polar

Examples of polar in the following topics:

• Polar Coordinates

  • Such definitions are called polar coordinates.
  • The angle is known as the polar angle, or radial angle, and is usually given as $\theta$.
  • The polar axis is usually drawn horizontal and pointing to the right .
  • Polar coordinates in $r$ and $\theta$ can be converted to Cartesian coordinates $x$ and $y$.
  • A set of polar coordinates.

• 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:
  • Particularly in this case, you can see that the representation of the function f became simpler in polar coordinates.
  • This figure illustrates graphically a transformation from cartesian to polar coordinates

• 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.
  • To find the area enclosed by the arcs and the radius and polar angles, you again use integration.
  • Evaluate arc segment area and arc length using polar coordinates and integration

• 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:

• Cylindrical and Spherical Coordinates

  • 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.
  • Spherical coordinates ($r$, $\theta$, $\varphi$) as often used in mathematics: radial distance $r$, azimuthal angle $\theta$, and polar angle $\varphi$.
  • A cylindrical coordinate system with origin $O$, polar axis $A$, and longitudinal axis $L$.

• Arc Length and Speed

  • Obviously some cases require polar coordinates instead of Cartesian.
  • In polar coordinates:

• Change of Variables

• Triple Integrals in Cylindrical Coordinates

  • because the z component is unvaried during the transformation, the $dx\, dy\, dz$ differentials vary as in the passage in polar coordinates: therefore, they become: $\rho \, d\rho \,d\varphi \,dz$.