Categories:
Mathematics ,
Physics .
Polar cylindrical coordinates
Polar cylindrical coordinates extend polar coordinates ( r , φ ) (r, \varphi) ( r , φ ) ( r , φ , z ) (r, \varphi, z) ( r , φ , z ) z z z cylindrical .
Cartesian coordinates ( x , y , z ) (x, y, z) ( x , y , z ) ( r , φ , z ) (r, \varphi, z) ( r , φ , z )
x = r cos φ y = r sin φ z = z \begin{aligned}
\boxed{
\begin{aligned}
x
&= r \cos{\varphi}
\\
y
&= r \sin{\varphi}
\\
z
&= z
\end{aligned}
}
\end{aligned} x y z = r cos φ = r sin φ = z Conversely, a point given in ( x , y , z ) (x, y, z) ( x , y , z ) ( r , φ , z ) (r, \varphi, z) ( r , φ , z ) a t a n 2 \mathtt{atan2} atan2
r = x 2 + y 2 φ = a t a n 2 ( y , x ) z = z \begin{aligned}
\boxed{
\begin{aligned}
r
&= \sqrt{x^2 + y^2}
\\
\varphi
&= \mathtt{atan2}(y, x)
\\
z
&= z
\end{aligned}
}
\end{aligned} r φ z = x 2 + y 2 = atan2 ( y , x ) = z Polar cylindrical coordinates form
an orthogonal curvilinear system ,
whose scale factors h r h_r h r h φ h_\varphi h φ h z h_z h z
h r e ^ r = e ^ x ∂ x ∂ r + e ^ y ∂ y ∂ r + e ^ z ∂ z ∂ r = e ^ x cos φ + e ^ y sin φ h φ e ^ φ = e ^ x ∂ x ∂ φ + e ^ y ∂ y ∂ φ + e ^ z ∂ z ∂ φ = − e ^ x r sin φ + e ^ y r cos φ h z e ^ z = e ^ x ∂ x ∂ z + e ^ y ∂ y ∂ z + e ^ z ∂ z ∂ z = e ^ z \begin{aligned}
h_r \vu{e}_r
&= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
\\
&= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
\\
h_\varphi \vu{e}_\varphi
&= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
\\
&= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi}
\\
h_z \vu{e}_z
&= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
\\
&= \vu{e}_z
\end{aligned} h r e ^ r h φ e ^ φ h z e ^ z = e ^ x ∂ r ∂ x + e ^ y ∂ r ∂ y + e ^ z ∂ r ∂ z = e ^ x cos φ + e ^ y sin φ = e ^ x ∂ φ ∂ x + e ^ y ∂ φ ∂ y + e ^ z ∂ φ ∂ z = − e ^ x r sin φ + e ^ y r cos φ = e ^ x ∂ z ∂ x + e ^ y ∂ z ∂ y + e ^ z ∂ z ∂ z = e ^ z By normalizing the local basis vectors
e ^ r \vu{e}_r e ^ r e ^ φ \vu{e}_\varphi e ^ φ e ^ z \vu{e}_z e ^ z
h r = 1 h φ = r h z = 1 e ^ r = e ^ x cos φ + e ^ y sin φ e ^ φ = − e ^ x sin φ + e ^ y cos φ e ^ z = e ^ z \begin{aligned}
\boxed{
\begin{aligned}
h_r
&= 1
\\
h_\varphi
&= r
\\
h_z
&= 1
\end{aligned}
}
\qquad\qquad
\boxed{
\begin{aligned}
\vu{e}_r
&= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
\\
\vu{e}_\varphi
&= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
\\
\vu{e}_z
&= \vu{e}_z
\end{aligned}
}
\end{aligned} h r h φ h z = 1 = r = 1 e ^ r e ^ φ e ^ z = e ^ x cos φ + e ^ y sin φ = − e ^ x sin φ + e ^ y cos φ = e ^ z Thanks to these scale factors, we can easily convert calculus from the Cartesian system
using the standard formulae for orthogonal curvilinear coordinates.
