From 6ce0bb9a8f9fd7d169cbb414a9537d68c5290aae Mon Sep 17 00:00:00 2001 From: Prefetch Date: Fri, 14 Oct 2022 23:25:28 +0200 Subject: Initial commit after migration from Hugo --- .../know/concept/curvilinear-coordinates/index.md | 380 +++++++++++++++++++++ 1 file changed, 380 insertions(+) create mode 100644 source/know/concept/curvilinear-coordinates/index.md (limited to 'source/know/concept/curvilinear-coordinates') diff --git a/source/know/concept/curvilinear-coordinates/index.md b/source/know/concept/curvilinear-coordinates/index.md new file mode 100644 index 0000000..9cae643 --- /dev/null +++ b/source/know/concept/curvilinear-coordinates/index.md @@ -0,0 +1,380 @@ +--- +title: "Curvilinear coordinates" +date: 2021-03-03 +categories: +- Mathematics +- Physics +layout: "concept" +--- + +In a 3D coordinate system, the isosurface of a coordinate +(i.e. the surface where that coordinate is constant while the others vary) +is known as a **coordinate surface**, and the intersections of +the surfaces of different coordinates are called **coordinate lines**. + +A **curvilinear** coordinate system is one where at least one of the coordinate surfaces is curved, +e.g. in cylindrical coordinates the line between $r$ and $z$ is a circle. +If the coordinate surfaces are mutually perpendicular, +it is an **orthogonal** system, which is generally desirable. + +A useful attribute of a coordinate system is its **line element** $\dd{\ell}$, +which represents the differential element of a line in any direction. +For an orthogonal system, its square $\dd{\ell}^2$ is calculated +by taking the differential elements of the old Cartesian $(x, y, z)$ system +and writing them out in the new $(x_1, x_2, x_3)$ system. +The resulting expression will be of the form: + +$$\begin{aligned} + \boxed{ + \dd{\ell}^2 + = \dd{x}^2 + \dd{y}^2 + \dd{z}^2 + = h_1^2 \dd{x_1}^2 + h_2^2 \dd{x_2}^2 + h_3^2 \dd{x_3}^2 + } +\end{aligned}$$ + +Where $h_1$, $h_2$, and $h_3$ are called **scale factors**, +and need not be constants. +The equation above only contains quadratic terms +because the coordinate system is orthogonal by assumption. + +Examples of orthogonal curvilinear coordinate systems include +[spherical coordinates](/know/concept/spherical-coordinates/), +[cylindrical polar coordinates](/know/concept/cylindrical-polar-coordinates/), +and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coordinates/). + +In the following subsections, +we derive general formulae to convert expressions +from Cartesian coordinates to the new orthogonal system $(x_1, x_2, x_3)$. + + +## Basis vectors + +Consider the the vector form of the line element $\dd{\ell}$, +denoted by $\dd{\vu{\ell}}$ and expressed as: + +$$\begin{aligned} + \dd{\vu{\ell}} + = \vu{e}_x \dd{x} + \vu{e}_y \dd{y} + \vu{e}_z \dd{z} +\end{aligned}$$ + +We can expand the Cartesian differential elements, e.g. $\dd{y}$, +in the new basis as follows: + +$$\begin{aligned} + \dd{y} + = \pdv{y}{x_1} \dd{x_1} + \pdv{y}{x_2} \dd{x_2} + \pdv{y}{x_3} \dd{x_3} +\end{aligned}$$ + +If we write this out for $\dd{x}$, $\dd{y}$ and $\dd{z}$, +and group the terms according to $\dd{x}_1$, $\dd{x}_2$ and $\dd{x}_3$, +we can compare it the alternative form of $\dd{\vu{\ell}}$: + +$$\begin{aligned} + \dd{\vu{\ell}} + = \vu{e}_1 \:h_1 \dd{x_1} + \vu{e}_2 \:h_2 \dd{x_2} + \vu{e}_3 \:h_3 \dd{x_4} +\end{aligned}$$ + +From this, we can read off $\vu{e}_1$, $\vu{e}_2$ and $\vu{e}_3$. +Here we only give $\vu{e}_1$, since $\vu{e}_2$ and $\vu{e}_3$ are analogous: + +$$\begin{aligned} + \boxed{ + h_1 \vu{e}_1 + = \vu{e}_x \pdv{x}{x_1} + \vu{e}_y \pdv{y}{x_1} + \vu{e}_z \pdv{y}{x_1} + } +\end{aligned}$$ + + +## Gradient + +In an orthogonal coordinate system, +the gradient $\nabla f$ of a scalar $f$ is as follows, +where $\vu{e}_1$, $\vu{e}_2$ and $\vu{e}_3$ +are the basis unit vectors respectively corresponding to $x_1$, $x_2$ and $x_3$: + +$$\begin{gathered} + \boxed{ + \nabla f + = \vu{e}_1 \frac{1}{h_1} \pdv{f}{x_1} + + \vu{e}_2 \frac{1}{h_2} \pdv{f}{x_2} + + \vu{e}_3 \frac{1}{h_3} \pdv{f}{x_3} + } +\end{gathered}$$ + +
+ + + +
+ + +## Divergence + +The divergence of a vector $\vb{V} = \vu{e}_1 V_1 + \vu{e}_2 V_2 + \vu{e}_3 V_3$ +in an orthogonal system is given by: + +$$\begin{aligned} + \boxed{ + \nabla \cdot \vb{V} + = \frac{1}{h_1 h_2 h_3} + \Big( \pdv{(h_2 h_3 V_1)}{x_1} + \pdv{(h_1 h_3 V_2)}{x_2} + \pdv{(h_1 h_2 V_3)}{x_3} \Big) + } +\end{aligned}$$ + +
+ + + +
+ + +## Laplacian + +The Laplacian $\nabla^2 f$ is simply $\nabla \cdot \nabla f$, +so we can find the general formula +by combining the two preceding results +for the gradient and the divergence: + +$$\begin{aligned} + \boxed{ + \nabla^2 f + = \frac{1}{h_1 h_2 h_3} + \bigg( + \pdv{}{x_1}\Big(\! \frac{h_2 h_3}{h_1} \pdv{f}{x_1} \!\Big) + + \pdv{}{x_2}\Big(\! \frac{h_1 h_3}{h_2} \pdv{f}{x_2} \!\Big) + + \pdv{}{x_3}\Big(\! \frac{h_1 h_2}{h_3} \pdv{f}{x_3} \!\Big) + \bigg) + } +\end{aligned}$$ + + +## Curl + +The curl of a vector $\vb{V}$ is as follows +in a general orthogonal curvilinear system: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \nabla \times \vb{V} + &= \frac{\vu{e}_1}{h_2 h_3} \Big( \pdv{(h_3 V_3)}{x_2} - \pdv{(h_2 V_2)}{x_3} \Big) + \\ + &+ \frac{\vu{e}_2}{h_1 h_3} \Big( \pdv{(h_1 V_1)}{x_3} - \pdv{(h_3 V_3)}{x_1} \Big) + \\ + &+ \frac{\vu{e}_3}{h_1 h_2} \Big( \pdv{(h_2 V_2)}{x_1} - \pdv{(h_1 V_1)}{x_2} \Big) + \end{aligned} + } +\end{aligned}$$ + +
+ + + +
+ + +## Differential elements + +The point of the scale factors $h_1$, $h_2$ and $h_3$, as can seen from their derivation, +is to correct for "distortions" of the coordinates compared to the Cartesian system, +such that the line element $\dd{\ell}$ retains its length. +This property extends to the surface $\dd{S}$ and volume $\dd{V}$. + +When handling a differential volume in curvilinear coordinates, +e.g. for a volume integral, +the size of the box $\dd{V}$ must be corrected by the scale factors: + +$$\begin{aligned} + \boxed{ + \dd{V} + = \dd{x}\dd{y}\dd{z} + = h_1 h_2 h_3 \dd{x_1} \dd{x_2} \dd{x_3} + } +\end{aligned}$$ + +The same is true for the isosurfaces $\dd{S_1}$, $\dd{S_2}$ and $\dd{S_3}$ +where the coordinates $x_1$, $x_2$ and $x_3$ are respectively kept constant: + +$$\begin{aligned} + \boxed{ + \begin{aligned} + \dd{S_1} &= h_2 h_3 \dd{x_2} \dd{x_3} + \\ + \dd{S_2} &= h_1 h_3 \dd{x_1} \dd{x_3} + \\ + \dd{S_3} &= h_1 h_2 \dd{x_1} \dd{x_2} + \end{aligned} + } +\end{aligned}$$ + +Using the same logic, the normal vector element $\dd{\vu{S}}$ +of an arbitrary surface is given by: + +$$\begin{aligned} + \boxed{ + \dd{\vu{S}} + = \vu{e}_1 h_2 h_3 \dd{x_2} \dd{x_3} + \vu{e}_2 h_1 h_3 \dd{x_1} \dd{x_3} + \vu{e}_3 h_1 h_2 \dd{x_1} \dd{x_2} + } +\end{aligned}$$ + +Finally, the tangent vector element $\dd{\vu{\ell}}$ takes the following form: + +$$\begin{aligned} + \boxed{ + \dd{\vu{\ell}} + = \vu{e}_1 h_1 \dd{x_1} + \vu{e}_2 h_2 \dd{x_2} + \vu{e}_3 h_3 \dd{x_3} + } +\end{aligned}$$ + + + +## References +1. M.L. Boas, + *Mathematical methods in the physical sciences*, 2nd edition, + Wiley. -- cgit v1.2.3