Vector fields in cylindrical and spherical coordinates

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] Vectors are defined in cylindrical coordinates by (ρ, φ, z), where (ρ, φ, z) is given in Cartesian coordinates by:

ρ cos ⁡ ϕ

ρ sin ⁡ ϕ

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.

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:

The time derivatives of the unit vectors are needed.

So the time derivative simplifies to:

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).

In mechanics, the terms of this expression are called: Vectors are defined in spherical coordinates by (r, θ, φ), where (r, θ, φ) is given in Cartesian coordinates by:

r sin ⁡ θ cos ⁡ ϕ

The spherical basis vectors are related to the Cartesian basis vectors by the Jacobian matrix:

Normalizing the Jacobian matrix so that the spherical basis vectors have unit length we get:

sin ⁡ θ cos ⁡ ϕ

cos ⁡ θ sin ⁡ ϕ