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[[File:3D Spherical.svg|thumb|240px|right|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 <math>\theta</math> is the angle between the ''z'' axis and the radius vector connecting the origin to the point in question, while <math>\phi</math> 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.<ref name="wolfram">[http://mathworld.wolfram.com/CylindricalCoordinates.html Wolfram Mathworld, spherical coordinates]</ref> | |||
== Cylindrical coordinate system == | |||
=== Vector fields === | |||
Vectors are defined in [[cylindrical coordinates]] by (''r'', θ, ''z''), where | |||
* ''r'' is the length of the vector projected onto the ''xy''-plane, | |||
* θ is the angle between the projection of the vector onto the ''xy''-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: | |||
:<math>\begin{bmatrix} r \\ \theta \\ z \end{bmatrix} = | |||
\begin{bmatrix} | |||
\sqrt{x^2 + y^2} \\ \operatorname{arctan}(y / x) \\ z | |||
\end{bmatrix},\ \ \ 0 \le \theta < 2\pi, | |||
</math> | |||
or inversely by: | |||
:<math>\begin{bmatrix} x \\ y \\ z \end{bmatrix} = | |||
\begin{bmatrix} r\cos\theta \\ r\sin\theta \\ z \end{bmatrix}.</math> | |||
Any [[vector field]] can be written in terms of the unit vectors as: | |||
:<math>\mathbf A = A_x \mathbf{\hat x} + A_y \mathbf{\hat y} + A_z \mathbf{\hat z} | |||
= A_r \mathbf{\hat r} + A_\theta \boldsymbol{\hat \theta} + A_z \mathbf{\hat z}</math> | |||
The cylindrical unit vectors are related to the cartesian unit vectors by: | |||
:<math>\begin{bmatrix}\mathbf{\hat r} \\ \boldsymbol{\hat\theta} \\ \mathbf{\hat z}\end{bmatrix} | |||
= \begin{bmatrix} \cos\theta & \sin\theta & 0 \\ | |||
-\sin\theta & \cos\theta & 0 \\ | |||
0 & 0 & 1 \end{bmatrix} | |||
\begin{bmatrix} \mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix}</math> | |||
* Note: the matrix is an [[orthogonal matrix]], that is, its [[Invertible matrix|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 (<math>\dot{\mathbf{A}}</math>). | |||
In cartesian coordinates this is simply: | |||
:<math>\dot{\mathbf{A}} = \dot{A}_x \hat{\mathbf{x}} + \dot{A}_y \hat{\mathbf{y}} + \dot{A}_z \hat{\mathbf{z}}</math> | |||
However, in cylindrical coordinates this becomes: | |||
:<math>\dot{\mathbf{A}} = \dot{A}_r \hat{\boldsymbol{r}} + A_r \dot{\hat{\boldsymbol{r}}} | |||
+ \dot{A}_\theta \hat{\boldsymbol{\theta}} + A_\theta \dot{\hat{\boldsymbol{\theta}}} | |||
+ \dot{A}_z \hat{\boldsymbol{z}} + A_z \dot{\hat{\boldsymbol{z}}}</math> | |||
We need the time derivatives of the unit vectors. | |||
They are given by: | |||
:<math>\begin{align} | |||
\dot{\hat{\mathbf{r}}} &= \dot\theta \hat{\boldsymbol{\theta}} \\ | |||
\dot{\hat{\boldsymbol{\theta}}} &= - \dot\theta \hat{\mathbf{r}} \\ | |||
\dot{\hat{\mathbf{z}}} &= 0 \end{align}</math> | |||
So the time derivative simplifies to: | |||
:<math>\dot{\mathbf{A}} = \hat{\boldsymbol{r}} (\dot{A}_r - A_\theta \dot{\theta}) | |||
+ \hat{\boldsymbol{\theta}} (\dot{A}_\theta + A_r \dot{\theta}) | |||
+ \hat{\mathbf{z}} \dot{A}_z</math> | |||
=== 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 mechanics|classical mechanical]] systems. | |||
The second time derivative of a vector field in cylindrical coordinates is given by: | |||
:<math>\mathbf{\ddot A} = \mathbf{\hat r} (\ddot A_r - A_\theta \ddot\theta - 2 \dot A_\theta \dot\theta - A_r \dot\theta^2) | |||
+ \boldsymbol{\hat\theta} (\ddot A_\theta + A_r \ddot\theta + 2 \dot A_r \dot\theta - A_\theta \dot\theta^2) | |||
+ \mathbf{\hat z} \ddot A_z</math> | |||
To understand this expression, we substitute A = P, where p is the vector (r, θ, z). | |||
This means that <math>\mathbf{A} = \mathbf{P} = r \mathbf{\hat r} + z \mathbf{\hat z}</math>. | |||
After substituting we get: | |||
:<math>\ddot\mathbf{P} = \mathbf{\hat r} (\ddot r - r \dot\theta^2) | |||
+ \boldsymbol{\hat\theta} (r \ddot\theta + 2 \dot r \dot\theta) | |||
+ \mathbf{\hat z} \ddot z</math> | |||
In mechanics, the terms of this expression are called: | |||
:<math>\begin{align} | |||
\ddot r \mathbf{\hat r} &= \mbox{central outward acceleration} \\ | |||
-r \dot\theta^2 \mathbf{\hat r} &= \mbox{centripetal acceleration} \\ | |||
