\begin{definition}[Definition:Orthogonal Curvilinear Coordinates]
Let $\tuple {q_1, q_2, q_3}$ denote a set of curvilinear coordinates.
Let the relation between those curvilinear coordinates and Cartesian coordinates be expressed as:
{{begin-eqn}}
{{eqn | l = x
      | r = \map x {q_1, q_2, q_3}
}}
{{eqn | l = y
      | r = \map y {q_1, q_2, q_3}
}}
{{eqn | l = z
      | r = \map z {q_1, q_2, q_3}
}}
{{end-eqn}}
where $\tuple {x, y, z}$ denotes the Cartesian coordinates.
Let these equations have the property that:
:$\dfrac {\partial x} {\partial q_i} \dfrac {\partial x} {\partial q_j} + \dfrac {\partial y} {\partial q_i} \dfrac {\partial y} {\partial q_j} + \dfrac {\partial z} {\partial q_i} \dfrac {\partial z} {\partial q_j} = 0$
wherever $i \ne j$.
Then $\tuple {q_1, q_2, q_3}$ are '''orthogonal curvilinear coordinates'''.
\end{definition}