Vector fields in cylindrical and spherical coordinates

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 $\theta$ 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.^[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:

{\begin{bmatrix}r\\\theta \\z\end{bmatrix}}={\begin{bmatrix}{\sqrt {x^{2}+y^{2}}}\\\operatorname {arctan}(y/x)\\z\end{bmatrix}},\ \ \ 0\leq \theta <2\pi ,

or inversely by:

{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}r\cos \theta \\r\sin \theta \\z\end{bmatrix}}.

Any vector field can be written in terms of the unit vectors as:

{\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}}}

The cylindrical unit vectors are related to the cartesian unit vectors by:

{\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}}

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 ( ${\dot {{\mathbf {A}}}}$ ). In cartesian coordinates this is simply:

{\dot {{\mathbf {A}}}}={\dot {A}}_{x}{\hat {{\mathbf {x}}}}+{\dot {A}}_{y}{\hat {{\mathbf {y}}}}+{\dot {A}}_{z}{\hat {{\mathbf {z}}}}

However, in cylindrical coordinates this becomes:

{\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}}}}}}

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

{\begin{aligned}{\dot {{\hat {{\mathbf {r}}}}}}&={\dot {\theta }}{\hat {{\boldsymbol {\theta }}}}\\{\dot {{\hat {{\boldsymbol {\theta }}}}}}&=-{\dot \theta }{\hat {{\mathbf {r}}}}\\{\dot {{\hat {{\mathbf {z}}}}}}&=0\end{aligned}}

So the time derivative simplifies to:

{\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}

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 {{\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}

To understand this expression, we substitute A = P, where p is the vector (r, θ, z).

This means that ${\mathbf {A}}={\mathbf {P}}=r{\mathbf {{\hat r}}}+z{\mathbf {{\hat z}}}$ .

After substituting we get:

{\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}

In mechanics, the terms of this expression are called:

{\begin{aligned}{\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{aligned}}

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 the vector in question (0 ≤ θ ≤ π), and
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:

{\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\leq \theta \leq \pi ,\ \ \ 0\leq \phi <2\pi ,

or inversely by:

{\begin{bmatrix}x\\y\\z\end{bmatrix}}={\begin{bmatrix}\rho \sin \theta \cos \phi \\\rho \sin \theta \sin \phi \\\rho \cos \theta \end{bmatrix}}.

Any vector field can be written in terms of the unit vectors as:

{\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 }}}

The spherical unit vectors are related to the cartesian unit vectors by:

{\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}}

Note: the matrix is an orthogonal matrix, that is, its inverse is simply its transpose.

So the cartesian unit vectors are related to the spherical unit vectors by:

\begin{bmatrix}\mathbf{\hat x} \\ \mathbf{\hat y} \\ \mathbf{\hat z} \end{bmatrix} = \begin{bmatrix} \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{bmatrix} \begin{bmatrix} \boldsymbol{\hat\rho} \\ \boldsymbol{\hat\theta} \\ \boldsymbol{\hat\phi} \end{bmatrix}

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 {{\dot A}}}={\dot A}_{x}{\mathbf {{\hat x}}}+{\dot A}_{y}{\mathbf {{\hat y}}}+{\dot A}_{z}{\mathbf {{\hat z}}}

However, in spherical coordinates this becomes:

{\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 }}}}}

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

{\begin{aligned}{\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{aligned}}

So the time derivative becomes:

{\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 )

References

↑ Wolfram Mathworld, spherical coordinates

This article is issued from Wikipedia - version of the 11/25/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.

Vector fields in cylindrical and spherical coordinates

Cylindrical coordinate system

Vector fields

Time derivative of a vector field

Second time derivative of a vector field

Spherical coordinate system

Vector fields

Time derivative of a vector field

See also

References