r \ddot\theta \boldsymbol{\hat\theta} &= \mbox{angular acceleration} \\ | |||
2 \dot r \dot\theta \boldsymbol{\hat\theta} &= \mbox{Coriolis effect} \\ | |||
\ddot z \mathbf{\hat z} &= \mbox{z-acceleration} | |||
\end{align}</math> | |||
See also: [[Centripetal force]], [[Angular acceleration]], [[Coriolis effect]]. | |||
== Spherical coordinate system == | |||
=== 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: | |||
:<math>\begin{bmatrix}\rho \\ \theta \\ \phi \end{bmatrix} = | |||
\begin{bmatrix} | |||
\sqrt{x^2 + y^2 + z^2} \\ \arccos(z / \rho) \\ \arctan(y / x) | |||
\end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi, | |||
</math> | |||
or inversely by: | |||
:<math>\begin{bmatrix} x \\ y \\ z \end{bmatrix} = | |||
\begin{bmatrix} \rho\sin\theta\cos\phi \\ \rho\sin\theta\sin\phi \\ \rho\cos\theta\end{bmatrix}.</math> | |||
Any vector field can be written in terms of the unit vectors as: | |||
:<math>\mathbf A = A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z} | |||
= A_\rho\boldsymbol{\hat \rho} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}</math> | |||
The spherical unit vectors are related to the cartesian unit vectors by: | |||
:<math>\begin{bmatrix}\boldsymbol{\hat\rho} \\ \boldsymbol{\hat\theta} \\ \boldsymbol{\hat\phi} \end{bmatrix} | |||
= \begin{bmatrix} \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{bmatrix} | |||
\begin{bmatrix} \mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix}</math> | |||
* 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: | |||
:<math>\mathbf{\dot A} = \dot A_x \mathbf{\hat x} + \dot A_y \mathbf{\hat y} + \dot A_z \mathbf{\hat z}</math> | |||
However, in spherical coordinates this becomes: | |||
:<math>\mathbf{\dot A} = \dot A_\rho \boldsymbol{\hat \rho} + A_\rho \boldsymbol{\dot{\hat \rho}} | |||
+ \dot A_\theta \boldsymbol{\hat\theta} + A_\theta \boldsymbol{\dot{\hat\theta}} | |||
+ \dot A_\phi \boldsymbol{\hat\phi} + A_\phi \boldsymbol{\dot{\hat\phi}}</math> | |||
We need the time derivatives of the unit vectors. | |||
They are given by: | |||
:<math>\begin{align} | |||
\boldsymbol{\dot{\hat \rho}} &= \dot\theta \boldsymbol{\hat\theta} + \dot\phi\sin\theta \boldsymbol{\hat\phi} \\ | |||
\boldsymbol{\dot{\hat\theta}} &= - \dot\theta \boldsymbol{\hat \rho} + \dot\phi\cos\theta \boldsymbol{\hat\phi} \\ | |||
\boldsymbol{\dot{\hat\phi}} &= - \dot\phi\sin\theta \boldsymbol{\hat\rho} - \dot\phi\cos\theta \boldsymbol{\hat\theta} \end{align}</math> | |||
So the time derivative becomes: | |||
:<math>\mathbf{\dot A} = \boldsymbol{\hat \rho} (\dot A_\rho - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta) | |||
+ \boldsymbol{\hat\theta} (\dot A_\theta + A_\rho \dot\theta - A_\phi \dot\phi \cos\theta) | |||
+ \boldsymbol{\hat\phi} (\dot A_\phi + A_\rho \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta)</math> | |||
== See also == | |||
* [[Del in cylindrical and spherical coordinates]] for the specification of [[gradient]], [[divergence]], [[Curl (mathematics)|curl]], and [[laplacian]] in various coordinate systems. | |||
==References== | |||
<references/> | |||
{{DEFAULTSORT:Vector Fields In Cylindrical And Spherical Coordinates}} | |||
[[Category:Vector calculus]] | |||
[[Category:Coordinate systems]] |
Revision as of 03:19, 28 December 2013
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
Vector fields
Vectors are defined in cylindrical coordinates by (r, θ, z), where
- r is the length of the vector projected onto the xy-plane,
- θ is the angle between the projection of the vector onto the xy-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 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 (). 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
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, we substitute A = P, where p is the vector (r, θ, z).
After substituting we get:
In mechanics, the terms of this expression are called:
See also: Centripetal force, Angular acceleration, Coriolis effect.
Spherical coordinate system
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, 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:
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 for the specification of gradient, divergence, curl, and laplacian in various coordinate systems.