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diff --git a/source/know/concept/cylindrical-parabolic-coordinates/index.md b/source/know/concept/cylindrical-parabolic-coordinates/index.md index 766c9b6..58358dd 100644 --- a/source/know/concept/cylindrical-parabolic-coordinates/index.md +++ b/source/know/concept/cylindrical-parabolic-coordinates/index.md @@ -8,82 +8,97 @@ categories: layout: "concept" --- -**Cylindrical parabolic coordinates** are a coordinate system -that describes a point in space using three coordinates $$(\sigma, \tau, z)$$. -The $$z$$-axis is unchanged from the Cartesian system, -hence it is called a *cylindrical* system. -In the $$z$$-isoplane, however, confocal parabolas are used. -These coordinates can be converted to the Cartesian $$(x, y, z)$$ as follows: +**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: $$\begin{aligned} \boxed{ - x = \frac{1}{2} (\tau^2 - \sigma^2 ) - \qquad - y = \sigma \tau - \qquad - z = z + \begin{aligned} + x + &= \frac{1}{2} (\tau^2 - \sigma^2) + \\ + y + &= \sigma \tau + \\ + z + &= z + \end{aligned} } \end{aligned}$$ -Converting the other way is a bit trickier. -It can be done by solving the following equations, -and potentially involves some fiddling with signs: +Conversely, a point given in $$(x, y, z)$$ can be converted +to $$(\sigma, \tau, z)$$ using these formulae: $$\begin{aligned} - 2 x - = \frac{y^2}{\sigma^2} - \sigma^2 - \qquad \qquad - 2 x - = - \frac{y^2}{\tau^2} + \tau^2 + \boxed{ + \begin{aligned} + \sigma + &= \sgn(x) \sqrt{\sqrt{x^2 + y^2} - x} + \\ + \tau + &= \sqrt{\sqrt{x^2 + y^2} + x} + \\ + z + &= z + \end{aligned} + } \end{aligned}$$ Cylindrical parabolic coordinates form an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), -so we would like to find its scale factors $$h_\sigma$$, $$h_\tau$$ and $$h_z$$. -The differentials of the Cartesian coordinates are as follows: +whose **scale factors** $$h_\sigma$$, $$h_\tau$$ and $$h_z$$ we need. +To get those, we calculate the unnormalized local basis: $$\begin{aligned} - \dd{x} = - \sigma \dd{\sigma} + \tau \dd{\tau} - \qquad - \dd{y} = \tau \dd{\sigma} + \sigma \dd{\tau} - \qquad - \dd{z} = \dd{z} + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{\sigma} + \vu{e}_y \pdv{y}{\sigma} + \vu{e}_z \pdv{z}{\sigma} + \\ + &= - \vu{e}_x \sigma + \vu{e}_y \tau + \\ + h_\tau \vu{e}_\tau + &= \vu{e}_x \pdv{x}{\tau} + \vu{e}_y \pdv{y}{\tau} + \vu{e}_z \pdv{z}{\tau} + \\ + &= \vu{e}_x \tau + \vu{e}_y \sigma + \\ + h_\sigma \vu{e}_\sigma + &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} + \\ + &= \vu{e}_z \end{aligned}$$ -We calculate the line segment $$\dd{\ell}^2$$, -skipping many terms thanks to orthogonality: - -$$\begin{aligned} - \dd{\ell}^2 - &= (\sigma^2 + \tau^2) \:\dd{\sigma}^2 + (\tau^2 + \sigma^2) \:\dd{\tau}^2 + \dd{z}^2 -\end{aligned}$$ - -From this, we can directly read off the scale factors $$h_\sigma^2$$, $$h_\tau^2$$ and $$h_z^2$$, -which turn out to be: +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: $$\begin{aligned} \boxed{ - h_\sigma = \sqrt{\sigma^2 + \tau^2} - \qquad - h_\tau = \sqrt{\sigma^2 + \tau^2} - \qquad - h_z = 1 + \begin{aligned} + h_\sigma + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_\tau + &= \rho + \equiv \sqrt{\sigma^2 + \tau^2} + \\ + h_z + &= 1 + \end{aligned} } -\end{aligned}$$ - -With these scale factors, we can use -the general formulae for orthogonal curvilinear coordinates -to easily to convert things from the Cartesian system. -The basis vectors are: - -$$\begin{aligned} + \qquad\qquad \boxed{ \begin{aligned} \vu{e}_\sigma - &= \frac{- \sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y + &= - \vu{e}_x \frac{\sigma}{\rho} + \vu{e}_y \frac{\tau}{\rho} \\ \vu{e}_\tau - &= \frac{\tau}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_x + \frac{\sigma}{\sqrt{\sigma^2 + \tau^2}} \vu{e}_y + &= \vu{e}_x \frac{\tau}{\rho} + \vu{e}_y \frac{\sigma}{\rho} \\ \vu{e}_z &= \vu{e}_z @@ -91,13 +106,54 @@ $$\begin{aligned} } \end{aligned}$$ -The basic vector operations (gradient, divergence, Laplacian and curl) are given by: +Thanks to these scale factors, we can easily convert calculus from the Cartesian system +using the standard formulae for orthogonal curvilinear coordinates. + + + +## Differential elements + +For line integrals, +the tangent vector element $$\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 - = \frac{\vu{e}_\sigma}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\sigma} - + \frac{\vu{e}_\tau}{\sqrt{\sigma^2 + \tau^2}} \pdv{f}{\tau} + = \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}$$ @@ -105,78 +161,135 @@ $$\begin{aligned} $$\begin{aligned} \boxed{ \nabla \cdot \vb{V} - = \frac{1}{\sigma^2 + \tau^2} - \Big( \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\sigma} + \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) + \pdv{V_z}{z} + = \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\sigma V_\sigma}{\rho^3} + + \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\tau V_\tau}{\rho^3} + + \pdv{V_z}{z} + } +\end{aligned}$$ + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \pdv{V_\sigma}{z} - \frac{1}{\rho} \pdv{V_z}{\sigma} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} + \frac{\sigma V_\tau}{\rho^3} + - \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\tau V_\sigma}{\rho^3} \bigg) + \end{aligned} } \end{aligned}$$ $$\begin{aligned} \boxed{ \nabla^2 f - = \frac{1}{\sigma^2 + \tau^2} \Big( \pdvn{2}{f}{\sigma} + \pdvn{2}{f}{\tau} \Big) + \pdvn{2}{f}{z} + = \frac{1}{\rho^2} \pdvn{2}{f}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{f}{\tau} + \pdvn{2}{f}{z} } \end{aligned}$$ + + +## 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 $$\nabla \cdot \overline{\overline{\vb{T}}}$$: + $$\begin{aligned} \boxed{ \begin{aligned} - \nabla \times \vb{V} - &= \vu{e}_\sigma \Big( \frac{\vu{e}_1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\tau} - \pdv{V_\tau}{z} \Big) + \nabla (\nabla \cdot \vb{V}) + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \mpdv{V_\tau}{\sigma}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\sigma}{z} + \\ + &\qquad\qquad + \frac{\tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{\sigma}{\rho^4} \pdv{V_\tau}{\tau} + + \frac{\rho^2 - 3 \sigma^2}{\rho^6} V_\sigma - \frac{3 \sigma \tau V_\tau}{\rho^6} \bigg) \\ - &+ \vu{e}_\tau \Big( \pdv{V_\sigma}{z} - \frac{1}{\sqrt{\sigma^2 + \tau^2}} \pdv{V_z}{\sigma} \Big) + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \mpdv{V_\sigma}{\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + + \frac{1}{\rho} \mpdv{V_z}{\tau}{z} \\ - &+ \frac{\vu{e}_z}{\sigma^2 + \tau^2} - \Big( \pdv{(V_\tau \sqrt{\sigma^2 + \tau^2})}{\sigma} - \pdv{(V_\sigma \sqrt{\sigma^2 + \tau^2})}{\tau} \Big) + &\qquad\qquad - \frac{\tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{\sigma}{\rho^4} \pdv{V_\sigma}{\tau} + - \frac{3 \sigma \tau V_\sigma}{\rho^6} + \frac{\rho^2 - 3 \tau^2}{\rho^6} V_\tau \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho} \mpdv{V_\sigma}{z}{\sigma} + \frac{1}{\rho} \mpdv{V_\tau}{z}{\tau} + \pdvn{2}{V_z}{z} + + \frac{\sigma}{\rho^3} \pdv{V_\sigma}{z} + \frac{\tau}{\rho^3} \pdv{V_\tau}{z} \bigg) \end{aligned} } \end{aligned}$$ -The differential element of volume $$\dd{V}$$ -in cylindrical parabolic coordinates is given by: - $$\begin{aligned} \boxed{ - \dd{V} = (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} \dd{z} + \begin{aligned} + \nabla \vb{V} + &= \quad \vu{e}_\sigma \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\sigma} + \frac{\tau V_\tau}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\sigma} - \frac{\tau V_\sigma}{\rho^3} \bigg) + + \vu{e}_\sigma \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\sigma} + \\ + &\quad\: + \vu{e}_\tau \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{V_\sigma}{\tau} - \frac{\sigma V_\tau}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{V_\tau}{\tau} + \frac{\sigma V_\sigma}{\rho^3} \bigg) + + \vu{e}_\tau \vu{e}_z \frac{1}{\rho} \pdv{V_z}{\tau} + \\ + &\quad\: + \vu{e}_z \vu{e}_\sigma \pdv{V_\sigma}{z} + + \vu{e}_z \vu{e}_\tau \pdv{V_\tau}{z} + + \vu{e}_z \vu{e}_z \pdv{V_z}{z} + \end{aligned} } \end{aligned}$$ -The differential elements of the isosurfaces are as follows, -where $$\dd{S_\sigma}$$ is the $$\sigma$$-isosurface, etc.: - $$\begin{aligned} \boxed{ \begin{aligned} - \dd{S_\sigma} &= \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} + (\vb{U} \cdot \nabla) \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{U_\sigma}{\rho} \pdv{V_\sigma}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\sigma}{\tau} + U_z \pdv{V_\sigma}{z} + + \frac{\tau}{\rho^3} U_\sigma V_\tau - \frac{\sigma}{\rho^3} U_\tau V_\tau \bigg) \\ - \dd{S_\tau} &= \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} + &\quad\: + \vu{e}_\tau \bigg( \frac{U_\sigma}{\rho} \pdv{V_\tau}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_\tau}{\tau} + U_z \pdv{V_\tau}{z} + + \frac{\sigma}{\rho^3} U_\tau V_\sigma - \frac{\tau}{\rho^3} U_\sigma V_\sigma \bigg) \\ - \dd{S_z} &= (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} + &\quad\: + \vu{e}_z \bigg( \frac{U_\sigma}{\rho} \pdv{V_z}{\sigma} + \frac{U_\tau}{\rho} \pdv{V_z}{\tau} + U_z \pdv{V_z}{z} \bigg) \end{aligned} } \end{aligned}$$ -The normal element $$\dd{\vu{S}}$$ of a surface and -the tangent element $$\dd{\vu{\ell}}$$ of a curve are respectively: - $$\begin{aligned} \boxed{ - \dd{\vu{S}} - = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\tau} \dd{z} - + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\sigma} \dd{z} - + \vu{e}_z (\sigma^2 + \tau^2) \dd{\sigma} \dd{\tau} + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_\sigma \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\sigma}{\tau} + \pdvn{2}{V_\sigma}{z} + + \frac{2 \tau}{\rho^4} \pdv{V_\tau}{\sigma} - \frac{2 \sigma}{\rho^4} \pdv{V_\tau}{\tau} - \frac{V_\sigma}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_\tau \bigg( \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_\tau}{\tau} + \pdvn{2}{V_\tau}{z} + - \frac{2 \tau}{\rho^4} \pdv{V_\sigma}{\sigma} + \frac{2 \sigma}{\rho^4} \pdv{V_\sigma}{\tau} - \frac{V_\tau}{\rho^4} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \frac{1}{\rho^2} \pdvn{2}{V_z}{\sigma} + \frac{1}{\rho^2} \pdvn{2}{V_z}{\tau} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} } \end{aligned}$$ $$\begin{aligned} \boxed{ - \dd{\vu{\ell}} - = \vu{e}_\sigma \sqrt{\sigma^2 + \tau^2} \dd{\sigma} - + \vu{e}_\tau \sqrt{\sigma^2 + \tau^2} \dd{\tau} - + \vu{e}_z \dd{z} + \begin{aligned} + \nabla \cdot \overline{\overline{\mathbf{T}}} + &= \vu{e}_\sigma \bigg( \frac{1}{\rho} \pdv{T_{\sigma \sigma}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \sigma}}{\tau} + \pdv{T_{z \sigma}}{z} + + \frac{\sigma T_{\sigma \sigma}}{\rho^3} + \frac{\tau T_{\sigma \tau}}{\rho^3} + + \frac{\tau T_{\tau \sigma}}{\rho^3} - \frac{\sigma T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_\tau \bigg( \frac{1}{\rho} \pdv{T_{\sigma \tau}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau \tau}}{\tau} + \pdv{T_{k \tau}}{z} + - \frac{\tau T_{\sigma \sigma}}{\rho^3} + \frac{\sigma T_{\sigma \tau}}{\rho^3} + + \frac{\sigma T_{\tau \sigma}}{\rho^3} + \frac{\tau T_{\tau \tau}}{\rho^3} \bigg) + \\ + &+ \vu{e}_z \bigg( \frac{1}{\rho} \pdv{T_{\sigma z}}{\sigma} + \frac{1}{\rho} \pdv{T_{\tau z}}{\tau} + \pdv{T_{zz}}{z} + + \frac{\sigma T_{\sigma z}}{\rho^3} + \frac{\tau T_{\tau z}}{\rho^3} \bigg) + \end{aligned} } \end{aligned}$$ + ## References 1. M.L. Boas, *Mathematical methods in the physical sciences*, 2nd edition, |