coordinate system
a method of representing points in a space of given dimensions by coordinates from an origin
Examples of coordinate system in the following topics:
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Three-Dimensional Coordinate Systems
- The three-dimensional coordinate system expresses a point in space with three parameters, often length, width and depth (, , and ).
- Each parameter is perpendicular to the other two, and cannot lie in the same plane. shows a Cartesian coordinate system that uses the parameters , , and .
- 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
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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 coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely .
- 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 , polar axis , and longitudinal axis .
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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 -coordinate system, where locations are described as horizontal () and vertical () distances from an arbitrary point.
- This is called the Cartesian coordinate system.
- The or Cartesian coordinate system is not always the easiest system to use for every problem.
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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 .
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Surfaces in Space
- To say that a surface is "two-dimensional" means that, about each point, there is a coordinate patch on which a two-dimensional coordinate system is defined.
- For example, the surface of the Earth is (ideally) a two-dimensional surface, and latitude and longitude provide two-dimensional coordinates on it (except at the poles and along the 180th meridian).
- In spherical coordinates, the surface can be expressed simply by .
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Physics and Engeineering: Center of Mass
- In this case, the distribution of mass is balanced around the center of mass and the average of the weighted position coordinates of the distributed mass defines its coordinates.
- In the case of a system of particles $P_i, i = 1, \cdots, n$, each with a mass, , which are located in space with coordinates $r_i, i = 1, \cdots, n$, the coordinates of the center of mass satisfy the following condition:
- If the mass distribution is continuous with respect to the density, , within a volume, , then the integral of the weighted position coordinates of the points in this volume relative to the center of mass, , is zero, that is:
- COM can be defined for both discrete and continuous systems.
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Center of Mass and Inertia
- In the case of a distribution of separate bodies, such as the planets of the Solar System, the center of mass may not correspond to the position of any individual member of the system.
- In the case of a system of particles $P_i, i = 1, \cdots , n$, each with mass that are located in space with coordinates $\mathbf{r}_i, i = 1, \cdots , n$, the coordinates of the center of mass satisfy the condition:
- If the mass distribution is continuous with the density within a volume , then the integral of the weighted position coordinates of the points in this volume relative to the center of mass is zero; that is:
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Double Integrals in Polar Coordinates
- In , 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 in Cartesian coordinates switch to their respective points in polar coordinates.
- The polar coordinates and can be converted to the Cartesian coordinates and by using the trigonometric functions sine and cosine:
- The Cartesian coordinates and can be converted to polar coordinates and with and 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
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Triple Integrals in Spherical Coordinates
- When the function to be integrated has a spherical symmetry, change the variables into spherical coordinates and then perform integration.
- When the function to be integrated has a spherical symmetry, it is sensible to change the variables into spherical coordinates and then perform integration.
- It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
- Points on -axis do not have a precise characterization in spherical coordinates, so can vary from to .
- Spherical coordinates are useful when domains in have spherical symmetry.
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Triple Integrals in Cylindrical Coordinates
- When the function to be integrated has a cylindrical symmetry, it is sensible to integrate using cylindrical coordinates.
- When the function to be integrated has a cylindrical symmetry, it is sensible to change the variables into cylindrical coordinates and then perform integration.
- Also in switching to cylindrical coordinates, the differentials in the integral become .
- Finally, it is possible to apply the final formula to cylindrical coordinates:
- Cylindrical coordinates are often used for integrations on domains with a circular base.