Differential elements
For line integrals,
the tangent vector element d ℓ \dd{\vb{\ell}} d ℓ
d ℓ = e ^ r d r + e ^ φ r d φ + e ^ z d z \begin{aligned}
\boxed{
\dd{\vb{\ell}}
= \vu{e}_r \dd{r}
+ \: \vu{e}_\varphi \: r \dd{\varphi}
+ \: \vu{e}_z \dd{z}
}
\end{aligned} d ℓ = e ^ r d r + e ^ φ r d φ + e ^ z d z For surface integrals,
the normal vector element d S \dd{\vb{S}} d S
d S = e ^ r r d φ d z + e ^ φ d r d z + e ^ z r d r d φ \begin{aligned}
\boxed{
\dd{\vb{S}}
= \vu{e}_r \: r \dd{\varphi} \dd{z}
+ \: \vu{e}_\varphi \dd{r} \dd{z}
+ \: \vu{e}_z \: r \dd{r} \dd{\varphi}
}
\end{aligned} d S = e ^ r r d φ d z + e ^ φ d r d z + e ^ z r d r d φ And for volume integrals,
the infinitesimal volume d V \dd{V} d V
d V = r d r d φ d z \begin{aligned}
\boxed{
\dd{V}
= r \dd{r} \dd{\varphi} \dd{z}
}
\end{aligned} d V = r d r d φ d z Common operations
The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
∇ f = e ^ r ∂ f ∂ r + e ^ φ 1 r ∂ f ∂ φ + e z ∂ f ∂ z \begin{aligned}
\boxed{
\nabla f
= \vu{e}_r \pdv{f}{r}
+ \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi}
+ \mathbf{e}_z \pdv{f}{z}
}
\end{aligned} ∇ f = e ^ r ∂ r ∂ f + e ^ φ r 1 ∂ φ ∂ f + e z ∂ z ∂ f ∇ ⋅ V = ∂ V r ∂ r + V r r + 1 r ∂ V φ ∂ φ + ∂ V z ∂ z \begin{aligned}
\boxed{
\nabla \cdot \vb{V}
= \pdv{V_r}{r} + \frac{V_r}{r}
+ \frac{1}{r} \pdv{V_\varphi}{\varphi}
+ \pdv{V_z}{z}
}
\end{aligned} ∇ ⋅ V = ∂ r ∂ V r + r V r + r 1 ∂ φ ∂ V φ + ∂ z ∂ V z ∇ × V = e ^ r ( 1 r ∂ V z ∂ φ − ∂ V φ ∂ z ) + e ^ φ ( ∂ V r ∂ z − ∂ V z ∂ r ) + e ^ z ( ∂ V φ ∂ r + V φ r − 1 r ∂ V r ∂ φ ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \times \vb{V}
&= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg)
\end{aligned}
}
\end{aligned} ∇ × V = e ^ r ( r 1 ∂ φ ∂ V z − ∂ z ∂ V φ ) + e ^ φ ( ∂ z ∂ V r − ∂ r ∂ V z ) + e ^ z ( ∂ r ∂ V φ + r V φ − r 1 ∂ φ ∂ V r ) ∇ 2 f = ∂ 2 f ∂ r 2 + 1 r ∂ f ∂ r + 1 r 2 ∂ 2 f ∂ φ 2 + ∂ 2 f ∂ z 2 \begin{aligned}
\boxed{
\nabla^2 f
= \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r}
+ \frac{1}{r^2} \pdvn{2}{f}{\varphi}
+ \pdvn{2}{f}{z}
}
\end{aligned} ∇ 2 f = ∂ r 2 ∂ 2 f + r 1 ∂ r ∂ f + r 2 1 ∂ φ 2 ∂ 2 f + ∂ z 2 ∂ 2 f Uncommon operations
Uncommon operations include:
the gradient of a divergence ∇ ( ∇ ⋅ V ) \nabla (\nabla \cdot \vb{V}) ∇ ( ∇ ⋅ V ) ∇ V \nabla \vb{V} ∇ V ( U ⋅ ∇ ) V (\vb{U} \cdot \nabla) \vb{V} ( U ⋅ ∇ ) V U \vb{U} U ∇ 2 V \nabla^2 \vb{V} ∇ 2 V ∇ ⋅ T ‾ ‾ \nabla \cdot \overline{\overline{\vb{T}}} ∇ ⋅ T
