Vector Fields In Cylindrical And Spherical Coordinates

Note: This page uses common physics notation for spherical coordinates, in which $\theta$ is the angle between the ''z'' axis and the radius vector connecting the origin to the point in question, while $\phi$ 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.
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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 ''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'' < 2''π''),
* ''z'' is the regular ''z''-coordinate.

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

$\begin \rho \\ \phi \\ z \end = \begin \sqrt \\ \operatorname(y / x) \\ z \end,\ \ \ 0 \le \phi < 2\pi,$

or inversely by:
$\begin x \\ y \\ z \end = \begin \rho\cos\phi \\ \rho\sin\phi \\ z \end.$

Any vector field can be written in terms of the unit vectors as:
$\mathbf A = A_x \mathbf + A_y \mathbf + A_z \mathbf = A_\rho \mathbf + A_\phi \boldsymbol + A_z \mathbf$
The cylindrical unit vectors are related to the Cartesian unit vectors by:
$\begin\mathbf \\ \boldsymbol \\ \mathbf\end = \begin \cos\phi & \sin\phi & 0 \\ -\sin\phi & \cos\phi & 0 \\ 0 & 0 & 1 \end \begin \mathbf \\ \mathbf \\ \mathbf \end$

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, the time derivatives should be calculated.
For this purpose Newton's notation will be used for the time derivative ($\dot$).
In Cartesian coordinates this is simply:
$\dot = \dot_x \hat + \dot_y \hat + \dot_z \hat$

However, in cylindrical coordinates this becomes:
$\dot = \dot_\rho \hat + A_\rho \dot + \dot_\phi \hat + A_\phi \dot + \dot_z \hat + A_z \dot$

The time derivatives of the unit vectors are needed.
They are given by:
$\begin \dot & = \dot \hat \\ \dot & = - \dot\phi \hat \\ \dot & = 0 \end$

So the time derivative simplifies to:
$\dot = \hat \left(\dot_\rho - A_\phi \dot\right) + \hat \left(\dot_\phi + A_\rho \dot\right) + \hat \dot_z$

Second time derivative of a vector field

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:
$\mathbf = \mathbf \left(\ddot A_\rho - A_\phi \ddot\phi - 2 \dot A_\phi \dot\phi - A_\rho \dot\phi^2\right) + \boldsymbol \left(\ddot A_\phi + A_\rho \ddot\phi + 2 \dot A_\rho \dot\phi - A_\phi \dot\phi^2\right) + \mathbf \ddot A_z$

To understand this expression, A is substituted for P, where P is the vector (''ρ'', ''φ'', ''z'').

This means that $\mathbf = \mathbf = \rho \mathbf + z \mathbf$.

After substituting, the result is given:
$\ddot\mathbf = \mathbf \left(\ddot \rho - \rho \dot\phi^2\right) + \boldsymbol \left(\rho \ddot\phi + 2 \dot \rho \dot\phi\right) + \mathbf \ddot z$

In mechanics, the terms of this expression are called:
$\begin \ddot \rho \mathbf &= \text \\ -\rho \dot\phi^2 \mathbf &= \text \\ \rho \ddot\phi \boldsymbol &= \text \\ 2 \dot \rho \dot\phi \boldsymbol &= \text \\ \ddot z \mathbf &= \text \end$

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 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 ''xy''-plane and the positive X-axis (0 ≤ ''φ'' < 2''π'').

(''r'', ''θ'', ''φ'') is given in Cartesian coordinates by:
$\beginr \\ \theta \\ \phi \end = \begin \sqrt \\ \arccos(z / \sqrt) \\ \arctan(y / x) \end,\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi,$
or inversely by:
$\begin x \\ y \\ z \end = \begin r\sin\theta\cos\phi \\ r\sin\theta\sin\phi \\ r\cos\theta\end.$

Any vector field can be written in terms of the unit vectors as:
$\mathbf A = A_x\mathbf + A_y\mathbf + A_z\mathbf = A_r\boldsymbol + A_\theta\boldsymbol + A_\phi\boldsymbol$
The spherical unit vectors are related to the Cartesian unit vectors by:
$\begin\boldsymbol \\ \boldsymbol \\ \boldsymbol \end = \begin \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\ \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\ -\sin\phi & \cos\phi & 0 \end \begin \mathbf \\ \mathbf \\ \mathbf \end$

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:
$\begin\mathbf \\ \mathbf \\ \mathbf \end = \begin \sin\theta\cos\phi & \cos\theta\cos\phi & -\sin\phi \\ \sin\theta\sin\phi & \cos\theta\sin\phi & \cos\phi \\ \cos\theta & -\sin\theta & 0 \end \begin \boldsymbol \\ \boldsymbol \\ \boldsymbol \end$

Time derivative of a vector field

To find out how the vector field A changes in time, the time derivatives should be calculated.
In Cartesian coordinates this is simply:
$\mathbf = \dot A_x \mathbf + \dot A_y \mathbf + \dot A_z \mathbf$

However, in spherical coordinates this becomes:
$\mathbf = \dot A_r \boldsymbol + A_r \boldsymbol + \dot A_\theta \boldsymbol + A_\theta \boldsymbol + \dot A_\phi \boldsymbol + A_\phi \boldsymbol$

The time derivatives of the unit vectors are needed. They are given by:
$\begin \boldsymbol &= \dot\theta \boldsymbol + \dot\phi\sin\theta \boldsymbol \\ \boldsymbol &= - \dot\theta \boldsymbol + \dot\phi\cos\theta \boldsymbol \\ \boldsymbol &= - \dot\phi\sin\theta \boldsymbol - \dot\phi\cos\theta \boldsymbol \end$

Thus the time derivative becomes:
$\mathbf = \boldsymbol \left(\dot A_r - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta \right) + \boldsymbol \left(\dot A_\theta + A_r \dot\theta - A_\phi \dot\phi \cos\theta\right) + \boldsymbol \left(\dot A_\phi + A_r \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta\right)$

See also

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

References

{{DEFAULTSORT:Vector Fields In Cylindrical And Spherical Coordinates
Vector calculus
Coordinate systems