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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''.]] | | Person who wrote the guide is called Eusebio. South Carolina is the size of his birth place. The [http://Best-lovedhobby.net/ best-loved hobby] for him on top of that his kids is when you need to fish and he's resulted in being doing it for a long time. Filing has been his profession for a short time. Go to his website to search out out more: http://prometeu.net<br><br> |
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| 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>
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| == Cylindrical coordinate system ==
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| === Vector fields ===
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| Vectors are defined in [[cylindrical coordinates]] by (''r'', θ, ''z''), where
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| * ''r'' is the length of the vector projected onto the ''xy''-plane,
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| * 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),
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| * ''z'' is the regular ''z''-coordinate.
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| (''r'', θ, ''z'') is given in [[cartesian coordinates]] by:
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| :<math>\begin{bmatrix} r \\ \theta \\ z \end{bmatrix} =
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| \begin{bmatrix}
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| \sqrt{x^2 + y^2} \\ \operatorname{arctan}(y / x) \\ z
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| \end{bmatrix},\ \ \ 0 \le \theta < 2\pi,
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| </math>
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| or inversely by:
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| :<math>\begin{bmatrix} x \\ y \\ z \end{bmatrix} =
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| \begin{bmatrix} r\cos\theta \\ r\sin\theta \\ z \end{bmatrix}.</math>
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| Any [[vector field]] can be written in terms of the unit vectors as:
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| :<math>\mathbf A = A_x \mathbf{\hat x} + A_y \mathbf{\hat y} + A_z \mathbf{\hat z}
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| = A_r \mathbf{\hat r} + A_\theta \boldsymbol{\hat \theta} + A_z \mathbf{\hat z}</math>
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| The cylindrical unit vectors are related to the cartesian unit vectors by:
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| :<math>\begin{bmatrix}\mathbf{\hat r} \\ \boldsymbol{\hat\theta} \\ \mathbf{\hat z}\end{bmatrix}
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| = \begin{bmatrix} \cos\theta & \sin\theta & 0 \\
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| -\sin\theta & \cos\theta & 0 \\
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| 0 & 0 & 1 \end{bmatrix}
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| \begin{bmatrix} \mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix}</math>
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| * Note: the matrix is an [[orthogonal matrix]], that is, its [[Invertible matrix|inverse]] is simply its [[transpose]].
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| === Time derivative of a vector field ===
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| To find out how the vector field A changes in time we calculate the time derivatives.
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| For this purpose we use [[Newton's notation]] for the time derivative (<math>\dot{\mathbf{A}}</math>).
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| In cartesian coordinates this is simply:
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| :<math>\dot{\mathbf{A}} = \dot{A}_x \hat{\mathbf{x}} + \dot{A}_y \hat{\mathbf{y}} + \dot{A}_z \hat{\mathbf{z}}</math>
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| However, in cylindrical coordinates this becomes:
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| :<math>\dot{\mathbf{A}} = \dot{A}_r \hat{\boldsymbol{r}} + A_r \dot{\hat{\boldsymbol{r}}}
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| + \dot{A}_\theta \hat{\boldsymbol{\theta}} + A_\theta \dot{\hat{\boldsymbol{\theta}}}
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| + \dot{A}_z \hat{\boldsymbol{z}} + A_z \dot{\hat{\boldsymbol{z}}}</math>
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| We need the time derivatives of the unit vectors.
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| They are given by:
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| :<math>\begin{align}
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| \dot{\hat{\mathbf{r}}} &= \dot\theta \hat{\boldsymbol{\theta}} \\
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| \dot{\hat{\boldsymbol{\theta}}} &= - \dot\theta \hat{\mathbf{r}} \\
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| \dot{\hat{\mathbf{z}}} &= 0 \end{align}</math>
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| So the time derivative simplifies to:
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| :<math>\dot{\mathbf{A}} = \hat{\boldsymbol{r}} (\dot{A}_r - A_\theta \dot{\theta})
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| + \hat{\boldsymbol{\theta}} (\dot{A}_\theta + A_r \dot{\theta})
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| + \hat{\mathbf{z}} \dot{A}_z</math>
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| === Second time derivative of a vector field ===
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| The second time derivative is of interest in [[physics]], as it is found in [[equations of motion]] for [[classical mechanics|classical mechanical]] systems.
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| The second time derivative of a vector field in cylindrical coordinates is given by:
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| :<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)
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| + \boldsymbol{\hat\theta} (\ddot A_\theta + A_r \ddot\theta + 2 \dot A_r \dot\theta - A_\theta \dot\theta^2)
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| + \mathbf{\hat z} \ddot A_z</math>
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| To understand this expression, we substitute A = P, where p is the vector (r, θ, z).
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| This means that <math>\mathbf{A} = \mathbf{P} = r \mathbf{\hat r} + z \mathbf{\hat z}</math>.
