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+---
+title: "Polar cylindrical coordinates"
+sort_title: "Polar cylindrical coordinates"
+date: 2021-07-26
+categories:
+- Mathematics
+- Physics
+layout: "concept"
+---
+
+**Polar cylindrical 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:
+
+$$\begin{aligned}
+    \boxed{
+        \begin{aligned}
+            r
+            &= \sqrt{x^2 + y^2}
+            \\
+            \varphi
+            &= \mathtt{atan2}(y, x)
+            \\
+            z
+            &= z
+        \end{aligned}
+    }
+\end{aligned}$$
+
+Polar cylindrical 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.