Vector fields in cylindrical and spherical coordinates

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 systemEdit

Vector fieldsEdit

Vectors are defined in cylindrical coordinates by (ρ, φ, z), where

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

(ρ, φ, z) is given in cartesian coordinates by:

 

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:

 

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

Time derivative of a vector fieldEdit

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 fieldEdit

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 (\rho, θ, z).

This means that  .

After substituting we get:

 

In mechanics, the terms of this expression are called:

 

Spherical coordinate systemEdit

Vector fieldsEdit

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

  • r 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).

(r, 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:

 

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

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

 

Time derivative of a vector fieldEdit

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 alsoEdit

ReferencesEdit