Structural Stability Theory and Practice

Appendix EDifferential Geometry in Curvilinear Coordinates

A three-dimensional curve in a rectangular coordinate system (x, y, z) can be represented by the locus of the end point of the position vector r in Figure E.1.

(E.1)

Let s be the arc length along the space curve, then

(E.2)

From the dot product of the foregoing derivative with itself we get

(E.3)

(E.4)

Hence,

(E.5)

This shows that dr/ds is a unit vector. The vector Δr/Δs in Figure E.1 becomes the vector tangent to the curve at the point P as Δs approaches zero. Therefore, t = dr/ds is a unit tangent vector. The vector

(E.6)

is also a tangent vector in the direction dr/ds but is not necessarily a unit vector.

E.1 Curvature

By Eq. (E.5)

Figure E1 Position vectors on a curve.

(E.7)

where prime denotes differentiation ...