of a general point (x, y, z)can be expressed in terms of the curvilinear coordinates
(u, v, w )by utilizing the transformation equations (8.68). The position vector r, when
expressed in terms of the curvilinear coordinates, becomes
r =r (u, v, w ) = x(u, v, w )ˆe 1 +y(u, v, w )ˆe 2 +z(u, v, w )ˆe 3 (8 .71)
and an element of arc length squared is ds^2 =dr ·dr. In the curvilinear coordinates
one finds r =r (u, v, w )as a function of the curvilinear coordinates and consequently
dr =
∂r
∂u du +
∂r
∂v dv +
∂r
∂w dw. (8 .72)
From the differential element dr one finds the element of arc length squared given
by
dr ·dr =ds^2 =
∂r
∂u ·
∂r
∂u du du +
∂r
∂u ·
∂r
∂v du dv +
∂r
∂u ·
∂r
∂w du dw
+
∂r
∂v ·
∂r
∂u dvdu +
∂r
∂v ·
∂r
∂v dvdv +
∂r
∂v ·
∂r
∂w dv dw
+
∂r
∂w ·
∂r
∂u dwdu +
∂r
∂w ·
∂r
∂v dwdv +
∂r
∂w ·
∂r
∂w dwdw.
(8 .73)
The quantities
g 11 =∂r
∂u
·∂r
∂u
g 21 =∂r
∂v
·∂r
∂u
g 31 =∂r
∂w
·∂r
∂u
g 12 =∂r
∂u
·∂r
∂v
g 22 =∂r
∂v
·∂r
∂v
g 32 =∂r
∂w
·∂r
∂v
g 13 =∂r
∂u
·∂r
∂w
g 23 =∂r
∂v
·∂r
∂w
g 33 =∂r
∂w
·∂r
∂w
(8 .74)
are called the metric components of the curvilinear coordinate system. The metric
components may be thought of as the elements of a symmetric matrix, since gij =gji,
i, j = 1 , 2 , 3. These metrices play an important role in the subject area of tensor
calculus.
The vectors ∂r∂u , ∂r∂v , ∂w∂r , used to calculate the metric components gij have the
following physical interpretation. The vector r =r (u, c 2 , c 3 ),where uis a variable and
v=c 2 , w =c 3 are constants, traces out a curve in space called a coordinate curve.
Families of these curves create a coordinate system. Coordinate curves can also be
viewed as being generated by the intersection of the coordinate surfaces v(x, y, z ) = c 2
and w(x, y, z ) = c 3 .The tangent vector to the coordinate curve is calculated with the
