Vector fields in cylindrical and spherical coordinates

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Spherical coordinates (r, θ, φ) as commonly used in physics: radial distance r, polar angle θ (theta), and azimuthal angle φ (phi). The symbol ρ (rho) is often used instead of r.

NOTE: This page uses common physics notation for spherical coordinates, in which is the angle between the z axis and the radius vector connecting the origin to the point in question, while is the angle between the projection of the radius vector onto the x-y plane and the x axis. Several other definitions are in use, and so care must be taken in comparing different sources.[1]

Cylindrical coordinate system[edit]

Vector fields[edit]

Vectors are defined in cylindrical coordinates by (r, θ, z), where

  • r is the length of the vector projected onto the xy-plane,
  • θ is the angle between the projection of the vector onto the xy-plane (i.e. r) and the positive x-axis (0 ≤ θ < 2π),
  • z is the regular z-coordinate.

(r, θ, z) is given in cartesian coordinates by:

Physics Coordinates.png

or inversely by:

Any vector field can be written in terms of the unit vectors as:

The cylindrical unit vectors are related to the cartesian unit vectors by:

Time derivative of a vector field[edit]

To find out how the vector field A changes in time we calculate the time derivatives. For this purpose we use Newton's notation for the time derivative (). In cartesian coordinates this is simply:

However, in cylindrical coordinates this becomes:

We need the time derivatives of the unit vectors. They are given by:

So the time derivative simplifies to:

Second time derivative of a vector field[edit]

The second time derivative is of interest in physics, as it is found in equations of motion for classical mechanical systems. The second time derivative of a vector field in cylindrical coordinates is given by:

To understand this expression, we substitute A = P, where p is the vector (r, θ, z).

This means that .

After substituting we get:

In mechanics, the terms of this expression are called:

See also: Centripetal force, Angular acceleration, Coriolis effect.

Spherical coordinate system[edit]

Vector fields[edit]

Vectors are defined in spherical coordinates by (ρ,θ,φ), where

  • ρ is the length of the vector,
  • θ is the angle between the positive Z-axis and the vector in question (0 ≤ θ ≤ π), and
  • φ is the angle between the projection of the vector onto the X-Y-plane and the positive X-axis (0 ≤ φ < 2π).

(ρ,θ,φ) is given in Cartesian coordinates by:

or inversely by:

Any vector field can be written in terms of the unit vectors as:

The spherical unit vectors are related to the cartesian unit vectors by:

So the cartesian unit vectors are related to the spherical unit vectors by:

Time derivative of a vector field[edit]

To find out how the vector field A changes in time we calculate the time derivatives. In cartesian coordinates this is simply:

However, in spherical coordinates this becomes:

We need the time derivatives of the unit vectors. They are given by:

So the time derivative becomes:

See also[edit]

References[edit]

  1. ^ Wolfram Mathworld, spherical coordinates