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| After substituting we get:
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| :<math>\ddot\mathbf{P} = \mathbf{\hat r} (\ddot r - r \dot\theta^2)
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| + \boldsymbol{\hat\theta} (r \ddot\theta + 2 \dot r \dot\theta)
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| + \mathbf{\hat z} \ddot z</math>
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| In mechanics, the terms of this expression are called:
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| :<math>\begin{align}
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| \ddot r \mathbf{\hat r} &= \mbox{central outward acceleration} \\
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| -r \dot\theta^2 \mathbf{\hat r} &= \mbox{centripetal acceleration} \\
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| r \ddot\theta \boldsymbol{\hat\theta} &= \mbox{angular acceleration} \\
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| 2 \dot r \dot\theta \boldsymbol{\hat\theta} &= \mbox{Coriolis effect} \\
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| \ddot z \mathbf{\hat z} &= \mbox{z-acceleration}
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| \end{align}</math>
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| See also: [[Centripetal force]], [[Angular acceleration]], [[Coriolis effect]].
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| == Spherical coordinate system ==
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| === Vector fields ===
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| Vectors are defined in [[spherical coordinates]] by (ρ,θ,φ), where
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| * ρ is the length of the vector,
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| * θ is the angle between the positive Z-axis and vector in question (0 ≤ θ ≤ π)
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| * φ is the angle between the projection of the vector onto the X-Y-plane and the positive X-axis (0 ≤ φ < 2π),
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| (ρ,θ,φ) is given in [[cartesian coordinates]] by:
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| :<math>\begin{bmatrix}\rho \\ \theta \\ \phi \end{bmatrix} =
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| \begin{bmatrix}
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| \sqrt{x^2 + y^2 + z^2} \\ \arccos(z / \rho) \\ \arctan(y / x)
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| \end{bmatrix},\ \ \ 0 \le \theta \le \pi,\ \ \ 0 \le \phi < 2\pi,
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| </math>
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| or inversely by:
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| :<math>\begin{bmatrix} x \\ y \\ z \end{bmatrix} =
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| \begin{bmatrix} \rho\sin\theta\cos\phi \\ \rho\sin\theta\sin\phi \\ \rho\cos\theta\end{bmatrix}.</math>
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| Any vector field can be written in terms of the unit vectors as:
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| :<math>\mathbf A = A_x\mathbf{\hat x} + A_y\mathbf{\hat y} + A_z\mathbf{\hat z}
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| = A_\rho\boldsymbol{\hat \rho} + A_\theta\boldsymbol{\hat \theta} + A_\phi\boldsymbol{\hat \phi}</math>
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| The spherical unit vectors are related to the cartesian unit vectors by:
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| :<math>\begin{bmatrix}\boldsymbol{\hat\rho} \\ \boldsymbol{\hat\theta} \\ \boldsymbol{\hat\phi} \end{bmatrix}
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| = \begin{bmatrix} \sin\theta\cos\phi & \sin\theta\sin\phi & \cos\theta \\
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| \cos\theta\cos\phi & \cos\theta\sin\phi & -\sin\theta \\
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| -\sin\phi & \cos\phi & 0 \end{bmatrix}
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| \begin{bmatrix} \mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix}</math>
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| * Note: the matrix is an [[orthogonal matrix]], that is, its inverse is simply its [[transpose]].
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| === Time derivative of a vector field ===
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| To find out how the vector field A changes in time we calculate the time derivatives.
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| In cartesian coordinates this is simply:
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| :<math>\mathbf{\dot A} = \dot A_x \mathbf{\hat x} + \dot A_y \mathbf{\hat y} + \dot A_z \mathbf{\hat z}</math>
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| However, in spherical coordinates this becomes:
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| :<math>\mathbf{\dot A} = \dot A_\rho \boldsymbol{\hat \rho} + A_\rho \boldsymbol{\dot{\hat \rho}}
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| + \dot A_\theta \boldsymbol{\hat\theta} + A_\theta \boldsymbol{\dot{\hat\theta}}
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| + \dot A_\phi \boldsymbol{\hat\phi} + A_\phi \boldsymbol{\dot{\hat\phi}}</math>
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| We need the time derivatives of the unit vectors.
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| They are given by:
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| :<math>\begin{align}
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| \boldsymbol{\dot{\hat \rho}} &= \dot\theta \boldsymbol{\hat\theta} + \dot\phi\sin\theta \boldsymbol{\hat\phi} \\
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| \boldsymbol{\dot{\hat\theta}} &= - \dot\theta \boldsymbol{\hat \rho} + \dot\phi\cos\theta \boldsymbol{\hat\phi} \\
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| \boldsymbol{\dot{\hat\phi}} &= - \dot\phi\sin\theta \boldsymbol{\hat\rho} - \dot\phi\cos\theta \boldsymbol{\hat\theta} \end{align}</math>
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| So the time derivative becomes:
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| :<math>\mathbf{\dot A} = \boldsymbol{\hat \rho} (\dot A_\rho - A_\theta \dot\theta - A_\phi \dot\phi \sin\theta)
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| + \boldsymbol{\hat\theta} (\dot A_\theta + A_\rho \dot\theta - A_\phi \dot\phi \cos\theta)
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| + \boldsymbol{\hat\phi} (\dot A_\phi + A_\rho \dot\phi \sin\theta + A_\theta \dot\phi \cos\theta)</math>
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| == See also ==
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| * [[Del in cylindrical and spherical coordinates]] for the specification of [[gradient]], [[divergence]], [[Curl (mathematics)|curl]], and [[laplacian]] in various coordinate systems.
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Vector Fields In Cylindrical And Spherical Coordinates}}
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| [[Category:Vector calculus]]
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| [[Category:Coordinate systems]]
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