cylindrical coordinate

(noun)

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

Related Terms

  • differential

Examples of cylindrical coordinate in the following topics:

  • 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 $dx\, dy\, dz$ differentials in the integral become $\rho \, d\rho \,d\varphi \,dz$.
    • 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.
  • 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.
    • 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 $x = \rho \cos \varphi; \, y = \rho \sin \varphi$.
    • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.
  • 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 $z$).
    • The cylindrical system uses two linear parameters and one radial parameter:
    • 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
  • Change of Variables

    • To do so, the function must be adapted to the new coordinates.
    • Changing to cylindrical coordinates may be useful depending on the setup of problem.
  • Double Integrals in Polar Coordinates

    • When domain has a cylindrical symmetry and the function has several specific characteristics, apply the transformation to 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 is the case because the function has a cylindrical symmetry.
  • 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.
    • It's possible to use therefore the passage in spherical coordinates; the function is transformed by this relation:
    • Points on $z$-axis do not have a precise characterization in spherical coordinates, so $\theta$ can vary from $0$ to $2 \pi$.
    • Finally, you obtain the final integration formula: It's better to use this method in case of spherical domains and in case of functions that can be easily simplified, by the first fundamental relation of trigonometry, extended in $R^3$; in other cases it can be better to use cylindrical coordinates.
    • Spherical coordinates are useful when domains in $R^3$ have spherical symmetry.
  • Cylindrical Shells

    • In the shell method, a function is rotated around an axis and modeled by an infinite number of cylindrical shells, all infinitely thin.
    • In the integrand, the factor $x$ represents the radius of the cylindrical shell under consideration, while  is equal to the height of the shell.
    • Therefore, the entire integrand, $2\pi x \left | f(x) - g(x) \right | \,dx$, is nothing but the volume of the cylindrical shell.
    • Each segment located at $x$, between $f(x)$and the $x$-axis, gives a cylindrical shell after revolution around the vertical axis.
    • Use shell integration to create a cylindrical shell and calculate the volume of a "solid of revolution" perpendicular to the axis of revolution.
  • 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 $r \sin \theta$.
  • 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 $O = (0,0,0)$.
    • The coordinate representation of vectors allows the algebraic features of vectors to be expressed in a convenient numerical fashion.
  • 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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