R

Definition

The variable 'r' represents the radial distance or the distance from the origin in polar, cylindrical, and spherical coordinate systems. It is a fundamental component that describes the location of a point in these coordinate systems, which are widely used in various areas of mathematics and physics.

5 Must Know Facts For Your Next Test

1. In polar coordinates, the variable 'r' represents the radial distance or the distance from the origin to a point on the plane.
2. The radial distance 'r' is a key component in the calculation of area and arc length in polar coordinates.
3. In cylindrical coordinates, the variable 'r' represents the distance from the z-axis to a point in the xy-plane.
4. The radial distance 'r' is used in the transformation between Cartesian and cylindrical coordinates, as well as in the evaluation of triple integrals in cylindrical coordinates.
5. In spherical coordinates, the variable 'r' represents the distance from the origin to a point in three-dimensional space.

Review Questions

• Explain the role of the variable 'r' in the context of polar coordinates and how it is used to describe the location of a point.
  • In polar coordinates, the variable 'r' represents the radial distance or the distance from the origin to a point on the plane. It is one of the two coordinates used to specify the location of a point, the other being the angle θ. The radial distance 'r' is a crucial component in the calculation of area and arc length in polar coordinates, as it directly affects the size and shape of the region being measured.
• Describe the relationship between the variable 'r' and the other coordinates in the cylindrical coordinate system.
  • In the cylindrical coordinate system, the variable 'r' represents the distance from the z-axis to a point in the xy-plane. This radial distance 'r' is used in conjunction with the angle θ from the positive x-axis and the height z to fully specify the location of a point in three-dimensional space. The transformation between Cartesian and cylindrical coordinates involves the radial distance 'r', and it is also an essential component in the evaluation of triple integrals in cylindrical coordinates.
• Analyze the significance of the variable 'r' in the context of spherical coordinates and its role in describing the position of a point in three-dimensional space.
  • In spherical coordinates, the variable 'r' represents the distance from the origin to a point in three-dimensional space. This radial distance 'r' is one of the three coordinates used to specify the location of a point, along with the angle θ from the positive z-axis and the angle φ from the positive x-axis in the xy-plane. The radial distance 'r' is a crucial factor in the transformation between Cartesian and spherical coordinates, as well as in the evaluation of triple integrals in spherical coordinates, where it directly affects the volume element and the overall integral.

Related terms

Polar Coordinates:

A two-dimensional coordinate system that uses the distance from the origin (r) and the angle from a reference direction (θ) to specify the location of a point.

Cylindrical Coordinates:

A three-dimensional coordinate system that uses the distance from the z-axis (r), the angle from the x-axis (θ), and the height (z) to specify the location of a point.

Spherical Coordinates:

A three-dimensional coordinate system that uses the distance from the origin (r), the angle from the positive z-axis (θ), and the angle from the positive x-axis in the xy-plane (φ) to specify the location of a point.