Radial Coordinate

Definition

The radial coordinate is a fundamental component of polar, cylindrical, and spherical coordinate systems. It represents the distance from a fixed origin point to a specific point in space, measured along a straight line.

5 Must Know Facts For Your Next Test

1. The radial coordinate is often represented by the symbol $r$ and is measured in the same units as the other coordinates in the system.
2. In polar coordinates, the radial coordinate $r$ and the angular coordinate $\theta$ are the two components used to locate a point on a plane.
3. Cylindrical coordinates use the radial coordinate $r$, the angular coordinate $\theta$, and the height coordinate $z$ to locate a point in three-dimensional space.
4. Spherical coordinates use the radial coordinate $r$, the angle $\theta$ from a reference direction, and the angle $\phi$ from a reference plane to locate a point in three-dimensional space.
5. The radial coordinate is essential in the evaluation of double integrals in polar coordinates, as it is used to define the integration limits and the Jacobian of the transformation.

Review Questions

• Explain how the radial coordinate is used in polar coordinate systems to locate a point on a plane.
  • In a polar coordinate system, the radial coordinate $r$ represents the distance from the origin to a specific point on the plane. The angle $\theta$ from a reference direction, along with the radial coordinate $r$, uniquely define the location of a point. The radial coordinate $r$ is measured along a straight line from the origin to the point, while the angle $\theta$ specifies the orientation of this line relative to a reference direction.
• Describe the role of the radial coordinate in the evaluation of double integrals in polar coordinates.
  • When evaluating double integrals in polar coordinates, the radial coordinate $r$ is used to define the integration limits and the Jacobian of the transformation. The Jacobian, which is the determinant of the Jacobian matrix, accounts for the change in the area element from the Cartesian coordinate system to the polar coordinate system. The radial coordinate $r$ is essential in this transformation, as it allows the integration to be performed in terms of the radial distance from the origin and the angular coordinate.
• Analyze the differences in the role of the radial coordinate in polar, cylindrical, and spherical coordinate systems, and explain how it contributes to the unique properties of each system.
  • The radial coordinate $r$ plays a slightly different role in each of the coordinate systems. In polar coordinates, $r$ is the sole radial component used to locate a point on a plane. In cylindrical coordinates, $r$ represents the radial distance from the $z$-axis, while the height coordinate $z$ is the third component. In spherical coordinates, $r$ is the distance from the origin to the point, $\theta$ is the angle from a reference direction, and $\phi$ is the angle from a reference plane. The radial coordinate $r$ is common to all three systems, but its specific interpretation and the additional coordinates used alongside it contribute to the unique properties and applications of each coordinate system.

Related terms

Polar Coordinates: A two-dimensional coordinate system that uses the radial distance from the origin and the angle from a reference direction to locate a point.

Cylindrical Coordinates: A three-dimensional coordinate system that uses the radial distance, angle from a reference direction, and height above a reference plane to locate a point.

Spherical Coordinates: A three-dimensional coordinate system that uses the radial distance from the origin, the angle from a reference direction, and the angle from a reference plane to locate a point.