• This page uses standard physics notation. For spherical coordinates, is the angle between the z axis and the radius vector connecting the origin to the point in question. is the angle between the projection of the radius vector onto the x-y plane and the x axis. Some (American mathematics) sources reverse this definition.

Vector fields

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

• r is the length of the vector projected onto the X-Y-plane,
• θ is the angle between the projection of the vector onto the X-Y-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:


or inversely by:


Any vector field

Vector field

In vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...

 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
  Orthogonal matrix
  In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
  
  , that is, its inverse is simply its transpose
  Transpose
  In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
  
  .

Time derivative of a vector field

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

For physical purposes we are usually interested in the second time derivative which tells us something about motions in 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:

People should recognize this, because we see:

See also: Centripetal force

Centripetal force

Centripetal force is a force that makes a body follow a curved path: it is always directed orthogonal to the velocity of the body, toward the instantaneous center of curvature of the path. The mathematical description was derived in 1659 by Dutch physicist Christiaan Huygens...

, Angular acceleration

Angular acceleration

Angular acceleration is the rate of change of angular velocity over time. In SI units, it is measured in radians per second squared , and is usually denoted by the Greek letter alpha .- Mathematical definition :...

, Coriolis effect

Coriolis effect

In physics, the Coriolis effect is a deflection of moving objects when they are viewed in a rotating reference frame. In a reference frame with clockwise rotation, the deflection is to the left of the motion of the object; in one with counter-clockwise rotation, the deflection is to the right...

.

Vector fields

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 vector in question (0 ≤ θ ≤ π)
• 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:

• Note: the matrix is an orthogonal matrix
  Orthogonal matrix
  In linear algebra, an orthogonal matrix , is a square matrix with real entries whose columns and rows are orthogonal unit vectors ....
  
  , that is, its inverse is simply its transpose
  Transpose
  In linear algebra, the transpose of a matrix A is another matrix AT created by any one of the following equivalent actions:...
  
  .

Time derivative of a vector field

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

• Del in cylindrical and spherical coordinates
  Del in cylindrical and spherical coordinates
  This is a list of some vector calculus formulae of general use in working with various curvilinear coordinate systems.- Note :* This page uses standard physics notation. For spherical coordinates, \theta is the angle between the z axis and the radius vector connecting the origin to the point in...
  
   for the specification of gradient
  Gradient
  In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
  
  , divergence
  Divergence
  In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
  
  , curl, and laplacian in various coordinate systems.