Definition:Curvilinear Coordinate System/Cartesian Representation
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Definition
The relation between curvilinear coordinates and Cartesian coordinates can be expressed as:
\(\ds x\) | \(=\) | \(\ds \map x {q_1, q_2, q_3}\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds \map y {q_1, q_2, q_3}\) | ||||||||||||
\(\ds z\) | \(=\) | \(\ds \map z {q_1, q_2, q_3}\) |
\(\ds q_1\) | \(=\) | \(\ds \map {q_1} {x, y, z}\) | ||||||||||||
\(\ds q_2\) | \(=\) | \(\ds \map {q_2} {x, y, z}\) | ||||||||||||
\(\ds q_3\) | \(=\) | \(\ds \map {q_3} {x, y, z}\) |
where:
- $\tuple {x, y, z}$ denotes the Cartesian coordinates
- $\tuple {q_1, q_2, q_3}$ denotes their curvilinear equivalents.
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: $(1.2)$
- 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates