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diff --git a/source/know/concept/cylindrical-polar-coordinates/index.md b/source/know/concept/cylindrical-polar-coordinates/index.md index 43b4684..3c54ef8 100644 --- a/source/know/concept/cylindrical-polar-coordinates/index.md +++ b/source/know/concept/cylindrical-polar-coordinates/index.md @@ -8,11 +8,11 @@ categories: layout: "concept" --- -**Cylindrical polar coordinates** are an extension of polar coordinates to 3D, -which describes the location of a point in space -using the coordinates $$(r, \varphi, z)$$. -The $$z$$-axis is unchanged from Cartesian coordinates, -hence it is called a *cylindrical* system. +**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 it is called *cylindrical*. Cartesian coordinates $$(x, y, z)$$ and the cylindrical system $$(r, \varphi, z)$$ are related by: @@ -20,78 +20,85 @@ and the cylindrical system $$(r, \varphi, z)$$ are related by: $$\begin{aligned} \boxed{ \begin{aligned} - x &= r \cos\varphi \\ - y &= r \sin\varphi \\ - z &= z + x + &= r \cos{\varphi} + \\ + y + &= r \sin{\varphi} + \\ + z + &= z \end{aligned} } \end{aligned}$$ Conversely, a point given in $$(x, y, z)$$ -can be converted to $$(r, \varphi, z)$$ -using these formulae: +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: $$\begin{aligned} \boxed{ - r = \sqrt{x^2 + y^2} - \qquad - \varphi = \mathtt{atan2}(y, x) - \qquad - z = z + \begin{aligned} + r + &= \sqrt{x^2 + y^2} + \\ + \varphi + &= \mathtt{atan2}(y, x) + \\ + z + &= z + \end{aligned} } \end{aligned}$$ -The cylindrical polar coordinates form +Cylindrical polar coordinates form an [orthogonal curvilinear system](/know/concept/orthogonal-curvilinear-coordinates/), -whose scale factors $$h_r$$, $$h_\varphi$$ and $$h_z$$ we want to find. -To do so, we calculate the differentials of the Cartesian coordinates: +whose **scale factors** $$h_r$$, $$h_\varphi$$ and $$h_z$$ we need. +To get those, we calculate the unnormalized local basis: $$\begin{aligned} - \dd{x} = \dd{r} \cos\varphi - \dd{\varphi} r \sin\varphi - \qquad - \dd{y} = \dd{r} \sin\varphi + \dd{\varphi} r \cos\varphi - \qquad - \dd{z} = \dd{z} -\end{aligned}$$ - -And then we calculate the line element $$\dd{\ell}^2$$, -skipping many terms thanks to orthogonality, - -$$\begin{aligned} - \dd{\ell}^2 - &= \dd{r}^2 \big( \cos^2(\varphi) + \sin^2(\varphi) \big) - + \dd{\varphi}^2 \big( r^2 \sin^2(\varphi) + r^2 \cos^2(\varphi) \big) - + \dd{z}^2 + h_r \vu{e}_r + &= \vu{e}_x \pdv{x}{r} + \vu{e}_y \pdv{y}{r} + \vu{e}_z \pdv{z}{r} + \\ + &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} + \\ + h_\varphi \vu{e}_\varphi + &= \vu{e}_x \pdv{x}{\varphi} + \vu{e}_y \pdv{y}{\varphi} + \vu{e}_z \pdv{z}{\varphi} \\ - &= \dd{r}^2 + r^2 \: \dd{\varphi}^2 + \dd{z}^2 + &= - \vu{e}_x \: r \sin{\varphi} + \vu{e}_y \: r \cos{\varphi} + \\ + h_z \vu{e}_z + &= \vu{e}_x \pdv{x}{z} + \vu{e}_y \pdv{y}{z} + \vu{e}_z \pdv{z}{z} + \\ + &= \vu{e}_z \end{aligned}$$ -Finally, we can simply read off -the squares of the desired scale factors -$$h_r^2$$, $$h_\varphi^2$$ and $$h_z^2$$: +By normalizing the **local basis vectors** +$$\vu{e}_r$$, $$\vu{e}_\varphi$$ and $$\vu{e}_z$$, +we arrive at these expressions: $$\begin{aligned} \boxed{ - h_r = 1 - \qquad - h_\varphi = r - \qquad - h_z = 1 + \begin{aligned} + h_r + &= 