Value of Curvilinear Coordinate Metric
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Definition
Let $A$ and $B$ be two points in space.
Let a curvilinear $3$-Space coordinate system $\QQ$ be applied on top of a Cartesian $3$-space.
Let $h_{i j}$ be the metric of $\QQ$.
Then:
- $\forall i, j \in \set {1, 2, 3}: {h_{i j} }^2 = \dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j}$
where $\partial$ denotes partial differentiation.
Proof
From the Cartesian representation of $\QQ$:
\(\text {(1)}: \quad\) | \(\ds x\) | \(=\) | \(\ds \map x {q_1, q_2, q_3}\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds y\) | \(=\) | \(\ds \map y {q_1, q_2, q_3}\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds z\) | \(=\) | \(\ds \map z {q_1, q_2, q_3}\) |
where:
- $\tuple {x, y, z}$ denotes the Cartesian coordinates
- $\tuple {q_1, q_2, q_3}$ denotes the corresponding curvilinear coordinates.
Then:
\(\ds \d x\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_1} \rd q_1 + \dfrac {\partial x} {\partial q_2} \rd q_2 + \dfrac {\partial x} {\partial q_3} \rd q_3\) | partially differentiating $(1)$ with respect to $x$ | |||||||||||
\(\ds \d y\) | \(=\) | \(\ds \dfrac {\partial y} {\partial q_1} \rd q_1 + \dfrac {\partial y} {\partial q_2} \rd q_2 + \dfrac {\partial y} {\partial q_3} \rd q_3\) | partially differentiating $(2)$ with respect to $y$ | |||||||||||
\(\ds \d z\) | \(=\) | \(\ds \dfrac {\partial z} {\partial q_1} \rd q_1 + \dfrac {\partial z} {\partial q_2} \rd q_2 + \dfrac {\partial z} {\partial q_3} \rd q_3\) | partially differentiating $(3)$ with respect to $z$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \d x^2\) | \(=\) | \(\ds \paren {\dfrac {\partial x} {\partial q_1} \rd q_1 + \dfrac {\partial x} {\partial q_2} \rd q_2 + \dfrac {\partial x} {\partial q_3} \rd q_3}^2\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {\partial x} {\partial q_1} \rd q_1}^2 + \paren {\dfrac {\partial x} {\partial q_2} \rd q_2}^2 + \paren {\dfrac {\partial x} {\partial q_3} \rd q_3}^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {\dfrac {\partial x} {\partial q_1} \rd q_1} \paren {\dfrac {\partial x} {\partial q_2} \rd q_2} + 2 \paren {\dfrac {\partial x} {\partial q_1} \rd q_1} \paren {\dfrac {\partial x} {\partial q_3} \rd q_3} + 2 \paren {\dfrac {\partial x} {\partial q_2} \rd q_2} \paren {\dfrac {\partial x} {\partial q_3} \rd q_3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_1} \rd q_1 \rd q_1 + \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_2} \rd q_2 \rd q_2 + \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_3} \rd q_3 \rd q_3\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_2} \rd q_1 \rd q_2 + \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_3} \rd q_1 \rd q_3 + \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_3} \rd q_2 \rd q_3\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_1} \rd q_2 \rd q_1 + \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_1} \rd q_3 \rd q_1 + \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_2} \rd q_3 \rd q_2\) | |||||||||||
\(\ds \d y^2\) | \(=\) | \(\ds \paren {\dfrac {\partial y} {\partial q_1} \rd q_1 + \dfrac {\partial y} {\partial q_2} \rd q_2 + \dfrac {\partial y} {\partial q_3} \rd q_3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {\partial y} {\partial q_1} \rd q_1}^2 + \paren {\dfrac {\partial y} {\partial q_2} \rd q_2}^2 + \paren {\dfrac {\partial y} {\partial q_3} \rd q_3}^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {\dfrac {\partial y} {\partial q_1} \rd q_1} \paren {\dfrac {\partial y} {\partial q_2} \rd q_2} + 2 \paren {\dfrac {\partial y} {\partial q_1} \rd q_1} \paren {\dfrac {\partial y} {\partial q_3} \rd q_3} + 2 \paren {\dfrac {\partial y} {\partial q_2} \rd q_2} \paren {\dfrac {\partial y} {\partial q_3} \rd q_3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_1} \rd q_1 \rd q_1 + \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_2} \rd q_2 \rd q_2 + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_3} \rd q_3 \rd q_3\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_2} \rd q_1 \rd q_2 + \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_3} \rd q_1 \rd q_3 + \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_3} \rd q_2 \rd q_3\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_1} \rd q_2 \rd q_1 + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_1} \rd q_3 \rd q_1 + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_2} \rd q_3 \rd q_2\) | |||||||||||
