Vector fields in cylindrical and spherical coordinates

Vector fields in cylindrical and spherical coordinates

= Cylindrical coordinate system =

Vector fields

Vectors are defined in cylindrical coordinates by (ρ,φ,z), where
* ρ is the length of the vector projected onto the X-Y-plane,
* φ is the angle of the projected vector with 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

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 ($mathbf\left\{dot A\right\}$).In cartesian coordinates this is simply::$mathbf\left\{dot A\right\} = dot A_x mathbf\left\{hat x\right\} + dot A_y mathbf\left\{hat y\right\} + dot A_z mathbf\left\{hat z\right\}$

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::

Spherical coordinate system

Vector fields

Vectors are defined in spherical coordinates by (r,θ,φ), where
* r is the length of the vector,
* θ is the angle with the positive Z-axis (0 <= θ <= π),
* φ is the angle with the X-Z-plane (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.

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::$mathbf\left\{dot A\right\} = dot A_x mathbf\left\{hat x\right\} + dot A_y mathbf\left\{hat y\right\} + dot A_z mathbf\left\{hat z\right\}$

However, in spherical coordinates this becomes::

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

So the time derivative becomes::

* Del in cylindrical and spherical coordinates for the specification of gradient, divergence, curl, and laplacian in various coordinate systems.

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