cylindrical coordinate

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: