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----
-title: "Cylindrical polar coordinates"
-sort_title: "Cylindrical polar coordinates"
-date: 2021-07-26
-categories:
-- Mathematics
-- Physics
-layout: "concept"
----
-
-**Cylindrical polar coordinates** extend polar coordinates $$(r, \varphi)$$ to 3D,
-by describing the location of a point in space
-using the variables $$(r, \varphi, z)$$.
-The $$z$$-axis is unchanged from the Cartesian system,
-hence the name *cylindrical*.
-
-[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
-and the cylindrical system $$(r, \varphi, z)$$ are related by:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            x
-            &= r \cos{\varphi}
-            \\
-            y
-            &= r \sin{\varphi}
-            \\
-            z
-            &= z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-Conversely, a point given in $$(x, y, z)$$
-can be converted to $$(r, \varphi, z)$$ using these formulae,
-where $$\mathtt{atan2}$$ is the 2-argument arctangent,
-which is needed to handle the signs correctly:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            r
-            &= \sqrt{x^2 + y^2}
-            \\
-            \varphi
-            &= \mathtt{atan2}(y, x)
-            \\
-            z
-            &= z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-Cylindrical polar coordinates form
-an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
-whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need.
-To get those, we calculate the unnormalized local basis:
-
-$$\begin{aligned}
-    h_r \vu{e}_r
-    &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r}
-    \\
-    &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
-    \\
-    h_\varphi \vu{e}_\varphi
-    &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi}
-    \\
-    &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi}
-    \\
-    h_z \vu{e}_z
-    &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
-    \\
-    &= \vu{e}_z
-\end{aligned}$$
-
-By normalizing the **local basis vectors**
-$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$,
-we arrive at these expressions:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            h_r
-            &= 1
-            \\
-            h_\varphi
-            &= r
-            \\
-            h_z
-            &= 1
-        \end{aligned}
-    }
-    \qquad\qquad
-    \boxed{
-        \begin{aligned}
-            \vu{e}_r
-            &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi}
-            \\
-            \vu{e}_\varphi
-            &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi}
-            \\
-            \vu{e}_z
-            &= \vu{e}_z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-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 $$\dd{\vb{\ell}}$$ for a curve is as follows:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{\vb{\ell}}
-        = \vu{e}_r \dd{r}
-        + \: \vu{e}_\varphi \: r \dd{\varphi}
-        + \: \vu{e}_z \dd{z}
-    }
-\end{aligned}$$
-
-For surface integrals,
-the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{\vb{S}}
-        = \vu{e}_r \: r \dd{\varphi} \dd{z}
-        + \: \vu{e}_\varphi \dd{r} \dd{z}
-        + \: \vu{e}_z \: r \dd{r} \dd{\varphi}
-    }
-\end{aligned}$$
-
-And for volume integrals,
-the infinitesimal volume $$\dd{V}$$ takes the following form:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{V}
-        = r \dd{r} \dd{\varphi} \dd{z}
-    }
-\end{aligned}$$
-
-
-
-## Common operations
-
-The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
-
-$$\begin{aligned}
-    \boxed{
-        \nabla f
-        = \vu{e}_r \pdv{f}{r}
-        + \vu{e}_\varphi \frac{1}{r} \pdv{f}{\varphi}
-        + \mathbf{e}_z \pdv{f}{z}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \nabla \cdot \vb{V}
-        = \pdv{V_r}{r} + \frac{V_r}{r}
-        + \frac{1}{r} \pdv{V_\varphi}{\varphi}
-        + \pdv{V_z}{z}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \times \vb{V}
-            &= \quad \vu{e}_r \bigg( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \bigg)
-            \\
-            &\quad\: + \vu{e}_\varphi \bigg( \pdv{V_r}{z} - \pdv{V_z}{r} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \pdv{V_\varphi}{r} + \frac{V_\varphi}{r} - \frac{1}{r} \pdv{V_r}{\varphi} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \nabla^2 f
-        = \pdvn{2}{f}{r} + \frac{1}{r} \pdv{f}{r}
-        + \frac{1}{r^2} \pdvn{2}{f}{\varphi}
-        + \pdvn{2}{f}{z}
-    }
-\end{aligned}$$
-
-
-
-## Uncommon operations
-
-Uncommon operations include:
-the gradient of a divergence $$\nabla (\nabla \cdot \vb{V})$$,
-the gradient of a vector $$\nabla \vb{V}$$,
-the advection of a vector $$(\vb{U} \cdot \nabla) \vb{V}$$ with respect to $$\vb{U}$$,
-the Laplacian of a vector $$\nabla^2 \vb{V}$$,
-and the divergence of a 2nd-order tensor $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla (\nabla \cdot \vb{V})
-            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \mpdv{V_\varphi}{r}{\varphi} + \mpdv{V_z}{r}{z}
-            + \frac{1}{r} \pdv{V_r}{r} - \frac{1}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
-            \\
-            &\quad\: + \vu{e}_\varphi \bigg( \frac{1}{r} \mpdv{V_r}{\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi}
-            + \frac{1}{r} \mpdv{V_z}{\varphi}{z} + \frac{1}{r^2} \pdv{V_r}{\varphi} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \mpdv{V_r}{z}{r} + \frac{1}{r} \mpdv{V_\varphi}{z}{\varphi} + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_r}{z} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \vb{V}
-            &= \quad \vu{e}_r \vu{e}_r \pdv{V_r}{r}
-            + \vu{e}_r \vu{e}_\varphi \pdv{V_\varphi}{r}
-            + \vu{e}_r \vu{e}_z \pdv{V_z}{r}
-            \\
-            &\quad\: + \vu{e}_\varphi \vu{e}_r \bigg( \frac{1}{r} \pdv{V_r}{\varphi} - \frac{V_\varphi}{r} \bigg)
-            + \vu{e}_\varphi \vu{e}_\varphi \bigg( \frac{1}{r} \pdv{V_\varphi}{\varphi} + \frac{V_r}{r} \bigg)
-            + \vu{e}_\varphi \vu{e}_z \frac{1}{r} \pdv{V_z}{\varphi}
-            \\
-            &\quad\: + \vu{e}_z \vu{e}_r \pdv{V_r}{z}
-            + \vu{e}_z \vu{e}_\varphi \pdv{V_\varphi}{z}
-            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            (\vb{U} \cdot \nabla) \vb{V}
-            &= \quad \vu{e}_r \bigg( U_r \pdv{V_r}{r} + \frac{U_\varphi}{r} \pdv{V_r}{\varphi} + U_z \pdv{V_r}{z}
-            - \frac{U_\varphi V_\varphi}{r} \bigg)
-            \\
-            &\quad\: + \vu{e}_\varphi \bigg( U_r \pdv{V_\varphi}{r} + \frac{U_\varphi}{r} \pdv{V_\varphi}{\varphi} + U_z \pdv{V_\varphi}{z}
-            + \frac{U_\varphi V_r}{r} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( U_r \pdv{V_z}{r} + \frac{U_\varphi}{r} \pdv{V_z}{\varphi} + U_z \pdv{V_z}{z} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla^2 \vb{V}
-            &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + \pdvn{2}{V_r}{z}
-            + \frac{1}{r} \pdv{V_r}{r} - \frac{2}{r^2} \pdv{V_\varphi}{\varphi} - \frac{V_r}{r^2} \bigg)
-            \\
-            &\quad\: + \vu{e}_\varphi \bigg( \pdvn{2}{V_\varphi}{r} + \frac{1}{r^2} \pdvn{2}{V_\varphi}{\varphi} + \pdvn{2}{V_\varphi}{z}
-            + \frac{2}{r^2} \pdv{V_r}{\varphi} + \frac{1}{r} \pdv{V_\varphi}{r} - \frac{V_\varphi}{r^2} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi}
-            + \pdvn{2}{V_z}{z} + \frac{1}{r} \pdv{V_z}{r} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \cdot \overline{\overline{\mathbf{T}}}
-            &= \quad \vu{e}_r \bigg( \pdv{T_{rr}}{r} + \frac{1}{r} \pdv{T_{\varphi r}}{\varphi} + \pdv{T_{zr}}{z}
-            + \frac{T_{rr}}{r} - \frac{T_{\varphi \varphi}}{r} \bigg)
-            \\
-            &\quad\: + \vu{e}_\varphi \bigg( \pdv{T_{r \varphi}}{r} + \frac{1}{r} \pdv{T_{\varphi \varphi}}{\varphi} + \pdv{T_{z \varphi}}{z}
-            + \frac{T_{r \varphi}}{r} + \frac{T_{\varphi r}}{r} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \pdv{T_{rz}}{r} + \frac{1}{r} \pdv{T_{\varphi z}}{\varphi} + \pdv{T_{zz}}{z}
-            + \frac{T_{rz}}{r} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-
-
-## References
-1.  M.L. Boas,
-    *Mathematical methods in the physical sciences*, 2nd edition,
-    Wiley.
-2.  B. Lautrup,
-    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
-    CRC Press.