Definition:Coordinate System/3-Space/Curvilinear

Definition

Recall that we can define Cartesian $3$-space by means of perpendicular planes.

Let us superimpose onto this coordinate system $3$ other one-parameter families of surfaces.

In each of these families, the surfaces need not be parallel and they need not be planes.

These families are such that every point in (ordinary) $3$-dimensional space can then be uniquely identified as the intersection of $3$ such surfaces: one surface from each family.

Hence every point is identified as the ordered triple $\tuple {q_1, q_2, q_3}$, where $q_1$, $q_2$ and $q_3$ are the parameters of each of the families.

Hence we can describe a curvilinear coordinate system.

Curvilinear Coordinate Surface

Let $\KK$ be a curvilinear coordinate system in $3$-space.

Let $\QQ_1$, $\QQ_2$ and $\QQ_3$ denote the one-parameter families that define $\KK$.

Each of the surfaces which is an element of $\QQ_i$ is known as a curvilinear coordinate surface.

Each curvilinear coordinate surface has its unique $q_i$ which is constant for that surface.

Sources

• 1970: George Arfken: Mathematical Methods for Physicists (2nd ed.) ... (previous) ... (next): Chapter $2$ Coordinate Systems $2.1$ Curvilinear Coordinates