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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

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Vector fields

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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 field

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To find out how the vector field A changes in time, the time derivatives should be calculated. For this purpose Newton's notation will be used for the time derivative (). In Cartesian coordinates this is simply:

However, in cylindrical coordinates this becomes:

The time derivatives of the unit vectors are needed. They are given by:

So the time derivative simplifies to:

Second time derivative of a vector field

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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, A is substituted for P, where P is the vector (ρ, φ, z).

This means that .

After substituting, the result is given:

In mechanics, the terms of this expression are called:

central outward acceleration
centripetal acceleration
angular acceleration
Coriolis effect
z-acceleration

Spherical coordinate system

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Vector fields

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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 ≤ θπ), and
  • φ is the angle between the projection of the vector onto the xy-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.

The Cartesian unit vectors are thus related to the spherical unit vectors by:


Time derivative of a vector field

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To find out how the vector field A changes in time, the time derivatives should be calculated. In Cartesian coordinates this is simply: However, in spherical coordinates this becomes: The time derivatives of the unit vectors are needed. They are given by: Thus the time derivative becomes:

See also

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References

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