Definition:Orthogonal Curvilinear Coordinates

Definition
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:

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.