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.

$\blacksquare$


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