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----
-title: "Cylindrical parabolic coordinates"
-sort_title: "Cylindrical parabolic coordinates"
-date: 2021-03-04
-categories:
-- Mathematics
-- Physics
-layout: "concept"
----
-
-**Cylindrical parabolic coordinates** extend parabolic coordinates $$(\sigma, \tau)$$ to 3D,
-by describing a point in space using the variables $$(\sigma, \tau, z)$$.
-The $$z$$-axis is the same as in the Cartesian system, (hence the name *cylindrical*),
-while the coordinate lines of $$\sigma$$ and $$\tau$$ are confocal parabolas.
-
-[Cartesian coordinates](/know/concept/cartesian-coordinates/) $$(x, y, z)$$
-and this system $$(\sigma, \tau, z)$$ are related by:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            x
-            &= \frac{1}{2} (\tau^2 - \sigma^2)
-            \\
-            y
-            &= \sigma \tau
-            \\
-            z
-            &= z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-Conversely, a point given in $$(x, y, z)$$ can be converted
-to $$(\sigma, \tau, z)$$ using these formulae:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \sigma
-            &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x}
-            \\
-            \tau
-            &= \sqrt{\sqrt{x^2 + y^2} + x}
-            \\
-            z
-            &= z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-Cylindrical parabolic coordinates form
-an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/),
-whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need.
-To get those, we calculate the unnormalized local basis:
-
-$$\begin{aligned}
-    h_\sigma \vu{e}_\sigma
-    &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma}
-    \\
-    &= - \vu{e}_x \sigma + \vu{e}_y \tau
-    \\
-    h_\tau \vu{e}_\tau
-    &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau}
-    \\
-    &= \vu{e}_x \tau + \vu{e}_y \sigma
-    \\
-    h_\sigma \vu{e}_\sigma
-    &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z}
-    \\
-    &= \vu{e}_z
-\end{aligned}$$
-
-By normalizing the **local basis vectors**
-$$\vu{e}_\sigma$$, $$\vu{e}_\tau$$ and $$\vu{e}_z$$,
-we arrive at these expressions,
-where we have defined the abbreviation $$\rho$$ for convenience:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            h_\sigma
-            &= \rho
-            \equiv \sqrt{\sigma^2 + \tau^2}
-            \\
-            h_\tau
-            &= \rho
-            \equiv \sqrt{\sigma^2 + \tau^2}
-            \\
-            h_z
-            &= 1
-        \end{aligned}
-    }
-    \qquad\qquad
-    \boxed{
-        \begin{aligned}
-            \vu{e}_\sigma
-            &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho}
-            \\
-            \vu{e}_\tau
-            &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho}
-            \\
-            \vu{e}_z
-            &= \vu{e}_z
-        \end{aligned}
-    }
-\end{aligned}$$
-
-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:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{\vb{\ell}}
-        = \vu{e}_\sigma \: \rho \dd{\sigma} + \: \vu{e}_\tau \: \rho \dd{\tau} + \: \vu{e}_z \dd{z}
-    }
-\end{aligned}$$
-
-For surface integrals,
-the normal vector element $$\dd{\vb{S}}$$ for a surface is given by:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{\vb{S}}
-        = \vu{e}_\sigma \: \rho \dd{\tau} \dd{z} + \: \vu{e}_\tau \: \rho \dd{\sigma} \dd{z} + \: \vu{e}_z \: \rho^2 \dd{\sigma} \dd{\tau}
-    }
-\end{aligned}$$
-
-And for volume integrals,
-the infinitesimal volume $$\dd{V}$$ takes the following form:
-
-$$\begin{aligned}
-    \boxed{
-        \dd{V}
-        = \rho^2 \dd{\sigma} \dd{\tau} \dd{z}
-    }
-\end{aligned}$$
-
-
-
-## Common operations
-
-The basic vector operations (gradient, divergence, curl and Laplacian) are given by:
-
-$$\begin{aligned}
-    \boxed{
-        \nabla f
-        = \vu{e}_\sigma \frac{1}{\rho} \pdv{f}{\sigma}
-        + \vu{e}_\tau \frac{1}{\rho} \pdv{f}{\tau}
-        + \vu{e}_z \pdv{f}{z}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \nabla \cdot \vb{V}
-        = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3}
-        + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3}
-        + \pdv{V_z}{z}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \times \vb{V}
-            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg)
-            \\
-            &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3}
-            - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \nabla^2 f
-        = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z}
-    }
-\end{aligned}$$
-
-
-
-## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$:
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla (\nabla \cdot \vb{V})
-            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau}
-            + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z}
-            \\
-            &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau}
-            + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg)
-            \\
-            &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau}
-            + \frac{1}{\rho} \mpdv{V_z}{\tau}{z}
-            \\
-            &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau}
-            - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z}
-            + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \vb{V}
-            &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg)
-            + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg)
-            + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma}
-            \\
-            &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg)
-            + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg)
-            + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau}
-            \\
-            &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z}
-            + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z}
-            + \vu{e}_z \vu{e}_z \pdv{V_z}{z}
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            (\vb{U} \cdot \nabla) \vb{V}
-            &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z}
-            + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg)
-            \\
-            &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z}
-            + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla^2 \vb{V}
-            &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z}
-            + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg)
-            \\
-            &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z}
-            - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg)
-            \\
-            &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-$$\begin{aligned}
-    \boxed{
-        \begin{aligned}
-            \nabla \cdot \overline{\overline{\mathbf{T}}}
-            &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z}
-            + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3}
-            + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg)
-            \\
-            &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z}
-            - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3}
-            + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg)
-            \\
-            &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z}
-            + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg)
-        \end{aligned}
-    }
-\end{aligned}$$
-
-
-
-## References
-1.  M.L. Boas,
-    *Mathematical methods in the physical sciences*, 2nd edition,
-    Wiley.