an electron moves from one atom to another, changing the charge on each but leaving the net charge of the system the same.