TABLE OF CONTENTS The position of a point P( x, y, z) in a Cartesian coordinate system is determined by the intersection of three mutually perpendicular planes x = k 1, y = k 2, z = k 3, where k i ( i = 1, 2, 3) are constants.
Let the Cartesian coordinates ( x, y, z) of any point be expressed as functions of three new quantities u 1, u 2, u 3, so that
Suppose that the equations (1) are solvable for u 1, u 2, u 3 in terms of x, y, z
The functions in (1) and (2) are assumed to be single valued and to have continuous derivatives so that the correspondence between ( x, y, z) and ( u 1, u 2, u 3) is unique.
Note. The point where this assumption is not applied requires special consideration.
? To each point P( x, y, z) in the region there corresponds a unique set of values ( u 1, u 2, u 3) called curvilinear coordinates of P( x, y, z).
The sets of equations (1) and (2) define a transformation of coordinates.
For a given point P( x, y, z) the set of equations (2) becomes
where c 1, c 2, c 3 are constants are called coordinates of surface.
