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+---
+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 $x_3$:
+
+$$\begin{gathered}
+ \boxed{
+ \nabla f
+ = \vu{e}_1 \frac{1}{h_1} \pdv{f}{x_1}
+ + \vu{e}_2 \frac{1}{h_2} \pdv{f}{x_2}
+ + \vu{e}_3 \frac{1}{h_3} \pdv{f}{x_3}
+ }
+\end{gathered}$$
+
+
+
+
+
+
+For a direction $\dd{\ell}$, we know that
+$\idv{f}{\ell}$ is the component of $\nabla f$ in that direction:
+
+$$\begin{aligned}
+ \dv{f}{\ell}
+ = \pdv{f}{x} \dv{x}{\ell} + \pdv{f}{y} \dv{y}{\ell} + \pdv{f}{z} \dv{z}{\ell}
+ = \nabla f \cdot \bigg( \dv{x}{\ell}, \dv{y}{\ell}, \dv{z}{\ell} \bigg)
+ = \nabla f \cdot \vu{u}
+\end{aligned}$$
+
+Where $\vu{u}$ is simply a unit vector in the direction of $\dd{\ell}$.
+We thus find the expression for the gradient $\nabla f$
+by choosing $\dd{\ell}$ to be $h_1 \dd{x_1}$, $h_2 \dd{x_2}$ and $h_3 \dd{x_3}$ in turn:
+
+$$\begin{gathered}
+ \nabla f
+ = \vu{e}_1 \dv{x_1}{\ell} \pdv{f}{x_1}
+ + \vu{e}_2 \dv{x_2}{\ell} \pdv{f}{x_2}
+ + \vu{e}_3 \dv{x_3}{\ell} \pdv{f}{x_3}
+\end{gathered}$$
+
+
+
+
+## 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:
+
+$$\begin{aligned}
+ \boxed{
+ \nabla \cdot \vb{V}
+ = \frac{1}{h_1 h_2 h_3}
+ \Big( \pdv{(h_2 h_3 V_1)}{x_1} + \pdv{(h_1 h_3 V_2)}{x_2} + \pdv{(h_1 h_2 V_3)}{x_3} \Big)
+ }
+\end{aligned}$$
+
+
+
+
+
+
+As preparation, we rewrite $\vb{V}$ as follows
+to introduce the scale factors:
+
+$$\begin{aligned}
+ \vb{V}
+ &= \vu{e}_1 \frac{1}{h_2 h_3} (h_2 h_3 V_1)
+ + \vu{e}_2 \frac{1}{h_1 h_3} (h_1 h_3 V_2)
+ + \vu{e}_3 \frac{1}{h_1 h_2} (h_1 h_2 V_3)
+\end{aligned}$$
+
+We start by taking only the $\vu{e}_1$-component of this vector,
+and expand its divergence using the following vector identity:
+
+$$\begin{gathered}
+ \nabla \cdot (\vb{U} \: f)
+ = \vb{U} \cdot (\nabla f) + (\nabla \cdot \vb{U}) f
+\end{gathered}$$
+
+Inserting the scalar $f = h_2 h_3 V_1$
+the vector $\vb{U} = \vu{e}_1 / (h_2 h_3)$,
+we arrive at:
+
+$$\begin{gathered}
+ \nabla \cdot \Big( \frac{\vu{e}_1}{h_2 h_3} (h_2 h_3 V_1) \Big)
+ = \frac{\vu{e}_1}{h_2 h_3} \cdot \Big( \nabla (h_2 h_3 V_1) \Big)
+ + \Big( \nabla \cdot \frac{\vu{e}_1}{h_2 h_3} \Big) (h_2 h_3 V_1)
+\end{gathered}$$
+
+The first right-hand term is easy to calculate
+thanks to our expression for the gradient $\nabla f$.
+Only the $\vu{e}_1$-component survives due to the dot product:
+
+$$\begin{aligned}
+ \frac{\vu{e}_1}{h_2 h_3} \cdot \Big( \nabla (h_2 h_3 V_1) \Big)
+ = \frac{\vu{e}_1}{h_1 h_2 h_3} \pdv{(h_2 h_3 V_1)}{x_1}
+\end{aligned}$$
+
+The second term is more involved.