1 + \\ + h_\varphi + &= r + \\ + h_z + &= 1 + \end{aligned} } -\end{aligned}$$ - -With these factors, we can easily convert things from the Cartesian system -using the standard formulae for orthogonal curvilinear coordinates. -The basis vectors are: - -$$\begin{aligned} + \qquad\qquad \boxed{ \begin{aligned} \vu{e}_r - &= \cos\varphi \:\vu{e}_x + \sin\varphi \:\vu{e}_y + &= \vu{e}_x \cos{\varphi} + \vu{e}_y \sin{\varphi} \\ \vu{e}_\varphi - &= - \sin\varphi \:\vu{e}_x + \cos\varphi \:\vu{e}_y + &= - \vu{e}_x \sin{\varphi} + \vu{e}_y \cos{\varphi} \\ \vu{e}_z &= \vu{e}_z @@ -99,7 +106,52 @@ $$\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}_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{ @@ -113,7 +165,7 @@ $$\begin{aligned} $$\begin{aligned} \boxed{ \nabla \cdot \vb{V} - = \frac{1}{r} \pdv{(r V_r)}{r} + = \pdv{V_r}{r} + \frac{V_r}{r} + \frac{1}{r} \pdv{V_\varphi}{\varphi} + \pdv{V_z}{z} } @@ -121,81 +173,124 @@ $$\begin{aligned} $$\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 - = \frac{1}{r} \pdv{}{r}\Big( r \pdv{f}{r} \Big) + = \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 \times \vb{V} - &= \vu{e}_r \Big( \frac{1}{r} \pdv{V_z}{\varphi} - \pdv{V_\varphi}{z} \Big) + \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) \\ - &+ \vu{e}_\varphi \Big( \pdv{V_r}{z} - \pdv{V_z}{r} \Big) + &\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) \\ - &+ \frac{\vu{e}_z}{r} \Big( \pdv{(r V_\varphi)}{r} - \pdv{V_r}{\varphi} \Big) + &\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}$$ -The differential element of volume $$\dd{V}$$ -takes the following form: - $$\begin{aligned} \boxed{ - \dd{V} - = r \dd{r} \dd{\varphi} \dd{z} + \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}$$ -So, for example, an integral over all of space is converted like so: - -$$\begin{aligned} - \iiint_{-\infty}^\infty f(x, y, z) \dd{V} - = \int_{-\infty}^{\infty} \int_0^{2\pi} \int_0^\infty f(r, \varphi, z) \: r \dd{r} \dd{\varphi} \dd{z} -\end{aligned}$$ - -The isosurface elements are as follows, where $$S_r$$ is a surface at constant $$r$$, etc.: - $$\begin{aligned} \boxed{ \begin{aligned} - \dd{S}_r = r \dd{\varphi} \dd{z} - \qquad - \dd{S}_\varphi = \dd{r} \dd{z} - \qquad - \dd{S}_z = r \dd{r} \dd{\varphi} + (\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}$$ -Similarly, the normal vector element $$\dd{\vu{S}}$$ for an arbitrary surface is given by: - $$\begin{aligned} \boxed{ - \dd{\vu{S}} - = \vu{e}_r \: r \dd{\varphi} \dd{z} - + \vu{e}_\varphi \dd{r} \dd{z} - + \vu{e}_z \: r \dd{r} \dd{\varphi} + \begin{aligned} + \nabla^2 \vb{V} + &= \quad \vu{e}_r \bigg( \pdvn{2}{V_r}{r} + \frac{1}{r} \pdv{V_r}{r} + \frac{1}{r^2} \pdvn{2}{V_r}{\varphi} + + \pdvn{2}{V_r}{z} - \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} \pdv{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{V_\varphi}{r^2} \bigg) + \\ + &\quad\: + \vu{e}_z \bigg( \pdvn{2}{V_z}{r} + \frac{1}{r} \pdv{V_z}{r} + + \frac{1}{r^2} \pdvn{2}{V_z}{\varphi} + \pdvn{2}{V_z}{z} \bigg) + \end{aligned} } \end{aligned}$$ -And finally, the tangent vector element $$\dd{\vu{\ell}}$$ of a given curve is as follows: - $$\begin{aligned} \boxed{ - \dd{\vu{\ell}} - = \vu{e}_r \dd{r} - + \vu{e}_\varphi \: r \dd{\varphi} - + \vu{e}_z \dd{z} + \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. |