ABSTRACT
In a three-dimensional (6.1a) orthogonal curvilinear coordinate system (6.1b) may be considered at any point an in”nitesimal parallelepiped (Figure 6.1a) with curved sides with scale factors (6.1c) along the unit vectors (6.1d):
i x x x x h h h h e e e ei i i= ≡ ( ) ≡ ( ) = ( )1 2 3 1 2 3 1 2 3 1 2 3, , : , , , , , , , , . (6.1a-d)
¡e unit vectors have constant (varying) direction in a rectilinear (curvilinear) coordinate system. ¡e position of the farthest vertex relative to the closest speci”es the in’nitesimal relative position vector:
d d d d d
x e h x e h x e h x e h xi i i
= = + + = ∑
In the case of an orthogonal coordinate system the inner product of two unit vectors is zero (unity) if they are (are not) distinct, and thus coincides with the identity matrix (5.278a-c) ≡ (6.3a,b):
e e
i j
i j i j ij⋅( ) = ≡
=
≠
δ
if
if
,
. (6.3a,b)
¡e modulus of (6.2) that speci”es the distance between the opposite vertices or arc length:
d d d d d ds x h x h x h x h xi i
( ) = = ( ) = ( ) + ( ) + ( ) = ∑2 2 2
does not involve cross-products:
d d d d d d d x x x h x h x e e h h x xi i j j
= ⋅ = ⋅( ) = = = ∑ , ,
∑ ∑= ( ) =
d . (6.5)
¡is is because the identity matrix (6.3a and b) leads to a sum of squares in the case of an orthogonal curvilinear coordinate system.