∇ ( ∇ ⋅ V ) = e ^ r ( ∂ 2 V r ∂ r 2 + 1 r ∂ 2 V φ ∂ r ∂ φ + ∂ 2 V z ∂ r ∂ z + 1 r ∂ V r ∂ r − 1 r 2 ∂ V φ ∂ φ − V r r 2 ) + e ^ φ ( 1 r ∂ 2 V r ∂ φ ∂ r + 1 r 2 ∂ 2 V φ ∂ φ 2 + 1 r ∂ 2 V z ∂ φ ∂ z + 1 r 2 ∂ V r ∂ φ ) + e ^ z ( ∂ 2 V r ∂ z ∂ r + 1 r ∂ 2 V φ ∂ z ∂ φ + ∂ 2 V z ∂ z 2 + 1 r ∂ V r ∂ z ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla (\nabla \cdot \vb{V})
&= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z}
+ \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
+ \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg)
\end{aligned}
}
\end{aligned} ∇ ( ∇ ⋅ V ) = e ^ r ( ∂ r 2 ∂ 2 V r + r 1 ∂ r ∂ φ ∂ 2 V φ + ∂ r ∂ z ∂ 2 V z + r 1 ∂ r ∂ V r − r 2 1 ∂ φ ∂ V φ − r 2 V r ) + e ^ φ ( r 1 ∂ φ ∂ r ∂ 2 V r + r 2 1 ∂ φ 2 ∂ 2 V φ + r 1 ∂ φ ∂ z ∂ 2 V z + r 2 1 ∂ φ ∂ V r ) + e ^ z ( ∂ z ∂ r ∂ 2 V r + r 1 ∂ z ∂ φ ∂ 2 V φ + ∂ z 2 ∂ 2 V z + r 1 ∂ z ∂ V r ) ∇ V = e ^ r e ^ r ∂ V r ∂ r + e ^ r e ^ φ ∂ V φ ∂ r + e ^ r e ^ z ∂ V z ∂ r + e ^ φ e ^ r ( 1 r ∂ V r ∂ φ − V φ r ) + e ^ φ e ^ φ ( 1 r ∂ V φ ∂ φ + V r r ) + e ^ φ e ^ z 1 r ∂ V z ∂ φ + e ^ z e ^ r ∂ V r ∂ z + e ^ z e ^ φ ∂ V φ ∂ z + e ^ z e ^ z ∂ V z ∂ z \begin{aligned}
\boxed{
\begin{aligned}
\nabla \vb{V}
&= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r}
+ \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
+ \vu{e}_r \vu{e}_z \pdv{V_z}{r}
\\
&\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
+ \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg)
+ \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi}
\\
&\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z}
+ \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z}
+ \vu{e}_z \vu{e}_z \pdv{V_z}{z}
\end{aligned}
}
\end{aligned} ∇ V = e ^ r e ^ r ∂ r ∂ V r + e ^ r e ^ φ ∂ r ∂ V φ + e ^ r e ^ z ∂ r ∂ V z + e ^ φ e ^ r ( r 1 ∂ φ ∂ V r − r V φ ) + e ^ φ e ^ φ ( r 1 ∂ φ ∂ V φ + r V r ) + e ^ φ e ^ z r 1 ∂ φ ∂ V z + e ^ z e ^ r ∂ z ∂ V r + e ^ z e ^ φ ∂ z ∂ V φ + e ^ z e ^ z ∂ z ∂ V z ( U ⋅ ∇ ) V = e ^ r ( U r ∂ V r ∂ r + U φ r ∂ V r ∂ φ + U z ∂ V r ∂ z − U φ V φ r ) + e ^ φ ( U r ∂ V φ ∂ r + U φ r ∂ V φ ∂ φ + U z ∂ V φ ∂ z + U φ V r r ) + e ^ z ( U r ∂ V z ∂ r + U φ r ∂ V z ∂ φ + U z ∂ V z ∂ z ) \begin{aligned}
\boxed{
\begin{aligned}
(\vb{U} \cdot \nabla) \vb{V}
&= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z}
- \frac{U_\varphi V_\varphi}{r} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z}