+First, we use the gradient formula to observe that:
+
+$$\begin{aligned}
+ \nabla x_1
+ = \frac{\vu{e}_1}{h_1}
+ \qquad \quad
+ \nabla x_2
+ = \frac{\vu{e}_2}{h_2}
+ \qquad \quad
+ \nabla x_3
+ = \frac{\vu{e}_3}{h_3}
+\end{aligned}$$
+
+Because $\vu{e}_2 \cross \vu{e}_3 = \vu{e}_1$ in an orthogonal basis,
+these gradients can be used to express the vector whose divergence we want:
+
+$$\begin{aligned}
+ \nabla x_2 \cross \nabla x_3
+ = \frac{\vu{e}_2}{h_2} \cross \frac{\vu{e}_3}{h_3}
+ = \frac{\vu{e}_1}{h_2 h_3}
+\end{aligned}$$
+
+We then apply the divergence and expand the expression using a vector identity.
+In all cases, the curl of a gradient $\nabla \cross \nabla f$ is zero, so:
+
+$$\begin{aligned}
+ \nabla \cdot \frac{\vu{e}_1}{h_2 h_3}
+ = \nabla \cdot \big( \nabla x_2 \cross \nabla x_3 \big)
+ = \nabla x_3 \cdot (\nabla \cross \nabla x_2) - \nabla x_2 \cdot (\nabla \cross \nabla x_3)
+ = 0
+\end{aligned}$$
+
+After repeating this procedure for the other components of $\vb{V}$,
+we get the desired general expression for the divergence.
+
+
+
+
+## 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:
+
+$$\begin{aligned}
+ \boxed{
+ \begin{aligned}
+ \nabla \times \vb{V}
+ &= \frac{\vu{e}_1}{h_2 h_3} \Big( \pdv{(h_3 V_3)}{x_2} - \pdv{(h_2 V_2)}{x_3} \Big)
+ \\
+ &+ \frac{\vu{e}_2}{h_1 h_3} \Big( \pdv{(h_1 V_1)}{x_3} - \pdv{(h_3 V_3)}{x_1} \Big)
+ \\
+ &+ \frac{\vu{e}_3}{h_1 h_2} \Big( \pdv{(h_2 V_2)}{x_1} - \pdv{(h_1 V_1)}{x_2} \Big)
+ \end{aligned}
+ }
+\end{aligned}$$
+
+
+
+
+
+
+The curl is found in a similar way as the divergence.
+We rewrite $\vb{V}$ like so:
+
+$$\begin{aligned}
+ \vb{V}
+ = \frac{\vu{e}_1}{h_1} (h_1 V_1) + \frac{\vu{e}_2}{h_2} (h_2 V_2) + \frac{\vu{e}_3}{h_3} (h_3 V_3)
+\end{aligned}$$
+
+We expand the curl of its $\vu{e}_1$-component using the following vector identity:
+
+$$\begin{gathered}
+ \nabla \cross (\vb{U} \: f)
+ = (\nabla \cross \vb{U}) f - \vb{U} \cross (\nabla f)
+\end{gathered}$$
+
+Inserting the scalar $f = h_1 V_1$
+and the vector $\vb{U} = \vu{e}_1 / h_1$, we arrive at:
+
+$$\begin{gathered}
+ \nabla \cross \Big( \frac{\vu{e}_1}{h_1} (h_1 V_1) \Big)
+ = \Big( \nabla \cross \frac{\vu{e}_1}{h_1} \Big) (h_1 V_1) - \frac{\vu{e}_1}{h_1} \cross \Big( \nabla (h_1 V_1) \Big)
+\end{gathered}$$
+
+Previously, when proving the divergence,
+we already showed that $\vu{e}_1 / h_1 = \nabla x_1$.
+Because the curl of a gradient is zero,
+the first term disappears, leaving only the second,
+which contains a gradient that turns out to be:
+
+$$\begin{aligned}
+ \nabla (h_1 V_1)
+ = \vu{e}_1 \frac{1}{h_1} \pdv{(h_1 V_1)}{x_1}
+ + \vu{e}_2 \frac{1}{h_2} \pdv{(h_1 V_1)}{x_2}
+ + \vu{e}_3 \frac{1}{h_3} \pdv{(h_1 V_1)}{x_3}
+\end{aligned}$$
+
+Consequently, the curl of the first component of $\vb{V}$ is as follows,
+using the fact that $\vu{e}_1$, $\vu{e}_2$ and $\vu{e}_3$
+are related to each other by cross products:
+
+$$\begin{aligned}
+ \nabla \cross \Big( \frac{\vu{e}_1}{h_1} (h_1 V_1) \Big)
+ = - \frac{\vu{e}_1}{h_1} \cross \Big( \nabla (h_1 V_1) \Big)
+ = - \frac{\vu{e}_3}{h_1 h_2} \pdv{(h_1 V_1)}{x_2} + \frac{\vu{e}_2}{h_1 h_3} \pdv{(h_1 V_1)}{x_3}
+\end{aligned}$$
+
+If we go through the same process for the other components of $\vb{V}$
+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,
+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.
--
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