From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001 From: Prefetch Date: Thu, 20 Oct 2022 18:25:31 +0200 Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex' --- .../know/concept/curvilinear-coordinates/index.md | 103 +++++++++++---------- 1 file changed, 52 insertions(+), 51 deletions(-) (limited to 'source/know/concept/curvilinear-coordinates/index.md') diff --git a/source/know/concept/curvilinear-coordinates/index.md b/source/know/concept/curvilinear-coordinates/index.md index 3012ca6..cb22e43 100644 --- a/source/know/concept/curvilinear-coordinates/index.md +++ b/source/know/concept/curvilinear-coordinates/index.md @@ -14,15 +14,15 @@ 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. +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** $\dd{\ell}$, +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. +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: $$\begin{aligned} @@ -33,7 +33,7 @@ $$\begin{aligned} } \end{aligned}$$ -Where $h_1$, $h_2$, and $h_3$ are called **scale factors**, +Where $$h_1$$, $$h_2$$, and $$h_3$$ are called **scale factors**, and need not be constants. The equation above only contains quadratic terms because the coordinate system is orthogonal by assumption. @@ -45,20 +45,20 @@ and [cylindrical parabolic coordinates](/know/concept/cylindrical-parabolic-coor In the following subsections, we derive general formulae to convert expressions -from Cartesian coordinates to the new orthogonal system $(x_1, x_2, x_3)$. +from Cartesian coordinates to the new orthogonal system $$(x_1, x_2, x_3)$$. ## Basis vectors -Consider the the vector form of the line element $\dd{\ell}$, -denoted by $\dd{\vu{\ell}}$ and expressed as: +Consider the the vector form of the line element $$\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}$$ -We can expand the Cartesian differential elements, e.g. $\dd{y}$, +We can expand the Cartesian differential elements, e.g. $$\dd{y}$$, in the new basis as follows: $$\begin{aligned} @@ -66,17 +66,17 @@ $$\begin{aligned} = \pdv{y}{x_1} \dd{x_1} + \pdv{y}{x_2} \dd{x_2} + \pdv{y}{x_3} \dd{x_3} \end{aligned}$$ -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 $\dd{\vu{\ell}}$: +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: +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: $$\begin{aligned} \boxed{ @@ -89,9 +89,9 @@ $$\begin{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 $x_3$: +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 $$x_3$$: $$\begin{gathered} \boxed{ @@ -107,8 +107,8 @@ $$\begin{gathered} ## Divergence -The divergence of a vector $\vb{V} = \vu{e}_1 V_1 + \vu{e}_2 V_2 + \vu{e}_3 V_3$ +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: $$\begin{aligned} @@ -149,7 +150,7 @@ $$\begin{aligned} @@ -227,7 +228,7 @@ we get the desired general expression for the divergence. ## Laplacian -The Laplacian $\nabla^2 f$ is simply $\nabla \cdot \nabla f$, +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: @@ -247,7 +248,7 @@ $$\begin{aligned} ## Curl -The curl of a vector $\vb{V}$ is as follows +The curl of a vector $$\vb{V}$$ is as follows in a general orthogonal curvilinear system: $$\begin{aligned} @@ -269,22 +270,22 @@ $$\begin{aligned} @@ -322,14 +323,14 @@ and add up the results, we get the desired expression for the curl. ## Differential elements -The point of the scale factors $h_1$, $h_2$ and $h_3$, as can seen from their derivation, +The point of the scale factors $$h_1$$, $$h_2$$ and $$h_3$$, as can seen from their derivation, is to correct for "distortions" of the coordinates compared to the Cartesian system, -such that the line element $\dd{\ell}$ retains its length. -This property extends to the surface $\dd{S}$ and volume $\dd{V}$. +such that the line element $$\dd{\ell}$$ retains its length. +This property extends to the surface $$\dd{S}$$ and volume $$\dd{V}$$. When handling a differential volume in curvilinear coordinates, e.g. for a volume integral, -the size of the box $\dd{V}$ must be corrected by the scale factors: +the size of the box $$\dd{V}$$ must be corrected by the scale factors: $$\begin{aligned} \boxed{ @@ -339,8 +340,8 @@ $$\begin{aligned} } \end{aligned}$$ -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: +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: $$\begin{aligned} \boxed{ @@ -354,7 +355,7 @@ $$\begin{aligned} } \end{aligned}$$ -Using the same logic, the normal vector element $\dd{\vu{S}}$ +Using the same logic, the normal vector element $$\dd{\vu{S}}$$ of an arbitrary surface is given by: $$\begin{aligned} @@ -364,7 +365,7 @@ $$\begin{aligned} } \end{aligned}$$ -Finally, the tangent vector element $\dd{\vu{\ell}}$ takes the following form: +Finally, the tangent vector element $$\dd{\vu{\ell}}$$ takes the following form: $$\begin{aligned} \boxed{ -- cgit v1.2.3