+ \frac{U_\varphi V_r}{r} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg)
\end{aligned}
}
\end{aligned} ( U ⋅ ∇ ) V = e ^ r ( U r ∂ r ∂ V r + r U φ ∂ φ ∂ V r + U z ∂ z ∂ V r − r U φ V φ ) + e ^ φ ( U r ∂ r ∂ V φ + r U φ ∂ φ ∂ V φ + U z ∂ z ∂ V φ + r U φ V r ) + e ^ z ( U r ∂ r ∂ V z + r U φ ∂ φ ∂ V z + U z ∂ z ∂ V z ) ∇ 2 V = e ^ r ( ∂ 2 V r ∂ r 2 + 1 r 2 ∂ 2 V r ∂ φ 2 + ∂ 2 V r ∂ z 2 + 1 r ∂ V r ∂ r − 2 r 2 ∂ V φ ∂ φ − V r r 2 ) + e ^ φ ( ∂ 2 V φ ∂ r 2 + 1 r 2 ∂ 2 V φ ∂ φ 2 + ∂ 2 V φ ∂ z 2 + 2 r 2 ∂ V r ∂ φ + 1 r ∂ V φ ∂ r − V φ r 2 ) + e ^ z ( ∂ 2 V z ∂ r 2 + 1 r 2 ∂ 2 V z ∂ φ 2 + ∂ 2 V z ∂ z 2 + 1 r ∂ V z ∂ r ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla^2 \vb{V}
&= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z}
+ \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z}
+ \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi}
+ \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg)
\end{aligned}
}
\end{aligned} ∇ 2 V = e ^ r ( ∂ r 2 ∂ 2 V r + r 2 1 ∂ φ 2 ∂ 2 V r + ∂ z 2 ∂ 2 V r + r 1 ∂ r ∂ V r − r 2 2 ∂ φ ∂ V φ − r 2 V r ) + e ^ φ ( ∂ r 2 ∂ 2 V φ + r 2 1 ∂ φ 2 ∂ 2 V φ + ∂ z 2 ∂ 2 V φ + r 2 2 ∂ φ ∂ V r + r 1 ∂ r ∂ V φ − r 2 V φ ) + e ^ z ( ∂ r 2 ∂ 2 V z + r 2 1 ∂ φ 2 ∂ 2 V z + ∂ z 2 ∂ 2 V z + r 1 ∂ r ∂ V z ) ∇ ⋅ T ‾ ‾ = e ^ r ( ∂ T r r ∂ r + 1 r ∂ T φ r ∂ φ + ∂ T z r ∂ z + T r r r − T φ φ r ) + e ^ φ ( ∂ T r φ ∂ r + 1 r ∂ T φ φ ∂ φ + ∂ T z φ ∂ z + T r φ r + T φ r r ) + e ^ z ( ∂ T r z ∂ r + 1 r ∂ T φ z ∂ φ + ∂ T z z ∂ z + T r z r ) \begin{aligned}
\boxed{
\begin{aligned}
\nabla \cdot \overline{\overline{\mathbf{T}}}
&= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z}
+ \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
\\
&\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z}
+ \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg)
\\
&\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z}
+ \frac{T_{rz}}{r} \bigg)
\end{aligned}
}
\end{aligned} ∇ ⋅ T = e ^ r ( ∂ r ∂ T rr + r 1 ∂ φ ∂ T φ r + ∂ z ∂ T zr + r T rr − r T φφ ) + e ^ φ ( ∂ r ∂ T r φ + r 1 ∂ φ ∂ T φφ + ∂ z ∂ T z φ + r T r φ + r T φ r ) + e ^ z ( ∂ r ∂ T rz + r 1 ∂ φ ∂ T φ z + ∂ z ∂ T zz + r T rz ) References
M.L. Boas,
Mathematical methods in the physical sciences , 2nd edition,
Wiley.
B. Lautrup,
Physics of continuous matter: exotic and everyday phenomena in the macroscopic world , 2nd edition,
CRC Press.