coordinate
a number representing the position of a point along a line, arc, or similar one-dimensional figure
Examples of coordinate in the following topics:
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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 coordinate is the same in both systems, and the correspondence between cylindrical and Cartesian are the same as for polar coordinates, namely .
- The spherical coordinates (radius , inclination , azimuth ) of a point can be obtained from its Cartesian coordinates (, , ) by the formulae:
- A cylindrical coordinate system with origin , polar axis , and longitudinal axis .
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Polar Coordinates
- This is called the Cartesian coordinate system.
- Such definitions are called polar coordinates.
- Polar coordinates in and can be converted to Cartesian coordinates and .
- A set of polar coordinates.
- The Cartesian coordinate is given by and the Cartesian coordinate is given by .
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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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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.
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Vectors in Three Dimensions
- The mathematical representation of a physical vector depends on the coordinate system used to describe it.
- 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 .
- The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.
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Conic Sections in Polar Coordinates
- Conic sections are sections of cones and can be represented by polar coordinates.
- In polar coordinates, a conic section with one focus at the origin is given by the following equation:
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Area and Arc Length in Polar Coordinates
- Area and arc length are calculated in polar coordinates by means of integration.
- Since it can be very difficult to measure the length of an arc linearly, the solution is to use polar coordinates.
- Using polar coordinates allows us to integrate along the length of the arc in order to compute its length.
- Evaluate arc segment area and arc length using polar coordinates and integration
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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 .