Length of Element of Arc in Orthogonal Curvilinear Coordinates
Theorem
Let $\tuple {q_1, q_2, q_3}$ denote a set of orthogonal curvilinear coordinates.
Let the relation between those orthogonal curvilinear coordinates and Cartesian coordinates be expressed as:
\(\ds x\) | \(=\) | \(\ds \map x {q_1, q_2, q_3}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \map y {q_1, q_2, q_3}\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \map z {q_1, q_2, q_3}\) |
where $\tuple {x, y, z}$ denotes the Cartesian coordinates.
Let $S$ be an infinitesimal arc.
Let $\d s$ be the length of $S$
Then:
\(\ds \d s^2\) | \(=\) | \(\ds {h_1}^2 {\rd q_1}^2 + {h_2}^2 {\rd q_2}^2 + {h_3}^2 {\rd q_3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_i {h_i}^2 {\rd q_i}^2\) | for $i \in \set {1, 2, 3}$ |
where:
- $\d q_i$ is the projection of $S$ onto the unit normal to the curvilinear coordinate surface determined by $q_i$, for $i \in \set {1, 3}$
- ${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$
Proof
By definition of the metric of $\tuple {q_1, q_2, q_3}$:
\(\ds \d s^2\) | \(=\) | \(\ds \d x^2 + \d y^2 + \d z^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j} {h_{i j} }^2 \rd q_i \rd q_j\) | for $i, j \in \set {1, 2, 3}$ |
From Value of Curvilinear Coordinate Metric:
- $\forall i, j \in \set {1, 2, 3}: {h_{i j} }^2 = \dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j}$
But we have that $\tuple {q_1, q_2, q_3}$ is orthogonal.
From Definition 1 of orthogonal curvilinear coordinates:
- $\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$
wherever $i \ne j$.
Hence we have:
\(\ds \d s^2\) | \(=\) | \(\ds {h_{1 1} }^2 {\rd q_1}^2 + {h_{2 2} }^2 {\rd q_2}^2 + {h_{3 3} }^2 {\rd q_3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_i {h_{i i} }^2 {\rd q_i}^2\) | for $i \in \set {1, 2, 3}$ |
Elements $h_{i i}$ are those elements of the metric which do not vanish when $\tuple {q_1, q_2, q_3}$ is orthogonal.
To streamline notation, we rename them $h_1$, $h_2$ and $h_3$.
Hence:
\(\ds \d s^2\) | \(=\) | \(\ds {h_1}^2 {\rd q_1}^2 + {h_2}^2 {\rd q_2}^2 + {h_3}^2 {\rd q_3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_i {h_i}^2 {\rd q_i}^2\) | for $i \in \set {1, 2, 3}$ |
where:
- ${h_i}^2 = \paren {\dfrac {\partial x} {\partial q_i} }^2 + \paren {\dfrac {\partial y} {\partial q_i} }^2 + \paren {\dfrac {\partial z} {\partial q_i} }^2$
$\blacksquare$
Also see
Sources
- 1961: Ian N. Sneddon: Special Functions of Mathematical Physics and Chemistry (2nd ed.) ... (previous) ... (next): Chapter $\text I$: Introduction: $\S 1$. The origin of special functions: $(1.3)$
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates: $(2.8)$