Volume Elements Revisited
Consider the volume element dV =dxdydz from cartesian coordinates and intro-
duce a change of variables
x=x(u, v, w ), y =y(u, v, w ), z =z(u, v, w )
from an x, y, z rectangular coordinate system to a u, v, w curvilinear coordinate sys-
tem. One finds the vector
r =r (u, v, w ) = x(u, v, w )ˆe 1 +y(u, v, w )ˆe 2 +z(u, v, w )ˆe 3
is the position vector of a general point within a region determined by the restrictions
placed upon the u, v, w variables. The surfaces
r (u, v, w 0 ), r(u, v 0 , w ), r (u 0 , v, w )
are called coordinates surfaces and the curves
r (u 0 , v 0 , w ), r(u 0 , v, w 0 ), r (u, v 0 , w 0 )
are called coordinate curves. The coordinate curves represent intersections of the
coordinate surfaces. The partial derivatives
∂r
∂u
, ∂r
∂v
, ∂r
∂w
represent tangent vectors to the coordinate curves and the quantities
hu=
∣∣
∣∣∂r
∂u
∣∣
∣∣, h v=
∣∣
∣∣∂r
∂v
∣∣
∣∣, h w=
∣∣
∣∣∂r
∂w
∣∣
∣∣
are called scale factors associated with the tangents to the coordinate curves. These
scale factors are used to calculate unit vectors
ˆeu=
1
hu
∂r
∂u
, ˆev=
1
hv
∂r
∂v
, eˆw=
1
hw
∂r
∂w
to the coordinate curves. If these unit vectors are all perpendicular to one another
the coordinate system is called an orthogonal coordinate system. The differential
dr =∂r
∂u
du +∂r
∂v
dv +∂r
∂w
dw