\(\ds \d z^2\) | \(=\) | \(\ds \paren {\dfrac {\partial z} {\partial q_1} \rd q_1 + \dfrac {\partial z} {\partial q_2} \rd q_2 + \dfrac {\partial z} {\partial q_3} \rd q_3}^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\dfrac {\partial z} {\partial q_1} \rd q_1}^2 + \paren {\dfrac {\partial z} {\partial q_2} \rd q_2}^2 + \paren {\dfrac {\partial z} {\partial q_3} \rd q_3}^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 2 \paren {\dfrac {\partial z} {\partial q_1} \rd q_1} \paren {\dfrac {\partial z} {\partial q_2} \rd q_2} + 2 \paren {\dfrac {\partial z} {\partial q_1} \rd q_1} \paren {\dfrac {\partial z} {\partial q_3} \rd q_3} + 2 \paren {\dfrac {\partial z} {\partial q_2} \rd q_2} \paren {\dfrac {\partial z} {\partial q_3} \rd q_3}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_1} \rd q_1 \rd q_1 + \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_2} \rd q_2 \rd q_2 + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_3} \rd q_3 \rd q_3\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_2} \rd q_1 \rd q_2 + \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_3} \rd q_1 \rd q_3 + \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_3} \rd q_2 \rd q_3\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_1} \rd q_2 \rd q_1 + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_1} \rd q_3 \rd q_1 + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_2} \rd q_3 \rd q_2\) |
By definition of the metric of $\QQ$:
\(\ds d^2\) | \(=\) | \(\ds \d x^2 + \d y^2 + \d z^2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i, j} h_{i j}^2 \rd q_i \rd q_j\) | for $i, j \in \set {1, 2, 3}$ |
Thus we can extract the appropriate terms with $\d q_i \rd q_j$:
\(\ds {h_{1 1} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_1} + \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_1} + \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_1}\) | ||||||||||||
\(\ds {h_{2 2} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_2} + \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_2} + \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_2}\) | ||||||||||||
\(\ds {h_{3 3} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_3} + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_3} + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_3}\) | ||||||||||||
\(\ds {h_{1 2} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_2} + \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_2} + \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_2}\) | ||||||||||||
\(\ds {h_{1 3} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_1} \dfrac {\partial x} {\partial q_3} + \dfrac {\partial y} {\partial q_1} \dfrac {\partial y} {\partial q_3} + \dfrac {\partial z} {\partial q_1} \dfrac {\partial z} {\partial q_3}\) | ||||||||||||
\(\ds {h_{2 1} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_1} + \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_1} + \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_1}\) | ||||||||||||
\(\ds {h_{2 3} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_2} \dfrac {\partial x} {\partial q_3} + \dfrac {\partial y} {\partial q_2} \dfrac {\partial y} {\partial q_3} + \dfrac {\partial z} {\partial q_2} \dfrac {\partial z} {\partial q_3}\) | ||||||||||||
\(\ds {h_{3 1} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_1} + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_1} + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_1}\) | ||||||||||||
\(\ds {h_{3 2} }^2\) | \(=\) | \(\ds \dfrac {\partial x} {\partial q_3} \dfrac {\partial x} {\partial q_2} + \dfrac {\partial y} {\partial q_3} \dfrac {\partial y} {\partial q_2} + \dfrac {\partial z} {\partial q_3} \dfrac {\partial z} {\partial q_2}\) |
Hence the result.
$\blacksquare$
Sources
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates: $(2.6)$